BLABASED DESIGN AND ANALYSIS OF VCOBASED …VCO-based Sigma-Delta Modulators SUPERVISOR Prof. Dr. ir. Gerd Vandersteen Vrije Universiteit Brussel, Belgium MEMBERS OF THE JURY Prof

BLA‐BASED DESIGN AND ANALYSIS OF VCO‐BASED

SIGMA‐DELTA MODULATORS

DRIES PEUMANSGERD VANDERSTEEN

ANALOGTO

DIGITAL

Fundamental Electricity

and Instrumentation (ELEC)

Academic year 2019-2020

Thesis submitted in fulfilment of the requirements for the degree ofDoctor of Engineering Sciences (Doctor in de Ingenieurswetenschappen) by

ir. Dries Peumans

BLA-based Design and Analysis ofVCO-based Sigma-Delta Modulators

SUPERVISOR Prof. Dr. ir. Gerd VandersteenVrije Universiteit Brussel, Belgium

MEMBERS OF THE JURY Prof. Dr. ir. Marijke Huysmans (President)Vrije Universiteit Brussel, Belgium

Prof. Dr. Roger Vounckx (Vice-President)Vrije Universiteit Brussel, Belgium

Prof. Dr. ing. Tim De Troyer (Secretary)Vrije Universiteit Brussel, Belgium

Prof. Dr. ir. Guillaume MercèreUniversity of Poitiers, France

Prof. Dr. ir. Tom DhaeneGhent University, Belgium

Prof. Dr. ir. Pieter RomboutsGhent University, Belgium

DOI: 10.13140/RG.2.2.15198.66884

University Press BVBARechtstro 2/001, 9185 Wachtebeke, Belgiumhttps://www.universitypress.be

Vrije Universiteit Brussel, dept. ELECPleinlaan 2, 1050 Brussels, Belgiumhttp://vubirelec.be

This work is funded by the Research Foundation - Flanders (FWO) and the Strategic Research Programof the VUB (SRP-19).

© May 2020 Dries Peumans and Gerd Vandersteen

All rights reserved. No parts of this document may be reproduced or transmitted in any form or byany means without the prior written permission of the authors.

Acknowledgements

Thank you

For your helpFor your time

Thank you

For what you didFor who you are

Thank you

More than a littleFrom the bottom of my heart

i

Contents

1. Introduction 11.1. Analogue-to-digital conversion . . . . . . . . . . . . . . . . . . . . 21.2. Paying for my ice-cream addiction . . . . . . . . . . . . . . . . . . . 41.3. Sigma-Delta modulation . . . . . . . . . . . . . . . . . . . . . . . . 61.4. Large-signal stability analysis . . . . . . . . . . . . . . . . . . . . . 101.5. Goals and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Linear approximations 152.1. The basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2. The Describing Functions . . . . . . . . . . . . . . . . . . . . . . . . 192.3. The Best Linear Approximation . . . . . . . . . . . . . . . . . . . . 212.4. Towards a combined approach? . . . . . . . . . . . . . . . . . . . . 27

3. An improved Describing Function applied to OTA-based circuits 293.1. Approximation of the static nonlinear behaviour . . . . . . . . . . 303.2. The improved Describing Function . . . . . . . . . . . . . . . . . . 323.3. Application of the DF to OTA-based circuits . . . . . . . . . . . . . 343.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4. Local modelling techniques 514.1. The basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2. Using Bootstrapped Total Least Squares for local modelling purposes 584.3. A first simulation example: the SISO case . . . . . . . . . . . . . . 694.4. Simulation example: the MIMO case . . . . . . . . . . . . . . . . . 744.5. Numerically efficient computation . . . . . . . . . . . . . . . . . . 804.6. Systems operating in feedback . . . . . . . . . . . . . . . . . . . . . 854.7. Measurements on the tailplane of a glider . . . . . . . . . . . . . . 864.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

iii

Contents

5. Stability analysis of VCO-based ΣΔ modulators using the describ-ing BLA 935.1. Stability assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2. VCO-based quantisation . . . . . . . . . . . . . . . . . . . . . . . . 965.3. Noise leakage reduction using local modelling techniques . . . . . 995.4. The Describing Best Linear Approximation . . . . . . . . . . . . . 1075.5. A DBLA-based stability analysis . . . . . . . . . . . . . . . . . . . . 1145.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6. Conclusions and future work 1276.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A. Mixing continuous-time and discrete-time signals 133

B. Interpretation of the LBTLS cost function 139

C. Vector Fitting applied to input-output data 141

List of Publications 145

List of Abbreviations 149

Bibliography 151

iv

1. Introduction

Digital technology plays a dominant role in many aspects of our daily lives.During the last decade, the number of applications in this area has explodedexponentially. We use computers and smartphones to surf the world wide web,communicate wirelessly over large distances, use social media and dating apps,put family portraits and holiday photos somewhere safe in the cloud ... Eventhough the digital application area is almost limitless, a translation is still neededfrom the physical, analogue world to the digital world. This translation is whereanalogue-to-digital converters come into play.

Analogue-to-digital converters link our physical world to digital computers andare therefore a vital element in the ever-continuing digitalisation of our world.They provide the means to translate electronic analogue quantities such as voltagesand currents to streams of bits and bytes. Ever since the conception of analogue-to-digital conversion, many different conversion techniques have been introduced,each with their respective strenghts and weaknesses. The state-of-the-art choicedepends on the requirements on the resolution and the speed of the converter.In recent-years, continuous-time ΣΔ modulators (pronounced as Sigma-Delta)attracted growing interest as they effectively narrow the boundary between theanalogue and digital world. Compared to conventional analogue-to-digital con-verters, ΣΔ modulators significantly reduce the sensitivity to nonidealities of mostof its building blocks. This improvement is achieved through a clever combinationof oversampling and digital post-processing.

Every analogue-to-digital converter has a certain dynamic range. It specifies theratio of the largest to the smallest input signal amplitude the converter can resolve.One of the main annoyances for ΣΔ designers is that the dynamic range is limiteddue to unstable behaviour which manifests itself when the input grows above acertain signal level. The goal of this thesis is to provide modelling tools for ΣΔdesigners to accurately predict this so-called modulator overloading level whenthe input signal is a modulated signal. Incorporating the resulting models in

1

1. Introduction

Figure 1.1.: The analogue-to-digital converter lies on the boundary between thecontinuous and discrete world. Its main purpose is to discretiseanalogue signals both in time and amplitude.

the design flow can shorten the time to market of the designed products sinceunstable behaviour is better and more early predicted during the design stage.

This chapter first discusses the fundamentals of analogue-to-digital conversion.Next, an introduction to continuous-time ΣΔ modulation is given. Then, state-of-the-art large-signal stability analysis methods of these modulators are summarised.The chapter concludes with the goals and the outline of the thesis.

1.1. Analogue-to-digital conversion

The physical world we live in is continuous in nature. The way we perceive andprocess information is based on analogue processes and signals which are inher-ently continuous in time and in amplitude. For example, wireless communicationsystems require analogue radio waves to be transmitted and received using an-tennas to convey information. Due to the continuous nature of these signals, it isunfortunately impossible to hook up a digital computer directly to these antennasto process the analogue radio waves. To accomplish this connection and limitthe loss of information between the analogue and digital world, an Analogue-to-Digital Converter (ADC) is required. It should execute two tasks: sampling(discretisation in time) and quantisation (discretisation in amplitude).

2

1.1. Analogue-to-digital conversion

Sampling converts a Continuous-Time (CT) signal into a Discrete-Time (DT)one by taking a snapshot of the analogue signal every )B seconds (Figure 1.1). Iteffectively reduces a CT signal to a sequence of analogue samples. To keep a uniquerelationship between the CT signal and the DT one, the useful signal bandwidth of the CT signal should be restricted. The Nyquist-Shannon sampling theorem[Nyqu 28, Shan 49] formalises this requirement and states that the bandwidth ofthe CT signal should be smaller than half the sampling frequency 5B (= 1/)B) ofthe ADC

< 5B/2 (1.1)

This ensures that the original CT signal can be perfectly reconstructed from thesampled DT signal. We should thus always carefully satisfy this condition beforeusing the ADC. Otherwise, aliasing errors are introduced that break the equival-ence relation between the CT signal and its DT counterpart and the informationin the signal can be badly corrupted.

Quantisation of the amplitude on the other hand discretises the amplitude of theCT signal by mapping a continuous voltage or current interval to a discrete set ofvalues. Each value is represented by a digital binary code (Figure 1.1). The numberof distinct values determines the resolution of the ADC and is mostly expressedby the number of bits used to code each sample. For example, a resolution of3 bits corresponds to 23 = 8 distinct values. Observe also from Figure 1.1 thatquantisation introduces errors. This error is called the quantisation noise. Its sizeis closely tied to the resolution: the higher the resolution, the smaller the noiselevel and vice versa.

When looking at the internal sampling frequency of the converter, an ADC canbe categorised into two main classes: the Nyquist-rate converters and the over-sampling converters. As the name suggests, the former class obeys the Nyquist-Shannon theorem and operates at a sampling frequency 5B that is equal or a bithigher than twice the bandwidth of the input signal. While being optimalin terms of conversion rate, Nyquist-rate converters impose stringent require-ments on the matching accuracy of the analog components (resistors, capacitors...) needed for the implementation. Oversampling converters relax these require-ments by using an internal sampling frequency that is much higher than twicethe Nyquist frequency. Subsequent combination (i.e. averaging) of multiple low-resolution samples allows to obtain one high-resolution sample. For example,oversampling a signal with a factor 4 theoretically results in a 1 bit increase inresolution [Schr 05]. The drawback of oversampling is that additional digitalsignal processing hardware, operating at high frequencies, is needed to increasethe resolution numerically after the actual low-resolution conversion process.

3

1. Introduction

ΣΔ modulator based ADCs belong to the class of oversampling converters. Toexplain the fundamental principles of ΣΔ modulation, an analogy is used. It isbased on an in-depth look into how I sustain my ice-cream addiction.

1.2. Paying for my ice-cream addiction

Based on the coffee shop problem in [Pava 17].

During the summer months, I like to bring a daily visit to the local ice-creamparlour near to my home. Every day I buy three scoops of ice cream that areneatly balanced on a cone and cost AC3.6. Being a traditional Italian ice-creamparlour, no credit cards are accepted and I need to pay this amount in cash. Theusual way of paying would be that every single day I pay the correct amountin coins. Or, similarly, I offer a banknote and the parlour owner returns me thechange. Everybody is used to this so-called "Nyquist" habit. But what if I told youthere exists another way to pay in cash banknotes that does not involve any cointransfer?

The ΣΔ way requires an agreement between the parlour owner and me. Being aregular customer, the parlour owner agrees to keep a money balance and addition-ally allows me to pay with AC5 banknotes only. The transaction rules concerningthe money balance are straightforward:

- If I owe more than AC2.5, I pay with a AC5 banknote.

- In the other case, I pay nothing.

The transactions using these rules are shown in Figure 1.2 for the first three days.At first glance, it seems unusual that you can pay the amount of AC3.6 by onlyusing AC5 banknotes since at any given day the money balance differs from AC0.So why bother introducing this ΣΔ way if apparently it does not succeed in everpaying the correct amount?

The subtlety lies in the fact that the payment on a single day is meaningless butthat on average the correct amount is obtained. To further investigate this ratherstrange behaviour, the agreement is translated into an equivalent signal flowschematic (Figure 1.3). In this schematic, D represents the cost of an ice creamcone (AC3.6) and H(=) represents the amount that I need to pay on the =th day (AC5

4

1.2. Paying for my ice-cream addiction

Figure 1.2.: The ΣΔway to pay for your daily dose of ice cream. You keep a balanceand everyday you verify whether or not you need to pay with a AC5bill.

or nothing). I−1 is an operator that implements a delay of one day such that wecan add D with the remaining balance of the previous day. If now the averagevalue of H is analysed as a function of the number of days (Figure 1.4), we cometo the conclusion that progressively a better approximation of D is obtained. Restto relate the ice-cream payment methods to the analogue-to-digital conversion.The example shows that while the ΣΔ way starts with a resolution of only AC5, ahigher resolution estimate (error < AC0.02) is obtained when averaging over 100consecutive days.

While oversampling is one of the fundamental principles that ΣΔ modulatorsuse to increase the resolution, it is not the only trick these modulators have uptheir sleeve to decrease the quantisation noise. They additionally introduce noiseshaping. This technique literally ’shapes’ the Power Spectral Density (PSD) ofthe quantisation noise such that its deteriorating effects in the frequency bandof interest are reduced. This concept is discussed more in depth in the followingsection.

5

1. Introduction

Figure 1.3.: The ΣΔ equivalent signalflow schematic.

25 50 75 1002.5

3.6

5

Number of days

Ave

rage

ofy

(AC)

Figure 1.4.: Visiting the ice cream par-lour more results in an in-creasingly better approx-imation of the requiredAC3.6.

1.3. Sigma-Delta modulation

The original concept of ΣΔ modulation dates back to the sixties. Inose et al.mentioned it for the first time in the published literature back in 1962 [Inos 62].Since its conception, a tremendous amount of research has been conducted on thetopic. Fueled by the technological downscaling, which provides faster and morepower efficient digital circuitry, many different architectures, system- and circuit-design techniques, integrated circuits ... were developed over a period of fivedecades [Cand 81, Bose 88, Cand 92, Cher 99a, Schr 05, Jose 13]. This continuingresearch and development effort resulted in a mature technology that is readilyimplemented in many contemporary industrial products.

Even though the technology behind the ΣΔ modulators has evolved quite signi-ficantly over the years, the original concept remains untouched and is still basedon the same two signal processing techniques: oversampling and noise shaping.Both techniques cleverly exploit the fact that the quantisation noise of a quantisercan be well approximated by random white noise with a uniform distribution.Suppose that the quantisation happens in amplitude steps Δ, then the quantisationnoise power f2

&is approximately equal to Δ2/12 and thus solely depends on the

quantiser’s resolution [Schr 05]. Using this approximation, the ideal behaviour ofdifferent types of ADCs can be conveniently understood.

6


Figure 1.5.: Architecture and noise behaviour of different ADC types. Analysingthe Power Spectral Density (PSD) shows that the ΣΔ modulator be-haves best in terms of in-band noise power. : signal bandwidth,

: quantisation noise.

7

1. Introduction

Take the Nyquist-rate converter in Figure 1.5 to start with. This converter consistsof an Anti-Alias Filter (AAF) to limit the bandwidth of the continuous-timesignal D(C) to 5B/2, a sampler, and a quantiser that introduces a certain amountof quantisation noise. Examining the PSD of the output H(=) shows that f2

&

( ) for Nyquist-rate converters falls completely inside the signal bandwidth( ). Therefore it creates an unwanted noisy contribution in the output of theADC. Oversampling converters mitigate this problem by increasing the samplingfrequency with an Oversampling Ratio (OSR). They spread the noise power over alarger bandwidth such that a smaller portion of the power remains in the signalbandwidth. ΣΔ modulators decrease this noise portion even further by activelyshaping, or in other words filtering, the quantisation noise. By enclosing thequantiser in a negative feedback loop, and combining it with a loopfilter, it ispossible to achieve a high-pass behaviour: noise at lower frequencies is reduced,while noise at higher frequencies is amplified. Depending on the architecture andthe order of the loopfilter, i.e. the number of integrators, a massive reduction ofthe in-band noise power can be achieved [Schr 05].

Example Consider the ΣΔ modulator from the ice-cream example (Figure 1.3).This modulator implements a certain Signal Transfer Function (STF) and NoiseTransfer Function (NTF). These transfer functions describe, respectively, theinfluence of the input signal and quantisation noise on the output. For thisspecific example they are equal to

STF(I) = 1 NTF(I) = 1− I−1 (1.2)

The STF reveals a one-to-one relationship between the input of the ΣΔ modu-lator and the corresponding output sequence. The NTF implements high-passnoise shaping. Due to this shaping, the in-band quantisation noise power isapproximately equal to [Schr 05]

f2& =

Δ2

36c2

( 1$('

)3(1.3)

This last expression clearly reveals the added value of noise shaping for over-sampling converters. Instead of just being inversely proportional to the OSR,this inverse proportionality is raised to the power 3 which greatly reduces thein-band quantisation noise power.

Depending on the system- and circuit-design techniques that are employed duringtheir construction, ΣΔ converters can be further classified into two additionaltypes: DT and CT converters. Discrete-time converters keep a clear distinction

8


between the sampling and quantisation operation (Figure 1.5). The vast majorityof ΣΔ modulators are DT and are mostly implemented using switched-capacitor[Dias 92] or switched-current [Craw 92] techniques. In recent years, however,communication systems are becoming increasingly eager to obtain higher datarates. As a consequence, they demand ΣΔ modulators handling increasing signalbandwidths (> 1MHz) while maintaining satisfactory noise shaping capabilities.To answer this demand, the focus of ΣΔ designers is shifting towards CT ΣΔ

modulators. These embed the sampling operation directly in the feedback loop(Figure 1.5) [Cher 99a, Bree 01, Ortm 06]. This embedding results in some ad-vantages when compared to the DT counterpart: the non-idealities introduced bythe sampler are subject to noise-shaping, implicit anti-alias filtering is provided,and the bandwidth requirements of the operational amplifiers that are present inthe filter are relaxed. In general, these CT ΣΔ modulators can therefore operate atsampling frequencies that are an order of magnitude larger than the ones obtainedwith DT ΣΔ modulators. Consequently, they can handle larger signal bandwidths.The main disadvantage of the embedding is that the theoretical analysis of theseCT ΣΔ modulators is made more difficult due to the mixing of CT and DT signals.For example, the STF and NTF of a DT ΣΔ modulator is straightforwardly determ-ined starting from the transfer function of the DT loopfilter [Schr 05]. The STFand NTF describe the influence of both the input signal and quantisation error onthe output

. (I) = STF(I)* (I) +NTF(I)&(I) (1.4)

where . (I), * (I) and &(I) are the Z-transforms of the output, input and quant-isation noise sequences respectively. Unfortunately, (1.4) is not applicable to CTΣΔ modulators due to the mixing of different types of signals. A more involvedmathematical analysis is then required to derive the equivalent STF and NTF inthe CT case (Appendix A).

A CT ΣΔ modulator is a diverse collection of components arranged in a feedbackconfiguration. It starts by filtering the CT input signal D(C) and the CT negativefeedback signal with the CT loopfilter (Figure 1.5). This loopfilter should be aslinear as possible to avoid degradation of the noise shaping capabilities of theΣΔ modulator. The output of the loopfilter is afterwards sampled and quant-ised. Sampling reduces the CT signal to a DT sequence, while quantisation isa nonlinear process due to the staircaselike transfer characteristic (Figure 1.6).Finally, a Digital-to-Analogue Convertor (DAC) is needed to close the feedbackloop. The combination of all these components complicates the analysis of CT ΣΔmodulators. Over the years, several (mathematical) analysis techniques have beenintroduced to (pre-)assess the performance of CT ΣΔ modulators. An overview ofthose techniques is the subject of next section.

9

1. Introduction

Figure 1.6.: Ideal transfer characteristic of quantiser with 3-bit resolution.

1.4. Large-signal stability analysis

Every system that incorporates one or more feedback loops is susceptible to un-stability and/or oscillating behaviour. If all the components embedded in theloop would be perfectly linear, the (unwanted) behaviour can be predicted. It caneven be accounted for by a careful design of the loopfilter. CT ΣΔ modulatorsunfortunately require the incorporation of a quantiser in the loop that is inher-ently nonlinear. Therefore, its behaviour becomes signal level dependent. Thisimposes considerable challenges on ΣΔ designers. After all, we would like the ΣΔmodulator to function for any allowable signal within the frequency band and theamplitude range of interest, regardless of the actual time-domain waveform.

Previously conducted research has revealed that the Describing Function (DF)theory provides a solid approach for an analytic prediction of the nonlinearbehaviour of ΣΔ modulators [Gelb 68, Arda 87, Lota 08, Romb 13]. In a nutshell,the DF theory approximates the behaviour of the nonlinear quantiser with alinearised static gain that is signal dependent and therefore varies as a function ofthe input power level. By doing so, linear system theory can be reused to accuratelydescribe the nonlinear system. For example, a first-order ΣΔ modulator very likelygenerates steady-state oscillations, also called limit cycles, when a constant inputsignal is applied [Schr 05]. This behaviour is counter-intuitive, since a completelylinear system would only generate a constant output in response to a constantinput. These limit cycles are undesirable and degrade the performance, especiallyif the oscillation frequency lies in the frequency band of interest. Using the DFtheory for a known architecture, enables one to anticipate the presence of theselimit cycles and to take countermeasures accordingly. The DF theory also predictsthe modulator overloading level. This overloading level is defined as the maximumamplitude level for which stable operation of the converter is guaranteed. It is acrucial parameter that determines the converter’s dynamic range.

10

1.4. Large-signal stability analysis

105 106 107 108

−150

−100

−50

0

Frequency [Hz]

Ou

tpu

t[d

BFS

/NB

W]

(a) Signal, nonlinear distortion, in-bandquantisation noise, out-of-band quantisa-tion noise.

10−2 10−1 10040

60

80

Input amplitude [V]

SNR

/SN

DR

[dB

]

(b) Signal-to-Noise Ratio (SNR), Signal-to-Noise and Distortion Ratio (SNDR).

Figure 1.7.: Illustration of a typical output spectrum and performance metrics(SNR and SNDR) of a continuous-time ΣΔ modulator [Ruiz 05].

In practice the DF theory provides a good approximation of the actual nonlinearbehaviour of the convertor. But it remains an approximation nonetheless. Due tothe complex behaviour of the ΣΔ modulator, designers therefore need to resort toextensive, time-consuming simulations to deduce nonlinear parameters such asthe modulator overloading level more accurately. Usually, a large-signal stabilityanalysis is performed. During this analysis, the ΣΔ modulator is excited with asingle-tone sinusoid (Figure 1.7a). The spectrum of the convertor’s output containsa dominant spectral line at the sinusoid’s frequency as can be expected. Noiseshaping is clearly present and does indeed push a large portion of the quantisationnoise out of the converter’s signal bandwidth (20MHz). Note also the effect ofnonlinear behaviour. It manifests itself at the harmonics of the fundamentalexcitation frequency (i.e. only the third harmonic here).

Remember that a nonlinear system shows a signal dependent behaviour. The per-formance of the ΣΔ modulator thus changes with the sinusoid’s input amplitude.To characterise this performance change, the Signal-to-Noise Ratio (SNR) andthe Signal-to-Noise and Distortion Ratio (SNDR) are simulated/measured overa range of input amplitudes (Figure 1.7b). The SNR and SNDR are obtained bydividing the signal power f2

Signal with the integrated in-band quantisation noise

power f2Noise with or without the inclusion of nonlinear distortions f2

Distortions

SNR =f2

Signal

f2Noise

SNDR =f2

Signal

f2Noise +f

2Distortions

(1.5)

11

1. Introduction

For low input amplitudes, the nonlinear distortion is buried under the quant-isation noise floor. Consequently, the SNR equals the SNDR. From a certainamplitude level on, the nonlinear distortion rises above the quantisation noisefloor and the SNDR starts to deviate from the SNR. This difference increases untilthe modulator overloading level is reached. The ΣΔ modulator then becomesunstable and both ratios collapse.

Note that the performed large-signal stability analysis heavily relies on the as-sumption that the results obtained for a single-tone sinusoid remain valid for anyexcitation signal of a similar power level. Nonlinear behaviour, however, is notthat easily swept under the rug. It does not only depend on the input amplitude(or equivalent power) level, but also depends on the shape of the time-domainwaveform [Sche 80]. Therefore, this behaviour should be characterised using thesame signals as those encountered during its real-world operation. By all means,it is not possible to predict the exact excitation signal. In most cases, though, wehave prior knowledge about the PSD and Probability Density Function (PDF) ofthe input signal class. For example, wireless communication systems that useOrthogonal Frequency Division Multiplexing (OFDM) or sound devices produceanalogue modulated signals whose PDF resembles a Gaussian PDF. Applyingsuch signals to the ΣΔ modulator induces a different nonlinear behaviour whencompared to the single-tone sinusoid. It should therefore be characterised accord-ingly. In this thesis, we explore different techniques that enable the analysis ofCT ΣΔ modulators using modulated signals that belong to the class of signals theconvertor is designed for.

1.5. Goals and outline

In the current state-of-the-art, the large-signal stability analysis of CT ΣΔ modu-lators mainly relies on single-tone or two-tone sinusoidal excitation tests. Here,the goal is to analyse the stability under the excitation of complex modulatedsignals. Therefore, we develop a power-dependent modelling technique for thelinear approximation of nonlinear systems excited by a specific class of signals.

12

1.5. Goals and outline

We would like this technique to meet the following specifications

Signal-dependent behaviourSimilar to the DF, it should be possible to predict the behaviour of the systemas a function of the applied input power. This will enable the prediction ofthe modulator overloading level.

No prior knowledgeThe technique should be able to extract a model of the system withoutrequiring an analytical description or access to the internal device model.

Applicable to mixed-signal systemsSampling is inherently part of any CT ΣΔ modulator. Any potential can-didate technique should manage to describe the CT to DT conversion ad-equately.

Minimal user interactionThe number of user adjustable external parameters should be kept to aminimum. This is required to obtain a technique that is automated as muchas possible.

We can already unveil that the so-called Best Linear Approximation (BLA) willplay a major role in our pursuit for a modelling technique that satisfies all thesespecifications. Similar to the DF theory, the BLA theory linearly approximatesthe behaviour of the nonlinear system by retrieving a dynamic model. Chapter 2describes both theories in detail and provides a comparison between them. Inchapters 3 and 4 we delve deeper into both theories separately. We come up withnew insights for both the DF and BLA theory. More specifically, in chapter 3 wepropose an improved DF for circuits that include operational transconductanceamplifiers and use the resulting DF to predict nonlinear effects. Chapter 4 dis-cusses local modelling techniques and how to extend these techniques to improvethe BLA estimation. Therefore, we deliberately shift our focus to strongly resonantmechanical systems. This shift is motivated by our desire to demonstrate that thedeveloped techniques are useful in a wide range of applications. Finally, the localmodelling techniques are used for the amplitude-dependent modelling techniquewe develop in chapter 5. There, we apply the modelling techniques for the stabilityanalysis of ΣΔ modulators.

13

2. Linear approximations

Mathematical models are only as goodas the assumptions that underpin them.

Miek Meeussen in aninspired philosophical mood

The ultimate goal of engineering is to create devices and systems to serve humanpurposes. To achieve this goal, engineers heavily rely on models. Models allowto predict the behaviour of their inventions in advance and allow to change theirdesign accordingly if the inventions do not behave as expected. Despite whatsome might believe, we usually do not just blindly put things together and wishthat the outcome simply works. Even though this process can be a lot of fun, itdoes not fit into a sound design strategy.

During the design phase, engineers often use linear dynamic models to describethe behaviour of the system. They provide an extensive set of useful mathematicalproperties to their users. The regrettable disadvantage is that practical systemsare not always linear. Practical implementation are often subjected to nonlinearbehaviour up to some level that degrades the performance of the resulting system.For nonlinear systems it is difficult, and in most cases even impossible, to deriveand evaluate an exact mathematical description.

Linear approximations have been introduced to make the nonlinear complexitytractable. They re-enable the use of linear system theory while incorporatingsignal dependency. These approximations are not unique. Over the years differenttheories of linear approximation were proposed. In this chapter, we discussboth the Describing Function (DF) theory and Best Linear Approximation (BLA)theory. Our discussion involves the basic principles, underlying assumptions andapplication examples for both theories.

15


Figure 2.1.: General representation of a system ℎ[•]. The system is excited by aCT input D(C) which induces a CT response H(C).

2.1. The basic principles

Before diving into the topic of linear approximation, we first need to formallyintroduce the concepts of linearity, nonlinearity, and how they affect the behaviourof a system. You can think of a system as a part of the physical world that interactswith its surroundings through a well-defined number of ports. The interaction isdescribed by signals that are either forced on (inputs) or appear at (outputs) theseports. The system is here represented by ℎ[•] (Figure 2.1). It is a more abstractrepresentation of a physical process, a circuit, or its components that are understudy. Most systems considered here have a Continuous-Time (CT) input port thatis excited by the signal D(C). The system processes this signal and this results in acorresponding CT output response H(C) at its output port. Formally, we representthe system as

H(C) = ℎ[D(C)] (2.1)

Example A car is a system that we can control using the steering wheel and itsthree pedals (clutch, brake and accelerator). All of these controls act as inputsto the system. The results of our driving actions, namely acceleration, velocity,traveled distance, etc. are outputs of the system happening in response to thesecontrol inputs.

A linear system is a system that satisfies the superposition and homogeneityprinciple. These principles state that if the input D1 (C) produces the responseH1 (C), and the input D2 (C) produces the response H2 (C); then the linear combina-tion UD1 (C) + VD2 (C) will produce UH1 (C) + VH2 (C) with U, V ∈ R. In this thesis, wefurthermore assume that all the considered systems possess the properties oftime-invariance and causality. Time-invariance states that the behaviour of thesystem does not change over time

H(C) = ℎ[D(C)] → H(C − g) = ℎ[D(C − g)] ∀g ∈ R (2.2)

16

2.1. The basic principles

while causality states that the response of the system taken on time instant C0only depends on the past and present values C ≤ C0 of its input. In the case of acausal linear time-invariant system, the behaviour of the system can be completelydescribed by its impulse response function 6(C) in the time domain. Equivalently,the system can be described in the frequency domain with the so-called TransferFunction (TF) ( 9l) that is obtained by applying the Fourier transform on 6(C)

( 9l) =+∞∫−∞

6(C) 4− 9lC dC (2.3)

Intuitively, the TF describes the magnitude and phase change that a single-tonesinusoid with frequency l would undergo when it passes through the system

D(C) = sin(lC) → H(C) = | ( 9l) | sin(lC + ∠ ( 9l)

)(2.4)

where | • | and ∠• are, respectively, the magnitude and phase operator.

Every system that does not satisfy this superposition principle is said to be nonlin-ear. For nonlinear systems, it is no longer possible to generalise the response toa specific class of input signals to another class of input signals. This makes theanalysis of nonlinear systems much more complicated than that of linear systems.

In reality, the linearity assumption is only approximately valid. Every real-lifesystem is intrinsically nonlinear in one way or another. Given this fact, is itsensible to approximate the behaviour of the nonlinear system by a linear one? Toanswer this question, researchers investigated several approximation techniques.Their common goal is to allow the reuse of linear system theory for the analysisand design of nonlinear systems.

A possible approach to study nonlinear systems uses simplified models thatsubstitute each nonlinear operation by an approximating linear one. Linearsystem theory can then be used to predict and analyse the behaviour of theapproximating system. As one could expect, the approximation error betweenthese linear results and the behaviour of the original nonlinear system dependsstrongly on the validity of each linear approximation.

Linear approximation can either be performed locally or globally. This boilsdown to small- and large-signal linearisation respectively. Suppose that we havea static saturating nonlinear function (Figure 2.2). Small-signal linearisationlinearises the function around a nominal operating point, in the case of the leftfigure this is set to be the origin. Clearly, this linearisation is only valid in the

17


Figure 2.2.: Small- (left) and large-signal (middle) linearisation result in a differ-ent behaviour as a function of the input signal’s properties (right).

proximity of the operating point. If we would apply a signal D with an input rangeexceeding the acceptable limits for small-signal operation, the validity of this kindof linear approximation is completely undermined. To cope with this limitationthe concept of large-signal least-squares linearisation was introduced. This type oflinearisation depends not only on the nonlinear function but also on the propertiesof the input signal. Different linear approximations are obtained when driven bysignals with a different waveform, or even when driven by the same waveformbut with a different input range. To demonstrate the usefulness of large-signallinearisation, we consider that we have a single-tone sinusoidal signal with twodifferent amplitudes 1 and 2 (Figure 2.2) for which a different approximatingfunction is obtained (− and −). In contrast to the small-signal case where thelinearised gain (the slope of the approximating function) is independent of theapplied amplitude, the large-signal linearisation does now result in an amplitude-dependent linearised gain and therefore enjoys a substantial advantage oversmall-signal linearisation at the cost of a larger effort to extract the model. Namelythere is no limit to the range of signal amplitudes that can be accommodated.Furthermore, signal-dependent behaviour is introduced as would be the casefor a full-blown nonlinear model. The main difficulty that is encountered withlarge-signal linearisation is that the resulting approximation depends on boththe system and the signal properties (waveform, magnitude, power, spectrum,probability density function, etc.). In practice, assumptions have to be made aboutthe current signal properties to be able to calculate the approximating function.This is where the Describing Functions and the Best Linear Approximation comeinto the picture.

18

2.2. The Describing Functions

2.2. The Describing Functions

The Describing Function allows to approximate the input-output relationship ofstatic nonlinear functions (including saturation and/or hysteresis phenomena) by alarge-signal linear gain that is a function of the input signal’s properties [Gelb 68].The DF only considers input signals that are combinations of three basic signalclasses: a constant, a sinusoid, and Gaussian noise. Note that the applicabilityof the DF within practical design and control problems relies on how well theactual input signals resemble a combination of these three basic signal classes.If a good resemblance exists, the DF allows to effectively and accurately analysethe nonlinear behaviour, estimate figures of merit such as gain compression, andobtain information about system properties (including the presence of limit cycles)without overcomplicating the mathematics involved.

Independently of the system and the input signal class, all DFs estimate a linearstatic gain #D that approximates the nonlinear function, in mean-squares sense,given the input signal D(C). Finding #D eventually boils down in solving thefollowing quadratic minimisation problem

#D = argmin#DF

ED|H(C) −#DF D(C) |2 (2.5)

where ED• is the expected value operator. This expected value is taken withrespect to the real random variables `1, . . . , `= present within D(C)

ED•(`1, . . . , `=) =∫"1

· · ·∫"=

•(`1, . . . , `=) ?(`1, . . . , `=) d`1 . . .d`= (2.6)

where ?(`1, . . . , `=) is the joint multivariate Probability Density Function (PDF),and "1, . . . , "= are, respectively, the domains of `1, . . . , `=. The minimisationproblem in (2.5) can be solved explicitly, resulting in the following optimal lineargain

#D =EDH(C) D(C)EDD(C)2

(2.7)

19


Remark that #D depends on the signal class, the joint PDF of the random variablespresent within D(C) and the time C. However, if a stationary signal is appliedto a static nonlinearity, the expected values in (2.7) become time-independent.Therefore, (2.7) can be calculated at any convenient time, which is mostly C = 0. Dueto this time-independence, (2.7) can be calculated analytically for elementary staticnonlinear functions. It therefore provides valuable insight into the approximativelinear behaviour of the nonlinear function.

Example Consider a symmetric one-bit quantiser that is defined as

H = ℎ(D) =+Δ if D ≥ 0−Δ if D < 0

(2.8)

and is excited by a single-tone signal D(C) = sin(lC + i). This single-tonesinusoid has a deterministic amplitude and angular frequency l. The phaseoffset i, however, is undetermined and is defined relative to an arbitrary timereference. The performance of a static nonlinear system, such as (2.8), isindependent of the choice of this time reference, and thus independent of i.Therefore, i is chosen to be a random variable that is uniformly distributedbetween 0 and 2c. The sinusoidal-input describing function #B () is obtainedby computing two expected values (C = 0)

EDH(0) D(0) =2c∫

0

H

( sin(i)

) sin(i)

12c

di =2Δc

(2.9)

and

EDD(0)2 =2c∫

0

2 sin(i)21

2cdi =

2

2(2.10)

Dividing (2.9) by (2.10) then results in an easily interpretable expression for #Bas a function of the input amplitude

#B () =4Δc

(2.11)

This is used within nonlinear analysis schemes to determine the amplitude andfrequency of e.g. limit cycles.

20

2.3. The Best Linear Approximation


The Best Linear Approximation uses a dynamic model to linearise the behaviourof the nonlinear system. It therefore considers dynamic nonlinear systems thatare excited by stationary random input signals. All signals that possess the samePower Spectral Density (PSD) and the same PDF result in the same BLA [Pint 12,Chapter 4].

In this thesis, we consider that the nonlinear systems have fading memory [Boyd 85].The systems that belong to this system class can be approximated arbitrarily wellin least-square sense by a Volterra series [Boyd 85, Pint 12]. An important prop-erty of the systems belonging to this system class is that the periodicity of theinput signal is preserved. Therefore, systems belonging to this nonlinear classare also called Periodic-In Same Period-Out (PISPO) systems. The periodicitypreservation rules out systems that generate sub-harmonics, contain bifurcations,or exhibit chaotic behaviour. It does, however, include systems that show hardnonlinear behaviour such as clipping, dead zones, quantisation, etc.

The BLA is obtained by estimating the `optimal´ Linear Time-Invariant (LTI)system that minimises in mean-square sense the error between the nonlinearresponse of the system and the response of the approximating LTI system. Thisestimation is most easily performed in the frequency domain, where it boils downto solving the following minimisation problem

BLA ( 9l) = argmin( 9l)

E* |. (l) − ( 9l) * (l) |2 (2.12)

where * (l) = F H(C) and . (l) = F D(C) are the input and output spectra ob-tained by taking the Fourier transform F •. The expected value E* • is takenwith respect to the random realisations of the input signal * (l)

E* • =∫"'

∫"

•@(*',* ) d*' d* (2.13)

where * =*' + 9* , and @(*',* ) is the PDF of the complex random variable *."' and " are, respectively, the domains of *' and * . Equation (2.12) can besolved explicitly and results in the following expression for the BLA

BLA ( 9l) =(HD (l)(DD (l)

=E* . (l) * (l)E* |* (l) |2

(2.14)

21


where (HD (l) is the cross-power spectrum between the output H(C) and the inputD(C), and (DD (l) is the auto-power spectrum of the input. • is the complexconjugate transpose operator.

Evaluating (2.14) analytically, as in the case of the DF, proves to be impossible fordynamic (nonlinear) systems. Therefore, the BLA is predominantly determinednon-parametrically from simulated or measured input-output spectra [Pint 12]. Inmany simulation or measurement scenarios, one only has access to time-domaindata. These spectra can then be obtained by performing the Discrete FourierTransform (DFT) on the time-domain signals. Suppose that we have a continuous-time signal G(C) that is sampled equidistantly with a sampling period )B for a totalof # samples at a sampling rate above the Nyquist rate. The DFT - (:) of thesequence G(=)B) is then defined as

- (:) =1#

#−1∑==0

G(=)B) 4− 9l:=)B (2.15)

In the remainder of this thesis, we assume that whenever the notation - (:) is used,we are dealing with the DFT of the signal G(C) as defined in (2.15). The frequencyl: that corresponds with the bin : is defined as l : = 2c:/(#)B). Remark that theDFT requires the time samples to be equally spaced in time. Variable time stepscan be used, but require the data to be interpolated to an equidistant time gridbefore computing the DFT. This will introduce interpolation errors and shouldtherefore be handled with care. Using the DFT spectra of the input * (:) andoutput . (:), the original BLA definition in (2.14) is recast to

BLA ( 9l : ) =E* . (:)* (:)E* |* (:) |2

(2.16)

Apart from estimating BLA ( 9l : ), the BLA theory additionally provides themeans to characterise the contributions that are not related to the linear beha-viour of the system. These contributions can be divided into two sources: anadditive noise source # (:) and an additive nonlinear distortion source (:) (Fig-ure 2.3). Performing a statistical analysis of the unmodelled residuals . (:) −BLA ( 9l : ) * (:) allows one to split the variance (power) contributed by thesesources as a function of l : . The estimation of these variances has one mainadvantage: the portion of the output spectrum . (:) that is perturbed by noiseand nonlinear distortions is identified, and the validity of the obtained linearapproximation can be evaluated both numerically and visually.

22


Figure 2.3.: Every nonlinear PISPO system that is excited by a random signal canbe linearly approximated by the BLA. The unmodelled dynamics areattributed to noise # (:) and nonlinear distortions (:).

Multisine excitations

Applying stochastic noise signals to excite the nonlinear PISPO system does notallow us to seperate the contributions of # (:) and (:). Establishing separabilityof both contributions requires to excite the system with a deterministic signal.The distortion term inherits (non-)determinism from the input signal while thenoise term remains stochastic. The idea is to synthesise a deterministic signal thatcan mimic the behaviour of the original stochastic signal. Then, the original signalis replaced by the synthesised signal to determine the BLA and to additionallyseparate the noise and the distortion contributions in the output spectrum.

A good candidate signal to implement this idea is the so-called random-phasemultisine signal. A random-phase multisine is defined in the time domain as thesum of commensurate tones

D(C) =∑:=1

: sin(2c: 50 C +i: ) (2.17)

where 50 is the base tone of the multisine. : and i: are, respectively, the amp-litude and phase of the : th tone. Choosing the amplitudes : deterministicallysets the PSD of the signal. The phases i: set the PDF. They are the realisationof a random process that sets the phases of one realisation of the multisine. Forone such a phase realisation, the multisine is a deterministic, periodic signal. Ithas fixed values for the amplitudes : and phases i: . When all realisations aretaken together, a random signal is created that belongs to the class of the originalrandom excitation (Figure 2.4). For a random-phase multisine, the phases i: areselected independently from a uniform distribution [0,2c) such that the multisinerealisations resemble filtered white Gaussian noise. Independent controllability ofboth the amplitudes and the phases is important to efficiently mimic the PSD andthe PDF of the original stochastic signals. After all, the same BLA is obtained fordifferent signals that share a common PSD and PDF. Ideally, an infinite number of

23


Figure 2.4.: Random-phase multisines have a Gaussian PDF with a discrete PSD.Different phase realisations are used to emulate a random signal.

tones should be present in the multisine to emulate a continuous PSD. Based onour practical experience, a good rule of thumb is to have at least 30 tones to obtainsatisfactory results. Another attractive property of the multisine signal is thatit is a periodic signal. The periodicity implies that a proper experimental setupensures that spectral leakage can be avoided completely in the steady-state inputand output spectra. This simplifies the BLA estimation techniques. Furthermore,the distortion term also becomes periodic with the same period (PISPO) whichadditionally ensures that it can be separated from the stochastic noise term.

Over the years, several BLA estimation techniques for systems excited with thesemultisines have been developed. The most straightforward one is the so-called ro-bust method [Pint 12, Section 4.3.1]. This method computes (2.16) by considering" independent phase realisations of the multisine signal. The changing phasesresult in a different time-domain signal and, consequently in a different nonlinearbehaviour. Great care needs to be taken to ensure that the changing phases do notalter the PDF of the multisine excitation. Otherwise, different BLAs are obtained.The method requires that " different steady-state inputs * (:) and outputs . (:)are obtained for these " independent phase realisations. Computing the averageand the variance over these realisations is then sufficient to estimate the BLA andthe combined variance of the noise and the nonlinear distortions. To computethe noise variance separately, one needs to perform a statistical noise analysisfor every phase realisation. A natural way to do this is to retrieve % periods ofthe steady-state input-output spectra for a certain phase realisation. After all,re-application of a multisine with the same phase realisation does not change the

24


nonlinear behaviour. Differences over these % spectra are exclusively caused bythe noise source # (:). It is then sufficient to compute the variance over these %spectra to compute the noise variance separately. In this PhD, the number % is alsocalled the number of periods of the multisine. For a more in-depth explanation ofthe robust method we refer to [Pint 12, Section 4.3].

Example The goal of this example is to showcase the capabilities of multisinesfor emulating filtered white Gaussian noise signals and for estimating theBLA. Therefore, we consider the nonlinear system displayed below, which is acascade of a linear dynamic Chebychev filter and a nonlinear static functionthat includes a deadzone and saturation. Furthermore, the output of the cascadeis perturbed by white Gaussian noise with a standard deviation equal to 0.01V.

Suppose that we would want to derive the BLA for a filtered white Gaussiannoise excitation (−) with a standard deviation f equal to 1V (Figure below).Instead of using the original signal, we will emulate the behaviour using arandom-phase multisine (−) with = 1000 tones.

−3σ 0 3σ

PDF

0 80 160−3σ

0

3σ

Time [s]

Time Domain

0 2.5 5 7.5 10Frequency [Hz]

Amplitude[dBV

]

Frequency Domain

Comparing the PDF and the time domain waveforms shows that the random-phase multisine indeed resembles the original Gaussian noise excitation. Themain difference becomes clear in the frequency domain where the multisinehas complete control over the PSD.

" = 100 phase realisations and % = 2 steady-state periods of the random-phasemultisine are acquired using a time-domain simulator with a fixed time-step.

25


One additional period is used during the simulation to ensure that the transientis completely damped out and steady-state is reached. The robust method isused to identify the BLA and estimates for the standard deviation of the noisef# (:) and nonlinear distortion contributions f (:) in the output spectrum:

0 2.5 5 7.5 10

−80

−60

−40|Y (k)|

σD(k)

σN(k)

Frequency [Hz]

Amplitude[dBV

]

Output spectrum

0 2.5 5 7.5 10−30

−20

−10

0

σ = 2VRMS

σ=

1VR

MS

Frequency [Hz]

|GB

LA(

jωk)|[dB]

BLA

A significant portion of the output spectrum is clearly perturbed by the non-linear distortion which in turn results in a ’noisy’ estimate of the mean output. (:) and the BLA. The noise on the other hand has, in this example, a marginalcontribution in the output spectrum. Additionally, we show the influence ofthe input amplitude on the resulting BLA by doubling the input Root MeanSquare (RMS) value f. The BLA clearly shows a significant reduction of theoverall gain. This gain reduction is caused by the static nonlinear function thatsaturates the power-elevated signal relatively more compared to the originalpower level. This example showed that the BLA is an effective tool to analysethe approximative linear behaviour of a dynamic nonlinear system, togetherwith a characterisation of the noise and distortion levels.

26

2.4. Towards a combined approach?

2.4. Towards a combined approach?

Taking a bird’s eye view, it seems that the DF and BLA start from a different per-spective (Table 2.1) and result in complementary application domains. Therefore,it is up to the user to decide which theory is best suited for the application at hand.The most notable difference between both theories is the required preliminarysystem knowledge. The DF theory needs an accurate mathematical representationof the system (white-box) to ensure that the computation of (2.7) yields a validlinear approximation. With the BLA theory, on the other hand, no prior know-ledge about the system is needed since the BLA is extracted non-parametricallyfrom simulated/measured data. In the particular case where the nonlinear systemis strictly static, both theories are related to each other as follows. Suppose thatwe have a strictly static nonlinear function that is excited by white Gaussiannoise with a certain power f2. The Bussgang theorem [Buss 75] then dictates that(HD (l) = (DD (l), where is a frequency-independent constant that dependsonly on the applied input power. In this case, the BLA reduces to a constant thatis, moreover, the same as the DF that is obtained for white Gaussian noise.

In the past, several solutions have been proposed to extend the original DF andBLA theory. The DF theory was generalised towards multisine excitations byrelating it to the more general Volterra theory [Peyt 91]. Furthermore, Nuij et al.introduced the higher-order sinusoidal input describing function that includesthe influence of the single-tone sinusoidal magnitude on the generated harmon-ics [Nuij 06]. Power-scalable parametric modelling techniques for the BLA havebeen proposed that rely on nonlinear optimisation algorithms [De L 06b, Vand 09].Regrettably, these extensions are not well adapted to be used for the analysis ofcontinuous-time ΣΔ modulators. They either require thorough knowledge aboutthe Volterra series expansion, or are not suited to model the mixed-signal natureof the ΣΔ modulator. In recent years, local modelling techniques [Pint 11] havebeen developed that enable the robust non-parametric estimation of the BLA.These techniques are a promising candidate to blend concepts from both the DFand the BLA and result in a combined approach for the linear approximation ofnonlinear systems. But before diving head-first into this combined approach, wefirst study some key aspects of both the DF (Chapter 3) and the BLA (Chapter 4)separately.

27


Describing FunctionsBest Linear

Approximation

Signals Constant(Two-tone) Sinusoid

Gaussian noise

Random processMultisine

Type of nonlinearities Static Dynamic

System knowledge White-box Black-box

Modelling dependency Amplitude Frequency

Table 2.1.: Depending on the needs of the user, a choice has to be made betweenthe Describing Functions or the Best Linear Approximation.

28

3. An improved DescribingFunction applied to OTA-basedcircuits

As a starting point for my research, I looked into the Describing Func-tion (DF) theory. This chapter is inspired by the following question: howcan this theory effectively be applied to the analysis of circuits incorporatingoperational transconductance amplifiers? The content of this chapter ispublished in [Peum 17].

Continuous-time active circuits using Operational Transconductance Amplifi-ers (OTA) and capacitors, also known as gm −C circuits, have attracted theinterest of designers due to their high-frequency capability (1MHz - 100MHzrange), easy tunability and structural flexibility [Sanc 00, Lina 91]. In recentyears, however, the Complementary Metal Oxide Semiconductor (CMOS) tech-nology downscaling has caused a substantial decline in the obtainable dynamicrange [Gras 15, Abde 15]. As a result, the Operational Transconductance Ampli-fier (OTA) behaves nonlinearly for smaller signal levels than before. This nonlinearbehaviour is a knife that cuts both ways: depending on the application at hand itis undesirable (e.g. causes nonlinear distortion in filters) or desired (e.g. electronicmixers).

The general description of weakly nonlinear systems (Volterra) has been used inthe past to predict the nonlinear effects within gm −C filters [Szcz 93, Fern 08].Although successful, this Volterra-based approach has one main disadvantage:interpretable results are obtained mainly for single-tone and two-tone excitations[Wamb 13]. On the other hand, modern wireless communication systems deal withmore complex, digitally modulated signals, e.g. Orthogonal Frequency DivisionMultiplexing (OFDM). As explained in Chapter 2, the signal-dependent nonlinearnature of the OTA does not enable the translation of the behaviour when excited

29

3. An improved Describing Function applied to OTA-based circuits

with a single-tone or a two-tone signal to the behaviour obtained for modulatedsignals [De L 06a].

Another solid approach to describe nonlinear systems is based on the earlierintroduced DF theory (Chapter 2). Current design flows merely use the DF toretrieve initial estimates of the system properties (i.e. oscillation frequency andamplitude). Obtaining accurate estimates proves difficult for most applicationsdue to the crude theoretical approximation that is used to describe the nonlinearbehaviour. To remedy the above mentioned issues, we propose an improvedDF which predicts the saturation behaviour for both single-tone and complexmodulated excitations. Designers can then more accurately predict the effectof the saturation nonlinearity without the need to perform a large number oftime-consuming simulations.

3.1. Approximation of the static nonlinearbehaviour

In the past, the DF has proven to be helpful in a myriad of different applications.The applicability of the DF theory stands or falls with the capability of the com-puted DF to accurately describe the actual nonlinear behaviour of the system.In general, two different approaches exist to compute the DF. Both of them re-quire access to simulated/measured data of the static nonlinear behaviour of thesystem. The first approach computes the DF directly by numerical integrationof the integrals in (2.7). The second approach first approximates the originalstatic nonlinear behaviour of the system with an analytic function. Thereafter,this analytic function is used to compute (2.7). We apply the second approachthroughout this chapter for the nonlinear analysis of saturation phenomena.

In literature, several analytic approximations exist that allow to model saturation.One commonly used approximation models the limiting behaviour of saturationby an abrupt change. It uses a piecewise linear function that maintains a per-fectly linear operation in the intermediate region (Figure 3.1a) [Gelb 68]. Thiselementary function is widely used, but cannot be applied in everyday practicedue to the poor modelling power that is caused by its simplicity (Figure 3.1b).Other saturating mathematical functions exist that are differentiable and thusexhibit smooth behaviour. Examples of such functions are the hyperbolic tangent

H(G) = tanh(G) and the error function H(G) = 2√c

G∫04−C

2 dC. While these functions

30

3.1. Approximation of the static nonlinear behaviour

u

y

Elementary

Actual

y

(a) Approximations

−0.6 −0.3 0 0.3 0.6−100

−70

−40

−10

Input u [V]

Rel

ativ

eer

ror

[dB

]

(b) Approximation error

Figure 3.1.: (a) The generally used elementary approximating function does notprovide a good fit of the actual saturation behaviour. The proposedapproximation function in (3.1) provides a far more better fit (' = 5).(b) Relative error for the elementary approximation, the hyperbolictangent, the error function, and the proposed approximation.

improve the modelling accuracy in the transition regions around |D | = 0.3V, theyare less accurate in the vicinity of the origin (Figure 3.1b). Furthermore, theyrequire a nonlinear optimisation scheme to be fitted onto the original saturationcharacteristic.

To tackle these problems, we propose to use the following approximating functioninstead

H(D) ='∑==0

U=

(D

√1+D2

)2=+1

(3.1)

where D is the input, H represents the approximation of the actual saturationcharacteristic H, and ' is the order of the approximation. The coefficients U= canbe obtained with a linear least-squares regression that tailors H to H. Only oddfunctions are considered for H since the even terms result in a zero contributionwhen evaluating the DF [Gelb 68].

The reason for choosing (3.1) is two-fold: the resulting model is linear in theparameters U=, which allows to avoid potential nonlinear estimation difficulties,and the small-signal linearised behaviour is obtained using the coefficient U0 ofthe expansion. Imposing that the modelled and simulated small-signal behaviour

31


coincide leads to the following relationship

limD→0

dH

dD= (( → U0 = (( (3.2)

where (( is the small-signal gain obtained by linearising the actual systemaround its operating point. The estimation algorithm that is used to estimatethe U= is a traditional linear least-squares minimizer for all the coefficients U=,except for U0, which is fixed to match (( . The result of fitting H on the previouslyintroduced saturation characteristic is illustrated for ' = 5 in Figure 3.1b. As itcan be observed from Figure 3.1a, H outperforms the earlier introduced analyticapproximations. A relative error with respect to H of −50dB is obtained over thewhole input range.

3.2. The improved Describing Function

The previous section dealt with the derivation of a model H which approximatesthe actual static nonlinear function closely. An accurate approximation is essentialsince the DF theory is applied directly to H rather than to the actual saturationnonlinearity H. We restrict ourselves to the single-tone and Gaussian noise caseshere, since they are the most interesting ones for the majority of applications.Neglecting the constant input is justified because most electronic systems stabilisetheir operating point using dedicated circuitry (e.g. through common-modefeedback).

Single-tone sinusoid

The Sinusoidal-Input Describing Function (SIDF) #B () can be obtained by ex-panding (2.7) into its integral form and substituting D with sin(q)

#B () ='∑==0

U=

c

c∫−c

( sin(q)√

1+ 2 sin2 (q)

)2=+1

sin(q)dq (3.3)

where q is considered to be uniformly distributed in the interval [−c, c]. #B ()can be simplified further by applying the substitution G = sin(q) and taking into

32

3.2. The improved Describing Function

= %[=]E () %

[=]F ()

0 1 11 2 +2 22 +2

2 4 +1332 +

83

34 +1732 +

83

3 6 +103154 +

128152 +

165

46 +164154 +

152152 +

165

Table 3.1.: Polynomials belonging to (3.5) for = ranging between 0 and 3.

account that we have a perfectly odd approximation characteristic

#B () ='∑==0

4U=c

1∫0

(G

√1+ 2 G2

)2=+1G

√1− G2

dG (3.4)

Unfortunately, no analytical solution for the integral of (3.4) exists. However,by using a symbolic integration package [Wolf], the integral can be written as afunction of the complete elliptic integral of the first and second kind, F (:) andE(:) respectively [Abra 48]

#B () ='∑==0

4U=(%[=]E () E(−

2) −% [=]F () F (−2)

)c2 (2 +1)=

(3.5)

Efficient numerical implementations exist for these elliptic integrals [Carl 95].They are readily available in most numerical math libraries. % [=]F (

2) and % [=]E (2)

are even polynomials of order = in the variable 2. These polynomials depend onthe order = and can be derived by performing arithmetic manipulations on theoriginal integral. Table 3.1 lists some of these polynomials [Wolf].

Gaussian noise

Real-world applications generally use complex modulated signals instead of single-tone waves. In most cases, these modulated signals can be analytically approx-imated by Gaussian noise such that otherwise untreatable signals can still bedealt with. The method used in the sinusoidal case can be adapted to retrieve the

33


Random-Input Describing Function (RIDF)

#A (f) ='∑==0

U=√

2cf3

+∞∫−∞

(D

√1+D2

)2=+1

D exp

(−D2

2f2

)dD (3.6)

where f is the standard deviation of the zero-mean Gaussian distributed noisesignal D. Again, a symbolic integration package [Wolf] is used to transform (3.6).This time the result includes the confluent hypergeometric function of the secondkind * (0, 1, I) [Slat 60]

#A (f) ='∑==0

(2=+1)!!U=2=√

2f*

(=+

12, 0,

12f2

)(3.7)

Here, (=)!! represent the double factorial operator defined by (=)!! = =(=−2) (=−4) ...1. As in the sinusoidal case, efficient numerical algorithms exist to evaluate(3.7) [Slat 60].

3.3. Application of the DF to OTA-based circuits

The derived RIDF and SIDF are applied to a gm−C filter and an oscillator. First, afully-differential OTA is designed in 0.18`m CMOS with a supply voltage+cc equalto 1.8V. The circuit is based on the master’s thesis of Stephane Bronckers [Bron 05].The RIDF is thereafter used to study the effects of modulated signals, modelled byrandom-phase multisines, on the shape of the transfer function of a Tow-Thomasbiquad filter configuration. The oscillator application involves the prediction ofthe amplitude and oscillation frequency of a quadrature OTA-based oscillatorusing the SIDF.

The operational transconductance amplifier

A fully differential wide bandwidth OTA ( 5−33 = 1.4GHz) was designed in a0.18`m CMOS technology. This OTA consists of three main stages (Figure 3.2a):

– An input stage that level shifts the input common-mode voltage from 0.9Vto 0.4V using two source followers. By doing so, the input and the outputcommon-mode voltage ranges are better matched.

34


– A transconductance stage that provides most of the transconductance 6<that is needed for the voltage-to-current conversion. Source degeneration('346) has been included to linearise the response of the OTA.

– An output stage that delivers the high output impedance using a foldedcascode configuration.

The common-mode voltage at the output nodes is stabilised at 0.9V, i.e. half ofthe power supply +22 , by an active common-mode circuit (Figure 3.2b).

Every practical OTA exhibits nonlinear saturation effects that limit the obtainabledynamic range. The static voltage-to-current relationship can be constructed toevaluate both the behaviour and the severeness of this saturation phenomenon.In the case of the OTA this analysis boils down to performing a DC sweep of thedifferential input voltage. The corresponding differential current which flowsthrough a shorted output is then analysed. The voltage-to-current relationshipthat is obtained here is the one that has been used as an example for the nonlinearbehaviour in Figure 3.1a. The input D and the output H are the differential inputvoltage and differential output current respectively. The small-signal linear gain(( of the OTA is equal to 625`S and is obtained by an AC analysis.

Verification of the Describing Functions

It is imperative to verify the performance of the approximative SIDF and the RIDFfirst. They have been derived earlier in Section 3.2 and are compared to the actualgain compression that is induced by the OTA. The actual compression is obtainedby Harmonic Balance (HB) simulations of the designed OTA. The simulations areperformed using the Advanced Design System (ADS) suite.

We excite the OTA independently with a single-tone sinusoid and a random-phasemultisine (with a flat amplitude spectrum) to compute the large-signal linearisedgain using (2.16) for a set of different values of and f. In the single-tone case,(2.16) simplifies to the division of the output spectrum by the input spectrum atthe excited frequency. For the multisine excitation, the robust method (Section 2.3)with " = 100 independent phase realisations is used.

35


(a) Operational transconductance amplifier.

(b) Common-mode feedback.

Figure 3.2.: Circuits used during the design of the OTA.

36


625

500

375

Ns(

A)[µ

S]

0 0.2 0.5 0.8

−80

−60

−40

−20

A [V]

Rel

ativ

eer

ror

[dB

]

(a) SIDF

625

500

375

σ [VRMS]

Nr(

σ)[µ

S]0 0.1 0.2 0.3

−80

−60

−40

−20

σ [VRMS]

Rel

ativ

eer

ror

[dB

]

(b) RIDF

Figure 3.3.: The proposed approximation H (' = 5) outperforms the nonlinearprediction capability of the elementary one for both the single-tone(a) and Gaussian noise excitation (b). The SIDF and RIDF are verifiedwith large-signal Harmonic Balance simulations (•).

Figure 3.3 shows #B () and #A (f) as a function of their respective input vari-able. The proposed DFs were also compared with the elementary approximatingfunction (Figure 3.1a). For low values of and f, the gain converges to thesmall-signal transconductance 6<. Compression is visible for higher values of theinput variables. The improved modelling of the nonlinear characteristic obtainedby H provides a far better fit than the elementary one. Still, the method remainsnumerically efficient to compute and evaluate. For example, simulating (2.16)with HB (" = 100 realisations) for a given Root Mean Square (RMS) value f on anIntel i7-4790 CPU (3.6 GHz) takes at least a minute, while the model evaluationof (3.7) takes less than a second.

37


Figure 3.4.: Differential Tow-Thomas biquad under study. To verify the RIDF, weexcite the system with a random-phase differential-mode multisinewith varying RMS level.

Tow-Thomas biquad

Early research [Bial 71] has shown that any transfer function needed for activefilter design can be established by the exclusive use of OTAs and capacitors,laying the foundation for gm−C filters. One widely used method to realise high-order filters is by cascading second-order gm −C biquads. A popular biquad isthe so-called Tow-Thomas biquad (Figure 3.4). This specific gm−C architectureconsists of two integrators, an ideal one and a lossy one, connected in a feedbackconfiguration [Sun 02].

The biquad exhibits the following differential low-pass transfer function (B, ®6)from input to output voltage

(B, ®6) =−60 61

12 B2 +631 B+61 62(3.8)

where B is the complex Laplace variable, 61 − 64 and 1 −2 are, respectively,the transconductances of the OTAs and the capacitors present in Figure 3.4. ®6represents the vector of all tranconductances combined, i.e. ®6 = (61, 62, 63, 64) ∈R4×1.

A straightforward method to evaluate the input amplitude-dependent behaviourof the biquad would be to replace all the transconductances 61 to 64 with theproposed #A (f). However, this approach is fundamentally wrong, the reasonbeing that each of the OTAs has a different f at its respective input which shouldbe taken into account properly. Looking at the existing literature, single feedbacksystems containing multiple nonlinearities were only investigated for sinusoidal

38


Algorithm 1 Iterative algorithm for the derivation of the correct f2:

and the cor-responding transconductance vector ®6.

1. For every node : , use the input power to initialise f2:

and ®6:

f2: = f

2D

®6 = #A (fD) using (3.7)

2. Use the current transconductance vector ®6 to compute the transfer functionsD→ : (B, ®6). Retrieve an update f2

:,new of f2:

by evaluating (3.10). Use f:,new

to obtain a new transconductance vector: ®6 = #A (f:,new).

3. Verify whether or not |f2:,new−f

2:|/f2

:remains below a specified maximum

relative error, and this for every node : . If it does, stop the iteration. If not,set f2

:= f2

:,new and reiterate starting from step 2.

signals due to their easy graphical interpretation [Gelb 68, Davi 71]. To take thesemultiple nonlinear OTAs into account for Gaussian noise, we start from the inputpower level f2

D and use linear system theory to retrieve an estimate of the powerf2:

at every node : present in the network. The following relationship can be usedfor the derivation of f2

:as a function of f2

D [Gelb 68]

f2: =

+∞∫−∞

|D→ : ( 92c 5 , ®6) |2 (DD (2c 5 )d 5 (3.9)

where (DD (2c 5 ) is the input power spectral density and D→ : (B, ®6) is the lineartransfer function from the input to node : expressed in symbolic form. (3.9) canbe further simplified if we consider that the input signal is white Gaussian noisemodelled by random-phase multisines as in (2.17)

f2: =

2f2D

∑A=1|D→ : ( 92cA 50, ®6) |2 (3.10)

The transfer functions D→: (B, ®6) can be derived using the modified nodal analysison the equivalent linear circuit [Giel 12]. This equivalent circuit is obtained by re-placing the OTAs with an ideal voltage-controlled current source. Symbolic circuitanalysis tools exist (e.g. [Mont]) which allow to automatically generate D→: (B, ®6).

39


These linear symbolic transfer functions are indirectly dependent on f2:

through®6, which means that (3.9) cannot be solved directly. To cope with this issue, wepropose to use an iterative scheme that deals with this dependency (Algorithm 1).Since Algorithm 1 is a nonlinear optimisation scheme, potential convergenceissues could arise. However, no convergence problems were encountered in anyof the considered cases. One possible method to mitigate convergence issues,if encountered, is to improve the estimates of f2

:during the initialisation by

performing an AC analysis on the circuit and subsequently calculating (3.10).

The iterative procedure of Algorithm 1 is applied to the Tow-Thomas biquadexample (Figure 3.4). The capacitors and transconductances are chosen such thata quality factor of 2 and a resonance frequency of 1MHz are obtained. As input,we apply a random-phase multisine with a frequency resolution 50 of 50kHz and200 excited tones. For comparison purposes, transistor-level HB simulations areperformed to extract the Best Linear Approximation (BLA) using (2.16). This isdone for RMS values ranging between 0.01VRMS and 0.3VRMS. We use the relativeerror (RE) between the outcome of Algorithm 1 and the estimated BLA as a figureof merit for the accuracy

RE( 5 ) = ( 92c 5 , ®6) − BLA ( 92c 5 )

BLA ( 92c 5 )

(3.11)

Unfortunately, the results of HB are distorted by nonlinear effects, which grow insize as the multisine’s power increases. The required total number of phase real-isations is determined using the standard deviation fBLA (:) of the BLA estimateBLA ( 9l: ), which is defined as

fBLA (:) =

√√√1

" (" −1)

"∑<=1

. [<] (:)* [<] (:)− BLA ( 9l: )

2 (3.12)

where " is the total number of phase realisations. * [<] (:) and . [<] (:) are,respectively, the input and output spectrum of the biquad corresponding to the<th multisine realisation. To guarantee that the relative error is a representativemeasure for the prediction capabilities of the RIDF, and is not dominated bythe adverse effects of the nonlinear distortion, we impose that the RMS fBLA

for all considered fD should be minimally 10dB below the relative error. Thisrequirement is fulfilled for " = 200 independent phase realisations.

40


-5

0

6

H[d

B]

105 106

−40

−20

0

Frequency [Hz]

Rel

ativ

eer

ror

[dB

]

Initialisation

Iteration 1

Iteration 3

HB

Initialisation

Iteration 1

Iteration 3

Figure 3.5.: Different iterations of Algorithm 1 show that the proposed algorithmconverges rapidly (only 3 iterations are needed) to acquire a goodestimate of the actual behaviour obtained through a steady-state ana-lysis (fD = 0.3VRMS).

To verify the performance of the proposed iterative scheme, different iterations ofAlgorithm 1 are showcased in Figure 3.5 for fD = 0.3VRMS. The results obtainedby the initialisation of the transfer function show that Algorithm 1 results in abig improvement of the model quality. In the case of the biquad only 3 iterationsof the algorithm are needed to obtain a good approximation of the steady-stateanalysis (a relative error fluctuating around −40dB is obtained). More iterationsare needed when more complex transfer functions are involved. For example,increasing the Q-factor of (B, ®6) in (3.8) from 2 to 4 already requires one moreiteration on average.

Figure 3.6 shows the influence of an increase of input RMS value on the transferfunction (B, ®6) of the biquad, and this for both the proposed and the elementaryapproximation. The following observations can be made

– Increasing fD does not exclusively result in gain compression. It also altersthe resonance frequency of the biquad (only slightly visible in the figure).

41


−40

−20

0 0.01VRMS

0.3VRMS

H[d

B]

105 106 107−60

−40

−20

0.01VRMS

0.3VRMS

Frequency [Hz]

Rel

ativ

eer

ror

[dB

]

(a) Proposed RIDF

−40

−20

0 0.01VRMS

0.3VRMS

H[d

B]

105 106 107−60

−40

−20

0.01VRMS

0.3VRMS

Frequency [Hz]

Rel

ativ

eer

ror

[dB

]

(b) Elementary RIDF

Figure 3.6.: The transfer function of the biquad changes when the input RMSvalue is increased from 0.01dBVRMS to 0.3dBVRMS. − : proposed RIDF,− : elementary RIDF, • and • : relative error for fD = 0.01dBVRMS, ×and × : relative error for fD = 0.3dBVRMS.

– Both RIDFs exhibit the same behaviour at low fD , as is shown by the equalrelative error for • and •.

– The relative error in the case of the elementary RIDF at 0.3dBVRMS (×)shows an increased deterministic peaking close to the resonance frequency.This indicates that the elementary RIDF does not well model the change inresonance frequency.

To further analyse the capabilities of both RIDFs, we compare the accuracy (re-lative error) and efficiency (simulation time) as a function of fD (Figure 3.7).The mean of the relative error taken over the frequency has been chosen to rep-resent the accuracy. For each value of fD a single figure of merit is extracted.Again, the results obtained by Algorithm 1 are verified against transistor-levelHB simulations. The average simulation time for one realisation of (2.16) with arandom-phase multisine was 1.34s on the same Intel i7-4790 (3.6GHz) machine.Analysing Figure 3.7a shows that the proposed RIDF is characterised by an approx-

42


0 0.1 0.2 0.3

−60

−40

−20

σu [VRMS]

Mea

nre

lati

veer

ror

[dB

]

y

Elementary

(2.13)

(a) Mean relative error

0 0.1 0.2 0.30

0.5

1

1.5

σu [VRMS]

CP

Uti

me

[s]

y

Elementary

(b) CPU time

Figure 3.7.: Comparison of the mean relative error (a) and CPU time (b) for theproposed and elementary RIDF as a function of fD .

imately constant relative error (≈ −40dB) as a function of fD . This is not the casefor the elementary RIDF where the mean relative error rises to a maximum valueof −22dB. The increase in accuracy is coupled with a small loss in computationalefficiency (Figure 3.7b). Investigation of Figure 3.7b shows that the CPU time isnot constant as a function of fD :

– The elementary RIDF shows two jumps in the computation time whichcorrespond with an increment of the iteration count in Algorithm 1.

– These jumps in iteration count are less apparent for the proposed RIDF.This behaviour is caused by the numerical evaluation of (3.7) for which thecomputation time depends on the input parameters of * (0, 1, I) [Pear 16].

The elementary RIDF takes maximally 0.19s to solve Algorithm 1 while the pro-posed RIDF has to compute maximally for 1.5s. These computation times are stillsignificantly lower than the HB simulation time for 200 realisations, which rangefrom 6 minutes to 9 minutes and 44 seconds.

From these comparisons, we conclude that the proposed RIDF exhibits a muchlower relative error than the elementary RIDF while still remaining numericallyefficient to compute.

43


Quadrature OTA-based oscillator

On-chip automatic tuning is essential to avoid that parasitic phenomena such asthermal variations, parasitic capacitances, fabrication tolerances and mismatchesinfluence the envisioned performance of a circuit. CMOS filter designers generallyadopt the master-slave system to obtain this tuning [Krum 88, Scha 89]. Master-slave tuning employs a Phase-Locked Loop (PLL) as a master to generate a propertuning signal for the slave system, i.e. the Tow-Thomas biquad. At the core ofthis PLL architecture, a Voltage-Controlled Oscillator (VCO) is required that hasto be carefully matched to the biquad. This is necessary to effectively counteractparasitic phenomena.

We want to verify how well the proposed SIDF (3.4) can predict the oscillationamplitude and frequency of a sinusoidal gm −C oscillator. For this purpose, weconsider the quadrature oscillator that is depicted in Figure 3.8 [Kard 92]. Itconsist of two parts: the linear part sets the oscillation frequency (61, 62, 1 and2), while the nonlinear one ensures the start-up of the oscillator and stabilisesits amplitude (63 and 64). The poles of such a quadrature oscillator are generallydescribed by the following characteristic equation

B2− 1 B+Ω20 = 0 (3.13)

where Ω0 sets the oscillation frequency. To achieve a persistent oscillation, 1should be equal to 0. 1 > 0 and 1 < 0, respectively, correspond with undamped ordamped oscillatory behaviour. To avoid the occurrence of an asymmetry in theoscillator, dummy OTAs were added as shown in Figure 3.8 to ensure that thecorresponding differential nodes are loaded by the same parasitic capacitance andpresent the same output conductance.

Both 1 and Ω0 are a function of the transconductances 68 , the total capacitances( tot8= 8 +parasitic) and the total output conductances (6 tot

>1 = 6>1 +36dummy> and

6 tot>2 = 6>2 +6>3 +6>4 +6dummy

> ) [Gala 05]

1 =(64 (1) −63 (1) −6 tot

>2 ( 9Ω0)) tot1 −6

tot1 (1) tot

2

tot1 tot

2(3.14)

Ω20 =

61 (1) 62 (2) tot

1 tot2

+6 tot>1 ( 9Ω0) (64 (1) −63 (1) +6 tot

>2 ( 9Ω0)) tot

1 tot2

(3.15)

44


Figure 3.8.: The differential quadrature oscillator under study. Dummy OTAs areadded (gray) such that each node of the oscillator sees the same para-sitic capacitance and presents the same output conductance. The SIDFpresumes a perfect sinusoidal operation at a fixed angular frequencyΩ0.

45


6>1 to 6>4 are the output conductances of the different OTAs in Figure 3.8. 6dummy>

is the output conductance of the inserted dummy OTAs. parasitic is the parasiticcapacitance that consists of a combination of the input and output capacitancesof the different OTAs connected to a specific node. 1 and 2 represent theamplitudes of the single-tone sinusoid taken at the different nodes of the circuit(Figure 3.8). Similarly to what we did in Section 3.3, we define ® as the vector ofthe different amplitudes, i.e. ® = (1, 2).

An ideal oscillator would not include the coefficient 1 (purely imaginary poles).However, the finite output impedances of the OTAs result in the presence of lossesand a departure from ideality. Therefore, a mechanism has to be put in place tocompensate for these losses and to allow for self-startup of the oscillation. Thenonlinear behaviour of 63 and 64 with respect to 1 is constructed such that 64is larger than 63. A proper degeneration resistor '346 is chosen to this end as inFig. 3.2a. Furthermore, 64 starts to compress at lower amplitude which ensuresself-startup and convergence of the initially unstable poles to the imaginary axis.

A pertinent question remains to be solved: how can we deduce the steady-state

oscillation parameters ®k = (k1, k2) = ®1=0

and Ω0 using the SIDF? The solution

is obtained starting from the observation that the characteristic equation (3.13)only reaches steady state if 1 equals to 0. By evaluating (3.14) as a function ofthe vector ®, and using the SIDF (68 (: ) = #B (: )), the amplitudes at which 1( ®)crosses the zero-axis are obtained. If the slope of 1( ®) at a specific intersectionis negative, then the obtained oscillation can be proven to be stable and uniquefor the corresponding amplitude [Davi 71]. Two issues still need to be figured outbefore applying the method:

– The amplitudes in ® are not independent from each other. Fixing onearbitrary amplitude : , results in fixed values for all the others. They canbe calculated using the closed-loop linearised transfer function from node :to the node under consideration. This requires to replace all the nonlinearsystems with their respective SIDF #B [Davi 71].

– The OTA is not a purely static nonlinear system. If this would be the case,the derivation of Ω0 would be greatly simplified: the output conductancewould become a frequency-independent constant in that case. However,assuming this adversely impacts the estimation accuracy of the oscillationfrequency. To take the frequency dependence of the output conductance ofthe OTA into account, we need to solve the implicit equation (3.15). Again,we propose to use an algorithm that iteratively introduces this frequencydependence (Algorithm 2).

46


Algorithm 2 Iterative scheme for the derivation of 1 and Ω0 as a function of 1.:→ :+1 ( 9Ω0) represents the linear transfer function between node : and the nextnode : +1. # :B (k: ) is the SIDF of the nonlinear element that connects node : and: +1.

1. Initialise Ω0 presuming no oscillation is present:

Ω0 = 0

2. Evaluate (3.14) for different values of 1. Set the oscillation amplitude k1equal to the amplitude for which 1 becomes 0 and has a negative slope.

3. Starting from k1, recursively determine the other amplitudes of ®k using thefollowing recursive formula:

k:+1← |:→:+1 ( 9Ω0) |# :B (k: )k: for : > 1

4. Retrieve an update of the oscillation frequency Ω0,new by evaluating (3.15)using the current values for ®k and Ω0.

5. Verify whether or not the relative variation |Ω0,new −Ω0 |/Ω0 lies below aspecified maximum allowed relative error. If it does, stop the iteration. If itdoes not, set Ω0 = Ω0,new and restart the iteration from step 2.

This iterative scheme is implemented and applied to the previously introducedquadrature oscillator. The capacitances and transconductances are chosen suchthat an oscillation frequency close to 1MHz is obtained. Figure 3.9 shows theresulting 1 and Ω0 as a function of 1 with 2 = k2. It required 3 iterations ofAlgorithm 2 to obtain a relative error smaller than 0.1%. To provide a point ofcomparison, the analysis has been applied using both the proposed and elementarySIDF. In the first stage, the steady-state oscillation amplitude k1 is deduced fromFigure 3.9a by determining where 1 crosses the zero-axis. If we compare this valueto the one retrieved with an oscillator analysis, we conclude that the elementarySIDF significantly underestimates the actual amplitude (Table 3.2). The proposed#B on the other hand results in an amplitude which almost coincides with theone predicted by the oscillator analysis (relative error of −45.1dB). Using theestimated amplitude ®k, we deduce the oscillation frequency that is predictedby the SIDF. Figure 3.9b shows that both SIDFs overestimate the oscillationfrequency. The prediction for H, however, is more accurate than the elementaryone (Table 3.2). This overestimation can be explained as follows. The DF theorydoes not account for the influence of harmonics [Gelb 68], which are significant inthe case of an oscillator. Simulations show that the prediction capability of the

47


0.3 ψ1 0.7

0

A1 [V ]

b

Elementary

y

Oscillatoranalysis

Unstable

Stable

(a) 1 as a function of 1.

0.3 ψ1 0.7A1 [V ]

Ω0

Elementary

y

Oscillatoranalysis

(b) Ω0 as a function of 1.

Figure 3.9.: Comparison between the elementary and proposed approximation.H provides a better estimation of the steady-state oscillation, both inamplitude and in frequency.

DF approach declines with an increasing oscillation amplitude. This behaviour iscaused by growing harmonics. They are no longer negligible when compared tothe fundamental tone.

The computation times in Table 3.2 show that the proposed SIDF does not signi-ficantly increase the computational cost when compared to the elementary one.Furthermore, both SIDFs remain computationally more efficient than the oscillatoranalysis using ADS.

48

3.4. Summary

Proposed SIDF Elementary SIDF HBCPUtime Building the SIDF 1.05s 1.05s

SIDF evaluation + 0.70s + 0.64s 5.29s

Total 1.75s 1.69s 5.29s

Errors

Maximum relative error ®k −45.1dB −34.6dBRelative error Ω0 −20.8dB −16.9dB

Table 3.2.: The proposed SIDF increases the accuracy of the obtained estimates for®k and Ω0. This marginally increases the computational cost comparedto the elementary SIDF.

3.4. Summary

In this chapter, we introduce an improved DF that accurately predicts the nonlin-ear behaviour of devices that exhibit saturation. The main advantage is that theproposed DF can be directly fit to the saturation characteristic. This eliminates theneed for often cumbersome assumptions about the shape of the saturation beha-viour. As it turns out, the proposed DF can be efficiently computed for single-tonesinusoidal and Gaussian distributed white noise excitations.

The DF has been applied on a gm−C Tow-Thomas filter and a quadrature OTA-based oscillator. In both cases, an iterative scheme using the DF was developedthat predicts the nonlinear effects of the saturating behaviour accurately withoutresorting to time-consuming simulations.

49

4. Local modelling techniques

During our exploration of the BLA theory, we developed an extensionfor currently existing local modelling techniques using the BootstrappedTotal Least Squares estimator. The results of this chapter are publishedin [Peum 18] and [Peum 19].

At first sight, the computation of the Best Linear Approximation (BLA) is basedon a relatively straightforward principle. Divide the cross-power spectrum ofthe input and output by the auto-power spectrum of the input and you obtainthe BLA for a nonlinear period-conserving system. The problem hence seems tobe solved? Unfortunately this is not the case. When using random signals as anexcitation, there are two principal hurdles to impede this simplicity. The first oneis related to the calculation of the expected values. A good approximation candemand a large simulation/measurement time due to the large number of inputrealisations that is needed to remove variability. The second hurdle is related toleakage/transient phenomena. They introduce a bias in the estimation.

In recent years, advanced local modelling techniques have been developed tocope with these challenges. The most prominent ones are the Local Polyno-mial Method (LPM) [Pint 10b, Pint 10a] and the Local Rational Method (LRM)[McKe 12, Voor 18]. Starting from the measured noisy input-output data, thesemethods locally approximate both the BLA and the leakage term in the frequencydomain by a polynomial (LPM) or a rational function (LRM) of low order. Bydoing so, accurate estimates of the non-parametric BLA have been obtained start-ing from a reduced amount of simulation/measurement time in a myriad ofapplications [Pint 12, Geer 17, Voor 18]. For the characterisation of stronglyresonant systems, however, there is still room for improvement. The LPM isill-suited for the identification of these systems since the local polynomial modelsinsufficiently capture the resonant behaviour, unless a disproportionately largesimulation/measurement time is used [Scho 13]. The LRM significantly enhancesthe modelling capacity in these resonances by using a local rational model. This

51


enhancement does not come for free: the rational model is derived using a biasedestimator [McKe 12, Voor 18], and the extension towards Multiple-Input Multiple-Output (MIMO) systems is not trivial due to the existence of various possibleparametrisations of the transfer function matrix [Glov 74, Kail 80]. However, theability of a method to estimate an unbiased local model remains essential to gen-erate confidence bounds [Pint 10b], while the identification of strongly resonantsystems with arbitrary MIMO excitations ensures a proper excitation for all theresonances of the system [Pint 12].

To improve the bias properties and extend currently existing local modellingtechniques towards MIMO, we develop a local rational modelling technique basedon the iterative Bootstrapped Total Least Squares (BTLS) esimator. It providesan unbiased estimate of the non-parametric BLA and is applicable to large-scaleMIMO systems as well. By doing so, both the BLA and its corresponding uncer-tainty can be accurately estimated at the expense of an increased computationalcost when compared to the LPM and the LRM. In this chapter, we introducethe basic ideas of local modelling techniques. We furthermore explain how theBTLS estimator functions together with the changes that are needed for the localmodelling context. Next, we provide simulation examples. To speed things up, wepropose a numerically efficient scheme for the calculation of a large-scale MIMOsystem of order 100 with as much as 100 input and 100 outputs. Finally, weconclude with the characterisation of the resonant behaviour of the tailplane of aglider using measurements.

4.1. The basic idea

The local modelling techniques studied in this chapter all start from the samefundamental equation in the frequency domain. It describes the input-outputbehaviour of a nonlinear Periodic-In Same Period-Out (PISPO) MIMO systemapproximated by the BLA (Figure 4.1)

Y(:) =GBLA (Ω : ) U(:) +L(Ω : ) +N(:) +D(:) (4.1)

where U(:) ∈ C =D×1 and Y(:) ∈ C =H×1 are the vectors of, respectively, the stacked=D input and stacked =H output Discrete Fourier Transform (DFT) spectra1. Ω:is the generalised frequency variable. Depending on the underlying process, Ω :

1All vectors and matrices are represented by boldface symbols.

52

4.1. The basic idea

equals 9l : for continuous-time systems, 4− 9l:)B for discrete-time systems and√9l : for diffusion phenomena.

The output is perturbed by noise N(:) ∈ C=H×1 and nonlinear distortions D(:) ∈C=H×1. We assume that the noise source N(:) has the following properties

1. N(:) is zero mean and circular complex distributed, which is equivalent tostating that EN(:) = 0 and EN(:)N) (A) = 0 for ∀:,A.

2. N(:) is uncorrelated with N(A) whenever : ≠ A : EN(:)N (A) = 0.

3. N(:) is uncorrelated with the input signal U(:): EN(:)U (:) = 0.This property is automatically fulfilled if the system under study is operatingin open loop.

If the input signals have a Gaussian Probability Density Function (PDF), thenonlinear distortion D(:) has the following properties

1. D(:) has zero mean: ED(:) = 0.

2. D(:) is uncorrelated with the input: ED(:)U (:) = 0.

3. D(:) has a smooth power spectrum ED(:)D (:) as a function of : .

An independent leakage contribution L(Ω : ) ∈ C=H×1 is present due to the factthat we use a (rectangular) finite-time window during the calculation of the DFTspectra in (4.1). Leakage occurs in the calculation of the DFT of aperiodic signalsor periodic signals that are observed during a non-integer number of periods. Itcan be proven [Pint 12, Chapter 6] that GBLA (Ω : ) and L(Ω : ) are both smoothfunctions of the frequency that share the same poles in their dynamics. Theleakage L(Ω : ) in itself can be divided into two different terms

L(Ω : ) = LT (Ω : ) +L (Ω : ) (4.2)

where LT is the noise leakage and L is the combined leakage introduced by [, _and J. These leakage terms become zero if the initial and final conditions of theexperiment become the same. Since T is a random variable, the noise leakage LT

is in general not zero. L can be made equal to zero if and only if periodic inputsignals are applied that are observed during an integer number of periods, and ifthe resulting observed outputs of the system have reached steady-state.

53


Figure 4.1.: The output-error framework describes the input-output behaviour ofa nonlinear PISPO system in the frequency domain using the BLA.

Due to the smoothness of both GBLA (Ω : ) and L(Ω : ), the method requires thatU(:) varies unpredictably as a function of the frequency such that the localmodelling technique can discriminate between GBLA (Ω : ) U(:) and L(Ω : ). Thisrequirement implies a kind of ’roughness’ condition on the =D input signals.Namely, the spectral difference

|*8 (: +1) −*8 (:) | 8 = 1,2 ... =D (4.3)

should not vanish to zero for ΔΩ : → 0 where ΔΩ : = Ω :+1−Ω : . Even though thelocal modelling techniques are applicable to both stochastic and deterministicexcitations, we focus only on signals that satisfy this ’roughness’ condition. Suchsignals are among others random noise, periodic noise, random-phase multisinesand (pseudo-)random binary sequences. Another requirement for these signals isthat they need to guarantee the identifiability of each local model in the MIMOsetting. A necessary condition for this requirement is that none of the inputs*8 (:)can be fully correlated with the other inputs *; (:) at each frequency. Mathematic-ally, this requirement is satisfied when EU(:)U (:) is of full rank [Pint 12].

The basic idea of the local modelling techniques is to approximate both GBLA (Ω : )and L(Ω : ) with a local model over Ω : . Since G(Ω : ) and L(Ω : ) are presumed tobe smooth functions of Ω : , essentially any continuous function can be used forthis purpose. Popular choices which showed outstanding results in the past arepolynomial matrices (LPM) and rational matrix parametrisations (LRM) [Pint 10b,Pint 10a, McKe 12, Voor 18]. To showcase the mechanism behind these localmodelling techniques, let us assume that we want to estimate GBLA (Ω : ) andL(Ω : ) for a Single-Input Single-Output (SISO) system by using a polynomial

54

4.1. The basic idea

Ωk−20

0

20

Inp

ut|U|[

dB

]

Ωk−10

10

30

Ou

tpu

t|Y|[

dB

]Ωk

−15

5

25

∆Ω

GBLA(Ωk)

GBLA(Ωk +δr)

|GB

LA|[

dB

]

Ωk−80

−25

30

∆Ω

L(Ωk) L(Ωk +δr)

|L|[

dB

]

Figure 4.2.: Starting from simulated/measured input-output data, a local modelfor both (Ω : + XA ) and ! (Ω : + XA ) is simultaneously estimated in thefrequency window ΔΩ () around the center frequency Ω : . When themodels are evaluated at the center frequency (XA = 0), estimates forboth the BLA BLA (Ω : ) and leakage ! (Ω : ) are obtained. This localmodelling strategy is repeated for all (Ω : ).

55


model (Figure 4.2). Using the local set of frequencies Ω:+A centered around Ω:2,both the BLA and leakage term in (4.1) can be approximated by polynomialmatrices

GBLA (Ω :+A ) =GBLA (Ω : + XA ) ≈ GBLA (Ω : ) +=∑<=1

g:< X<A (4.4)

L(Ω :+A ) = L(Ω : + XA ) ≈ L(Ω : ) +=!∑<=1ℓ:< X

<A (4.5)

where A = −=ΔΩ, ...,−1,0,1, ..., =ΔΩ and XA ∈ C quantifies the local frequency vari-ation around Ω : . The number of frequency points considered for the localestimation within the local frequency window ΔΩ is then equal to 2=ΔΩ + 1.GBLA (Ω : ) ∈ C=H×=D and L(Ω : ) ∈ C=H×1 are, respectively, the local BLA and leak-age estimates while g:< ∈ C=H×=D and ℓ:< ∈ C=H×1 are the unknown polynomialcoefficients that model the dynamic variations within each local model. Sincethese dynamic variations are limited in the local modelling context, the polyno-mial orders = and =! do not exceed 4 in most use cases [Pint 12, Chapter 7].Furthermore, the same order is mostly used for GBLA and L (= = =!) such thatidentical modelling capacities are provided for both terms. The coefficients g:<and ℓ:<, together with GBLA (Ω : ) and L(Ω : ), are retrieved by substituting (4.4)and (4.5) in (4.1) and consequently solving the resulting set of linear equations. Byshifting the local window over the whole frequency range, it is possible to retrievea non-parametric estimate for GBLA (Ω : ), L(Ω : ) and the combined covariancematrix CovN+D over the complete excited bandwidth.

To improve on the modelling capabilities around resonances, one could opt to uselocal rational models instead of polynomials. Identifying these rational modelsfor MIMO systems remains challenging for two reasons. The rational functionmatrix parametrisation is not unambiguously defined [Kail 80] and the estima-tion procedure is more complex since the model is inherently nonlinear in theparameters. Many different parametrisations of the rational function matrix ex-ist, e.g. the state-space representation, left and right matrix fraction and partialfraction descriptions [Kail 80]. One criterion that plays an important role in theselection process is whether or not the original nonlinear estimation problemcan be reduced to a linear least squares one. As it turns out, the Left MatrixFraction (LMF) description is the only parametrisation that satisfies this criterionwithin the input-output setting [Pint 12, Section 6.6].

2An asymmetric window is used at the borders of the frequency band.

56

4.1. The basic idea

The LMF rewrites the rational function matrices GBLA and L as the ratio of twomatrix polynomials

GBLA (Ω : + XA ) ≈ A−1: (XA )B: (XA ) =

( =∑<=0

a:< X<A)−1 ( =∑

<=0b:< X<A

)(4.6)

L(Ω : + XA ) ≈ A−1: (XA )C: (XA ) =

( =∑<=0

a:< X<A)−1 ( =∑

<=0c:< X<A

)(4.7)

where a:< ∈ C=H×=H represents the complex matrix coefficients of the denomin-ator and b:< ∈ C=H×=D and c:< ∈ C=H×1 represent the complex matrix and vectorcoefficients of the numerator of, respectively, GBLA and L. The same denomin-ator polynomial Ak is used for both GBLA and L since both terms share commonpoles [Pint 12, Chapter 6]. Again, the polynomial orders =, = and = are mostlychosen equal to each other (with a maximum order of 4). This choice ensuresthat the local rational function does not favor the modelling of the resonance orthe anti-resonance regions [Pint 12, Section 7.2.2.2]. The rationale behind thecomplex coefficients is that the reformulation of G and L as a function of XA doesnot yield real spectra. There is no complex conjugate mirrored spectrum aroundΩ : . Substituting (4.6) and (4.7) in (4.1) and multiplying the result with A: (XA )gives the following linearised model equations for A ∈ [−=ΔΩ, =ΔΩ] ∩ Z

A: (XA )Y(: + A) = B: (XA )U(: + A) +C: (XA ) +A: (XA )N(: + A) +A: (XA )D(: + A) (4.8)

which can be easily solved with a linear least squares estimation procedure. Thisprocedure only provides a consistent estimate if the noise and the nonlineardistortion in the local frequency window originate from a white noise sourcethat is filtered by the denominator polynomial A: (XA ). In all other cases, A: (XA )wrongly shapes these terms and this results in a biased estimate of the modelcoefficients. For low signal-to-noise ratios (< 20dB), this bias becomes increasinglydominant and undermines the general applicability of the LMF parametrisationfor local modelling purposes [Geer 16].

57


4.2. Using Bootstrapped Total Least Squares forlocal modelling purposes

A wide variety of estimators has been developed in the past to tackle the estimationof a rational transfer function with the LMF parametrisation [Pint 94]. Most ofthese frequency domain estimators have been studied and successfully applied ina global identification setting. The objective then consists in capturing the wholedynamic behaviour of GBLA in one single parametric rational transfer functionmatrix. The identification of this matrix is difficult for lightly-damped mechanicalsystems but can be made possible even when excessive orders (> 20) are demandeddue to a large number of resonant modes [Bult 05].

Within the global identification setting, accurate non-parametric models arestill required for initialisation and validation purposes. Local modelling tech-niques contribute significantly here as they improve the non-parametric estima-tion. Using a linear least squares estimator within this local modelling contextseemed favorable at first: the estimator is fast, non-iterative and computation-ally efficient implementations exist [Pint 12]. Unfortunately, this estimator isalso biased and introduces an erroneous noise shaping that becomes increasinglysignificant for large-scale MIMO systems. An increasing number of excited fre-quencies is then required to identify all the local model parameters. Iterativemethods have been introduced to improve the least squares estimate with varyingprecision [Pint 98]. The most straightforward one is the Sanathanan-Koernermethod [Sana 63, Desc 06]. It overcomes the lack of sensitivity towards the low-frequency errors by iteratively reweighting the least squares solution with thedenominator polynomial that is obtained in the previous iteration. However,recent application of this method for local modelling purposes [Geer 16, Voor 18]has resulted in estimation errors that are larger than the non-iterative LRM. Thisis mainly due to convergence issues and over-fitting. These problems render thismethod ill-suited to perform a local model estimation.

More advanced techniques exist that remove the bias from the Sanathanan-Koernermethod. An alternative iterative linear least squares estimator can be used, thatis based on the instrumental variables approach [Blom 10]. While this approacheffectively removes the bias of the Sanathanan-Koerner method, it results in adecrease of the numerical conditioning when elementary polynomials as in (4.6)and (4.7) are used [Gils 13, Herp 14]. This poor conditioning can potentiallyresult in an erroneous estimate. Additionally, gradient-based methods exist thatrequire a Gauss-Newton or Levenberg-Marquardt optimisation scheme to solve

58

4.2. Using Bootstrapped Total Least Squares for local modelling purposes

the nonlinear estimation problem directly [More 78, Knoc 09]. These schemesare not recommended either in the local modelling context as they require goodinitial estimates to avoid local minima. An accurate generation of these initialvalues most often requires an additional identification step. This increases thecomplexity of the estimation procedure even further.

To overcome the downsides that are associated with the above-mentioned estima-tion techniques, we develop a technique that efficiently estimates a local rationalmodel without introducing a bias on the estimate. We retrieve this unbiasedrational model by using the BTLS estimator. It has primarily been used in the pastfor global system identification purposes [Van , Pint 98]. The BTLS estimator is aniterative estimator that produces consistent estimates during each iteration step.Iteration is used to improve the weighting and thereby the stochastic efficiency ofthe estimator. The global minimum of the BTLS cost function is obtained usingthe Generalised Singular Value Decomposition (GSVD), which is a standard linearalgebra decomposition. Asymptotically, this cost tends towards the MaximumLikelihood (ML) cost for an infinite number of iterations. Furthermore, the BTLSestimator is self-starting. When compared with gradient-based methods, thiseliminates the dependency on other estimators for initialisation purposes. In theremainder of the thesis, the name Local Bootstrapped Total Least Squares (LBTLS)is adopted to describe the application of the BTLS estimator to the local modellingcontext.

The main purpose of this section is to explain the successive steps of the algorithmand their accompanying assumptions. To do so, the model equations are con-structed first. Next, the minimisation problem is defined. Afterwards, we deriveexpressions for the weighting matrix and the column covariance matrix that areused by the LBTLS estimator to estimate GBLA and L. In a final stage, we derive theuncertainty bounds for these estimated properties given the uncertainty boundson the measurements.

59


The model equations

Multivariable systems often contain common pole dynamics which originate fromthe division by det(: ) that is needed to compute −1

:= adjugate: /det(: ), if

: is invertible. We propose to use the common-denominator parametrisation byimposing structure on a:<

a:< = 0< (:) I=H (4.9)

where 0< (:) ∈ C, and I=H ∈ C=H×=H is the identity matrix of size =H × =H . Remarkthat the structure imposed in (4.9) is entirely equivalent to the following common-denominator transfer-matrix representation. It relates every output . 9 ( 9 = 1... =H)of the MIMO system with the input*8 (8 = 1... =D) and the leakage term (for claritythe index : is omitted in the polynomial models)

. 9 (: + A) ==D∑8

98 (XA ,))(XA ,))

*8 (: + A) + 9 (XA ,))(XA ,))

(4.10)

where (XA ,)) is the common-denominator polynomial, 98 (XA ,)) is the numeratorpolynomial from input *8 to output . 9 , and 9 (XA ,)) is the numerator polynomialthat captures the leakage contribution for the output . 9 . These polynomials aredefined as

(XA ,)) ==∑<=0

0< X<A 98 (XA ,)) =

=∑<=0

198< X

<A 9 (XA ,)) =

=∑<=0

29< X

<A (4.11)

where 0<, 1 98< , 2 9< ∈ C are the unknown coefficients that need to be estimated. Thevector ) ∈ C [ ( (=+1) =D+=+1) =H+=+1]×1 represents the stacked vector of all unknowncoefficients. It can be constructed by combining the coefficients as follows

) =

©«

),.1

),.1

),.2

),.2...

),.=H),.=H0001...

0=

ª®®®®®®®®®®®®®®®®®®®®®¬

∈ C=\×1 with

),.9 = vec©«1910 1

920 . . . 1

9=D0

1911 1

921 . . . 1

9=D1

...... . . .

...

191= 1

92= . . . 1

9=D=

ª®®®®®¬

),.9 =

©«29

029

1...

29=

ª®®®®®¬

(4.12)

60


where vec• is the vector operator that stacks all the columns of a matrix on topof each other and =\ is the total number of coefficients. Multiplying (4.10) with(XA ,)) and substituting the polynomials by their description shown in (4.12), theoriginal set of model equations can be restructured to a matrix format. For everyfrequency Ω :+A that is present in the local window, the complex error betweenmeasurement and model becomes

e(Ω :+A ,)) = J(Ω :+A ) ) = [J0 (Ω :+A ) +J(Ω :+A )] ) ≈ 0 (4.13)

where J is the stochastic part of J induced by the additive noise and nonlineardistortion that perturbs the noiseless part J0. The Jacobian matrix J(Ω :+A ) has thefollowing structure

J(Ω:+A ) =

©«U) ⊗P= P= −.1P=

U) ⊗P= P= −.2P=. . .

. . ....

U) ⊗P= P= −.=HP=

ª®®®®®¬) ≈ 0

(4.14)⊗ is the Kronecker product [Brew 78], •) is the transpose operator, and P= =(1 XA X2

A . . . X=A

). Adopting the proposed common denominator structure

not only results in a parametrisation with less parameters than the general LMFdescription (4.6) and (4.7). It also grants us the possibility to introduce a sparsestructure in e(Ω :+A ,)) for MIMO systems. This sparse representation facilitatesthe time-efficient computation of the LBTLS estimator (Section 4.5).

Example Consider a 2×2 MIMO system with = = = = = = 1. The Jacobianmatrix J(Ω :+A ) then has the following structure

( 11︷︸︸︷*1 *1 XA

12︷︸︸︷*2 *2 XA

1︷︸︸︷1 XA 0 0 0 0 0 0

−.1︷︸︸︷−.1 −.1 XA

0 0 0 0 0 0 ︸︷︷︸21

*1 *1 XA ︸︷︷︸22

*2 *2 XA ︸︷︷︸2

1 XA ︸︷︷︸−.2

−.2 −.2 XA

)

61


Parameter estimation

At its core, the LBTLS estimator uses the Weighted Generalised Total LeastSquares (WGTLS) method to estimate the local rational model [Pint 98]. Thismethod is a natural extension of the least squares estimation in the sense thaterrors on both the dependent and independent variables are taken into account. Ityields a consistent estimate provided that the input-output covariances are exactlyknown or consistently estimated [Van 91]. One of the advantages of the WGTLSis that it uses the linearised model equations of (4.10). It thus results in the sameJacobian matrix (4.14). Starting from J in (4.10), the WGTLS estimates the localrational model by solving the following minimisation problem

argminJ

| |W (J− J)C−1 | |

subject to J) = 0 and ) ) = 1(4.15)

where | | • | | is the Frobenius norm, W ∈ C [=H (2=ΔΩ+1) ]×[=H (2=ΔΩ+1) ] is a left weightingmatrix, and C ∈ C=\×=\ is a square root of the column covariance matrix C, ofthe matrix pair WJ. The solution ) can be efficiently calculated by the GSVDof the matrix pair (WJ, C). This is equivalent to minimising the cost function(4.15) [Paig 86, Pint 98]. Remark that the application of the GSVD does not requirethe computation of the inverse of C. This automatically implies that the estimationcan be performed even when C is singular. It is crucial that C is proportional tothe square root of the true (or consistently estimated) column covariance matrix

C = TC1/2,

such that C C = C, (4.16)

where T is an arbitrary orthogonal matrix. This weighting results in an exact com-pensation for the noise and distortion shaping that is present in WJ. Consequently,a consistent estimate is obtained [Pint 98].

One of the main differences between the WGTLS method and the linear leastsquares estimation lies in the manner used to remove the parameter ambiguity inthe matrix. Typically, this parameter ambiguity is resolved by fixing one of thecoefficients, mostly a:0, to the identity matrix. Unfortunately, this constraint canlead to an ill-conditioned set of equations. This is especially true if the value ofthe fixed parameter is close to zero in reality. The WGTLS method circumventsthis issue by constraining the Euclidian norm of the parameters to one () ) = 1).Every parameter can then be taken into account during the estimation procedurewhile keeping the numerical dynamics reasonable.

62


In general, the choice of the weighting matrix W does not affect the consistency ofthe WGTLS method. It mainly modifies the stochastic efficiency of the estimator.Similar to what is done for the Sanathanan-Koerner method, the LBTLS estimatorimproves the efficiency by iteratively updating W with an improved approxima-tion of the ML weighting [Pint 12, Section 9.12.3]. The improved estimate of theweighting is obtained at each iteration and it results in a progressively improvingestimate of the model parameters. This iterative scheme brings the efficiency ofthe LBTLS estimator close to the ML efficiency. While doing so, it maintains thefavorable global minimisation properties of the WGTLS method [Pint 12]. Themain advantage of the LBTLS estimator is that consistency is guaranteed at eachiteration step while this is not the case for the Sanathanan-Koerner method. Thisconsistency preserving property does not only ensure that the iteration can bestopped at any time, but also allows to limit the number of iterations. All simula-tion cases and practical applications considered here suggest that 5 iterations aresufficient to get very close to the ML efficiency.

Derivation of the left weighting matrix and columncovariance matrix

Originally, the BTLS estimator was developed in a global identification context.The noise and the distortion were imposed on the input U and the output Y.Although the local modelling context only considers the output-error framework,this section considers a complete errors-in-variables framework. More generalexpressions for both W and CC can therefore be obtained.

During each iteration of the LBTLS estimator, the weighting W is adapted to ap-proximate the ML weighting increasingly better [Bult 05]. In the multi-dimensionalcase, it is impossible to apply the full ML weighting without violating the linearitycondition of the parameters [Pint 98]. By adopting the common denominatorstructure, however, a frequency-dependent scalar weighting function F 9 (X) canbe constructed for each output . 9 that uses the covariance matrix of the inputC* (:) ∈ C=D×=D , the covariance matrix of the output C. (:) ∈ C=H×=H , and thecross-covariance between the input and output C.* ∈ C=H×=D

63


F 9 (XA , )) ==D∑8=1

98 (XA , ))C[8, 8 ]*(: + A) 98 (XA , ))

+=D∑8=1

=D∑;=8+1

2Real(9; (XA , ))C

[;, 8 ]*(: + A) 98 (XA , ))

)−=D∑8=1

2Real( (XA , ))C[ 9 , 8 ].*

(: + A) 98 (XA , )))

+ (XA , ))C[ 9 , 9 ].(: + A) (XA , ))

(4.17)

where ) is the parameter vector that is obtained in the previous iteration, and•[U, V ] is the operator that selects the element belonging to row U and column V ofthe corresponding matrix. Starting values for the iterative procedure are obtainedby setting the weighting F 9 (XA , )) = 1. Observe that F 9 (XA , )) does not include thecovariance contributions between the different outputs. This is inherent to theproposed common denominator structure in (4.10). It results in a loss of stochasticefficiency when compared to the ML efficiency.

The weighting matrix W(XA , )) is constructed in such a way that WJ accommodatesthe proposed scalar weighting function F 9 (XA , )) in the following manner

WJ =©«√F−1

1 U) ⊗P=√F−1

1 P= −√F−1

1 .1P=. . .

. . ....√

F−1=HU) ⊗P=

√F−1=HP= −

√F−1=H.=HP=

ª®®®®¬(4.18)

Starting from the constructed WJ derived earlier in (4.18), it is possible to de-rive an explicit expression for the column covariance matrix C, . The columncovariance matrix evaluated at Ω :+A is defined as

C, (Ω :+A , )) = CC = E(WΔJ) (WΔJ) (4.19)

where WΔJ is the part of WJ that exclusively depends on the additive noise andthe nonlinear distortion that is present at the input and output vectors.

64


It has the following structure

WΔJ =©«√F−1

1 V)*⊗P= 0 −

√F−1

1 +.1P=. . .

. . ....√

F−1=HV)* ⊗P= 0 −

√F−1=H+.=HP=

ª®®®®¬(4.20)

where V* =N* +D* and V. =N. +D. . Combining (4.18) and (4.20) results in thefollowing expression for CC

C, (Ω:+A , )) =

©«|F1 |−1Λ −|F1 |−1Φ1

. . ....

|F=H |−1Λ −|F=H |−1Φ=H−|F1 |−1Φ

1 . . . −|F=H |−1Φ=H|F1 |−1Σ1 + · · · + |F=H |−1Σ=H

ª®®®®®¬(4.21)

with

Λ = E((V)* (: + A)) ⊗ (P= )

).

(V)* (: + A) ⊗P=

)= E

(V)* (: + A))V)* (: + A)

⊗

((P= )P=

)(4.22)

= C)* (: + A) ⊗((P= )P=

)and

Φ 9 = (C[ 9 , :].*(: + A))) ⊗

((P= )P=

)(4.23)

Σ 9 = C[ 9 , 9 ].(: + A)

((P=)P=

)(4.24)

To obtain an expression for the covariance as a function of the noise covariances,we applied in (4.22)-(4.24) the mixed-product property of the kronecker product(d ⊗f).(g ⊗ h) = (dg) ⊗ (fh). •[ 9 , :] is the operator that selects the 9 th row from amatrix.

To ensure both the consistency and the stochastic efficiency, the construction ofF 9 (X, )) and C, requires a consistent estimate of the true covariances C* , C.and C.* . Unfortunately, information about these quantities is not available apriori. This can undermine the consistency of the LBTLS estimator. The lack ofinformation is mitigated by considering the following two aspects of the originalproblem formulation:

65


1. In the output-error framework, C* and C.* are both equal to zero as theinput is noise-free.

2. Following the lines of the LPM and the LRM, it also makes sense to assumehere that the disturbing output noise and the nonlinear distortion are whitein every small local frequency window. This assumption simplifies theoutput covariance matrix to

C[ 9 , 9 ].(: + A) = f2

9 and C[8, 9 ].(: + A) = 0 for 8 ≠ 9 (4.25)

where f29

is the unknown power of the white noise source that perturbsoutput 9 . On the condition that the noise and nonlinear distortions inthe local window can be well approximated by this assumption, the LBTLSestimator remains unbiased (Appendix B). Additionally, the exact knowledgeof f2

9is not necessary since it acts as a scaling factor. While this changes the

interpretation of the cost function value (Appendix B), it does not have aninfluence on the retrieved parameter solution.

Transformation from the errors-in-variables framework to the output-error frame-work (C* = C.* = 0) greatly shortens the expressions which are eventually to beused by the LBTLS estimator to derive consistent local model parameters. First ofall, the weighting F 9 (XA ,)) in (4.17) only has to compensate for the output noiseshaping that is induced by the common-denominator part (XA ,)). Secondly, Λand Φ 9 become zero and the column covariance matrix in (4.21) is simplified sinceit only considers noise on the common-denominator coefficients.

Calculation of the uncertainty bounds on the parameters

Aside from providing a consistent estimate of the BLA and the leakage term, theLBTLS estimator provides the uncertainty bounds on the estimated propertieswithout imposing strong requirements on the Signal-to-Noise Ratio (SNR). Theclassical LRM, which uses the least-squares estimation procedure, performs a lin-earisation to approximate the output covariance matrix [Voor 18]. A minimal SNRof 20dB is then recommended to make the bias error sufficiently small [Geer 16].Earlier conducted research revealed that this requirement can be softened forthe ML estimator in the errors-in-variables framework [Pint 07]. Only in fre-quency regions where the input and output SNR are both very low (smaller than3dB), a deviation between the sample ML estimator and the Cramér-Rao lowerbound (smallest possible stochastic uncertainty for an unbiased estimator) can

66


be observed. This situation hardly occurs in most applications. Since the LBTLSestimator belongs to the class of approximate ML estimators, a SNR of at least 3dBfor both the input and output in the same frequency region is a strict minimum toguarantee a correct estimation of the uncertainty bounds.

The uncertainty bounds on the estimated parameters can be calculated startingfrom the equivalent cost function +LBTLS of the parameters estimation problem in(4.15)

+LBTLS () ,[,_) ==ΔΩ∑

A=−=ΔΩ9 (Ω :+A ,))C−1

Y ()) 9(Ω :+A ,))

=

=ΔΩ∑A=−=ΔΩ

e (Ω :+A ,))W (XA , ))C−1

Y ())W(XA , )) e(Ω :+A ,))

= ( ())I−1[ ())(())

= 1 ()) 1())

(4.26)

subject to ) ) = 1. CY ()) can be further expanded in terms of C,

CY ()) ==ΔΩ∑;==ΔΩ

) C, (Ω :+; , )) ) (4.27)

) is the parameter vector that is obtained in the last iteration of the LBTLSestimator. The variables (()), I[ ()), and 1()) are defined as

( = [9) (Ω :−=ΔΩ ) . . . 9) (Ω :+=ΔΩ )]) (4.28)

I[ = Block diagonal(IY . . . IY) (4.29)

1(\) = I−1/2[ ())(()) (4.30)

The last identity of (4.26) shows that the LBTLS can be recast in the classical formof a nonlinear least-squares problem. In general, the covariance matrix of ) forsuch a nonlinear least-squares problem is equal to [Pint 12, Section 9.7]

Cov) =

[+ ′′LBTLS () ,[0,_0)]−1E[+ ′LBTLS () ,[,_)]

+ ′LBTLS () ,[,_)[+ ′′LBTLS () ,[0,_0)]−1

(4.31)

67


where •′ and •′′ are, respectively, the first- and second-order derivative of +LBTLS

towards the parameter vector ). [0 and _0 are the noise- and distortion-freespectra of, respectively, the input and the output. Unfortunately, (4.31) cannot becomputed explicitly for the LBTLS estimator.

To enable the computation of the covariance matrix of the estimated LBTLSparameters, we introduce an approximation. Since the LBTLS estimate is mostlyclose to the optimal ML solution, the covariance matrix (4.31) is approximated bythe following expression [Pint 12, Section 9.11]

Cov) ≈ [+ ′′LBTLS () ,[0,_0)]−1 = [ ())())]−1 (4.32)

where ()) = m1/m) ()) is the Jacobian matrix of (4.26) evaluated in ). Thisapproximation assumes that a sufficient SNR (> 6dB) is available for the measuredoutput [Pint 12, Section 12.3.4]. Furthermore, we assume that the model orders=, = and = are chosen large enough such that the model errors are smaller thanthe noise contribution.

The covariance matrix of the estimated parameters requires the computation ofthe Jacobian matrix . The analytic calculation is impossible, since it requiresto compute the partial derivative of I−1/2

[ ()) to the parameter vector ). Analternative is to approximate numerically using a finite-difference approach.However, this approach is not always very accurate and can be a time-consumingprocess. To cope with the above-mentioned issues, Guillaume et al. proposed tosubstitute the original Jacobian matrix with a so-called pseudo-Jacobian matrix+ [Guil 96]

[:, 8 ]+ (Ω :+A , )) = CY ())−1/2

(m 9(Ω :+A , ))

m ) [8 ]−

12mCY ())m ) [8 ]

CY ())−1 9(Ω :+A , )))

(4.33)

This pseudo-Jacobian matrix has two distinct advantages. First, only the deriv-atives of 9 and IY to the parameter ) occur. Since both of these quantities areanalytic functions of ), these derivatives can be easily computed without the needfor a finite-difference approximation. Second, it has been proven that and +yield the same stationary point and thus the same minimum of (4.26). Due tothese advantages, we use the pseudo-Jacobian matrix + for the computation ofthe covariance matrix of the LBTLS estimator in (4.32).

68

4.3. A first simulation example: the SISO case

Starting from the calculated Cov), it is now possible to obtain the correspondinguncertainty on the BLA estimate. This derivation starts from a linearisationof the parametric BLA GBLA (Ω :+A , )) around its expected value obtained at theparameter values \ = E)

GBLA (Ω :+A , )) ≈GBLA (Ω :+A ,E)) +mGBLA (Ω :+A ,))

m )

E) () −E)) (4.34)

By applying the vec• operator on GBLA and by combining it with the followingproperty of the Kronecker product: vecdf g = (g) ⊗ d) vecf, an expression forthe covariance matrix of the vectorised BLA matrix is obtained as follows

CovvecGBLA (Ω :+A , )) ≈(I=\ ⊗

mGBLA (Ω :+A ,))m )

)

)Cov)

(I=\ ⊗

mGBLA (Ω :+A ,))m )

)

)(4.35)

We additionally have approximated the unknown expected value of the parametersE) with the estimated parameters ).

The variances of GBLA [ 9 , 8 ] can subsequently be derived from (4.35) by merely con-sidering the diagonal values of CovvecGBLA (Ω :+A , )). These variances are usedlater in this chapter to visualise the uncertainty bounds on the estimated GBLA.One has to be aware that they do not take into account the correlation between thedifferent transfer functions that is described in CovvecGBLA (Ω :+A , )) by theoff-diagonal elements. Therefore, these (diagonal) variances potentially representan over- or underestimation of the actual multidimensional uncertainty volume.


With this first example, we want to illustrate the performance of the LPM, theLRM, the Vector Fitting (VF), and the LBTLS methods on an elementary SISOsystem. The VF technique is a reformulation of the iterative Sanathanan-Koernermethod that employs rational basis functions instead of polynomials as weightingfunctions [Gust 99, Hend 06]. To provide a fair comparison, we had to translatethe original VF technique from the transfer function level to the input-outputlevel (Appendix C).

69


Figure 4.3.: The open-loop simulation configuration excited the linear dynamicsystem ( 9l) with a known random-phase multisine input * (:)and afterwards adds filtered (band-limited) Gaussian white noise# (:) = ( 9l) (:) to retrieve the noisy output . (:).

We construct a continuous-time (Ω = 9l) open-loop system (Figure 4.3) that pos-sesses the following dynamic responses for and

( 9l) =0.1 (( 9l)2 +0.1 9l+625)( 9l)2 +0.1 9l+100

( 9l) =39.5 ( 9l)2

(( 9l)2 +3.8 9l+177.1) (( 9l))2 +5.1 9l+316.8)

(4.36)

The input * (:) is chosen to be a full flat random-phase multisine with a RootMean Square (RMS) value of 1+RMS. 1000 spectral lines are excited in a bandwidthranging from 0.005Hz to 5Hz. The simulation calculates the output response. (:) directly in the frequency domain which eliminates the leakage contribution! ( 9l : ). We can therefore focus entirely on the modelling capacities of the differentmethods for the transfer function BLA ( 9l : ) and the output noise variance f2

#(:).

The noise signal (:) originates from a zero-mean complex circular Gaussianwhite noise source with a RMS value of f = 0.01+RMS. The results of 1000 Monte-Carlo runs are averaged to deduce the general behaviour of all the methods. Theestimates are obtained with =ΔΩ = 5, = = 4, =? = 1, and = = = = 2.

A visual analysis of the bias on the averaged estimated transfer function ( 9l : )(Figure 4.4) proved to be difficult. The LBTLS estimator, the LRM, and theVF technique all provide a good fit. The LPM, however, is characterised by asubstantial approximation error in the vicinity of the strong resonance that ispresent at 1.6Hz. This discrepancy is caused by the lack of flexibility of the

70


0 1 2 3 4

40

−40

0

−80

GLBTLS

|G− GLBTLS|

Am

pli

tud

e[d

B]

1 2 3 4 5

40

−40

0

−80

GLPM

|G− GLPM|

Am

pli

tud

e[d

B]

0 1 2 3 4

40

−40

0

−80

GLRM

|G− GLRM|

Am

pli

tud

e[d

B]

1 2 3 4 5

40

−40

0

−80

GVF

|G− GVF|

Am

pli

tud

e[d

B]

Frequency [Hz]

Figure 4.4.: The estimated transfer functions are averaged over 1000 Monte-Carloruns on the measured output spectrum. The results are comparedwith the true ( 9l : ) (−).

Frequency

σ2 N

Ωk

|G(Ωk +δr)|

|G(Ωk)|

Figure 4.5.: Applying LBTLS on a single Monte-Carlo experiment results in out-liers for the noise variance estimate f2

#due to the presence of pole-

zero cancellations in the local model (Ω : + XA ).

71


polynomial model. It insufficiently captures the strongly resonant behaviour of ( 9l).

To better verify the modelling capacity of the different methods, the estimatednoise variance f2

#is compared with the theoretically applied noise variance

| ( 9l : ) |2f2

. As it turns out, the LBTLS estimator is plagued by outliers in thenoise variance estimate (Figure 4.5) due to pole-zero cancellations that are presentin the local model (Ω : + XA ). These cancellations do not influence the quality ofthe transfer function estimate. The noise variance estimate, however, depends onthe residual error over the whole local frequency range and therefore is negativelyimpacted by them.

To avoid the pole-zero cancellations, we propose to apply a (simple) model orderselection procedure during the local model estimation. By choosing the localmodel order as low as possible, while keeping the model accurate, the occurenceof pole-zero cancellations is reduced to a minimum. Model selection is performedwith the Akaike Information Criterion (AIC) to deduce the ’correct’ model or-der [Akai 74]. The AIC selects the ’best’ model among a group of candidatemodels. This model describes the local system dynamics accurately. At the sametime it remains parsimonious in the number of parameters because a penalty isadded for each parameter added to the model. The group of candidate models (contains all possible combinations of model orders limited to the maximum order= , =?, =, and = . For each possible model order belonging to (, the correspond-ing local model is derived using each of the local estimators. We retain the modelthat minimises the AIC cost function [Ljun 98]

+AIC (U, V, W) =+LM

(1+

=\

2=ΔΩ +1

)with +LM =

=ΔΩ∑A=−=ΔΩ

|. (: + A) − . (: + A) |2(4.37)

where . (:) is the predicted output generated with the estimated local model, and=\ is the number of unknown complex parameters.

Combining the local modelling techniques with the AIC procedure enables anequitable comparison between the noise variance estimates f2

#and the true noise

variance | ( 9l: ) |2f2

(Figure 4.6). The following observations can be made

1. The 68% confidence interval of the LBTLS estimator is centered around thetrue noise variance. This indicates that this estimator is unbiased over thewhole frequency range.

72


1 2 3 4

−70

−80

−90

−100

−110

LBTLS

|σ2 N|[

dBV

2 ]

2 3 4 5

−70

−80

−90

−100

−110

LPM

|σ2 N|[

dBV

2 ]

1 2 3 4

−70

−80

−90

−100

−110

LRM

|σ2 N|[

dBV

2 ]

2 3 4 5

−70

−80

−90

−100

−110

VF

|σ2 N|[

dBV

2 ]

Frequency [Hz]

Figure 4.6.: The shaded areas represent the 68% confidence interval of the estim-ated noise variances obtained by averaging 1000 Monte-Carlo runs.The results are compared with the true noise variance (−).

2. For the LPM this unbiasedness disappears in the vicinity of the resonancedue to modelling errors.

3. The LRM resolves this modelling issue partly. It visibly introduces a bias inthe estimated noise variance at higher frequencies and still shows peakingaround the resonance.

4. VF removes the peaking phenomenon around the resonance, but maintainsa bias at higher frequencies.

From these observations it can be concluded that LBTLS is the estimator thatoverall shows the best performance in retrieving an unbiased estimate of both ( 9l : ) and the noise variance f2

#. This unbiased behaviour is conditioned to

the absence of pole-zero cancellations. They need to be prevented by applying amodel order selection procedure.

73


4.4. Simulation example: the MIMO case

This section takes a closer look at two different properties of the LBTLS estimatorfor a generic lightly-damped MIMO system. First, the influence of iteration on theLBTLS estimates is studied. Secondly, the results are compared with existing localmodelling techniques [Pint 10b, Voor 18]. Three different estimators are used inthis comparison: the LPM, the iterative LRM employing the Sanathanan-Koernermethod, and the iterative LBTLS estimator. In the remainder of this thesis, thenumber indicated next to the methods designates the number of iterations usedduring the estimation.

The simulated system possesses the dynamic behaviour of a lightly dampedmechanical system with the following model structure

G[ 9 , 8 ] ( 9l : ) ==?∑==1

?2=

r [ 9 , 8 ]=(9l : − ?=

) + (?2=)

(r [ 9 , 8 ]= )(9l : − ?=

) (4.38)

where =? is the number of natural modes of the system, ?= are the pole locationsand A= ∈ C=H×=D are the residue matrices of the poles. All the poles have the samerelative damping equal to 1.2510−3 and are equally spaced in the simulated fre-quency window. Observe that the residue matrix of each mode is multiplied with?2= or (?2

=) such that a high dynamic range is obtained. The residue matrices arechosen in a random way to ensure that each natural mode is expressed differentlyfor all input-output combinations. To not overcomplicate the simulation example,the leakage contribution L is not included in the model. The output noise perturb-ation N(:) originates from a zero-mean complex circular Gaussian white noisesource. It is added in the frequency domain in such a way that a SNR of 40dB isobtained at each frequency. In this simulation example, we consider a system with2 inputs, 3 outputs and =? = 9 natural modes (Figure 4.7).

Influence of iterating with the LBTLS estimator

To evaluate the impact of the number of iterations, we compare the LBTLS-5estimate with the LBTLS-1 estimate using the initial values only. For each input*8 (:), we use a complex Gaussian white noise excitation with a RMS value of1+RMS in a bandwidth of 1kHz to excite the system. G is estimated in the frequencyband [5Hz, 1000Hz] from the input-output data using the LBTLS estimator with= = = = 1. Visual inspection of the results with = = = = 2, shows that unwanted

74


0 0.5

75

100

125

150

|G[1,1] |

[dB

]

0.5 1

75

100

125

150

|G[1,2] |

[dB

]

0 0.5

75

100

125

150

|G[2,1] |

[dB

]

0.5 1

75

100

125

150

|G[2,2] |

[dB

]

0 0.5

75

100

125

150

|G[3,1] |

[dB

]

0.5 1

75

100

125

150

|G[3,2] |

[dB

]

Frequency [kHz]

Figure 4.7.: Magnitude plot of the true system matrix.

75


0.0 0.5 1.0

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Entire range

0.55 0.60 0.65

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Zoomed-in

Figure 4.8.: Applying the proposed LBTLS estimator to a system with 2 inputsand 3 outputs for a varying iteration count reveals that the largestgain in accuracy by iteration occurs in the boundary regions. TheR2MSE of the estimated G[1, 2] with LBTLS-1 (light blue +) and LBTLS-5 (dark blue ×) is compared with the reference standard deviation ofthe transfer function (grey −).

pole-zero cancellations occur in the regions with a small dynamic variation. Thisin turn increases the modelling error. The width =ΔΩ of the frequency windowis chosen automatically by the algorithm based on the number of Degrees OfFreedom (DOF) @ of the residual matrix as follows [Pint 12, Section 7.2.2.2]

@ = 2=ΔΩ +1−=\ =⇒ =ΔΩ = ceil @ +=\ −1

2

(4.39)

In this simulation example, @ is chosen fixed and equal to 8+=H . Fixing @ ensuresthat the obtained non-parametric estimates can be straightforwardly used forfurther global identification purposes [Pint 12, Section 12.3.2]. Otherwise, agood bookkeeping of the varying DOF should be kept to be able to calculate theasymptotic properties of these global estimators.

To assess the goodness of fit of the estimated model, we compare the referencestandard deviation derived from the exact C. (:) with the Relative Root MeanSquare Error (R2MSE) of the estimated values, which is defined as

R2MSE(M [ 9 , 8 ] ( 9l: )

)=

√√√√1=exp

=exp∑;=1

M; [ 9 , 8 ] ( 9l: ) −M [ 9 , 8 ] ( 9l: )M [ 9 , 8 ] ( 9l: )

2 (4.40)

76


where G; ( 9l : ) is the acquired estimate for the ;th Monte-Carlo run on the noisyoutput spectrum. In this example, =exp = 100 Monte-Carlo realisations are per-formed. The reference expected value of the standard deviation is computedusing the noiseless regressor matrix of the LPM and the exactly known outputnoise covariance matrix C. , which is possible in this simulation context [Pint 12,Section 7.2.2.4]. For clarity, only one input-output combination is shown. It isselected randomly from G (Figure 4.8). A good fit is characterised by a R2MSE thatapproximates the reference standard deviation well. Remark that this standard de-viation of G[1, 2] contains resonances at 230Hz, 510Hz, 633Hz and 749Hz. Thesecorrespond exactly to the anti-resonances of G[1, 2] (Figure 4.7). These resonancesare an artefact of the division of the standard deviation by G[1, 2] . Due to the lowermagnitude of G[1, 2] around the anti-resonance frequencies, potentially misleadingresonances are created. Fortunately, these artefacts do not affect the interpretationof the results. Comparing the R2MSE of the estimated G over the different Monte-Carlo runs reveals that the LBTLS estimator, even without any iteration, providesan accurate estimate of the parameters \. Nevertheless, iterating results in animprovement of the accuracy. This accuracy improvement is especially visiblenear the boundaries of the considered frequency spectrum.

Comparison with the LPM and the LRM

We use the same simulation settings as in the previous section to analyse andcompare the performance of both the LPM [Pint 10b] and the LRM [Voor 18] withthe earlier obtained LBTLS estimate (Figure 4.9). In the case of the LPM a 4th

order polynomial is used with @ = 8+=H . The same properties are used both for theLRM and the LBTLS estimator. Furthermore, we apply the Sanathanan-Koernermethod with 5 iterations to the LRM to obtain a fair comparison with the iterativeLBTLS estimator.

As could be expected, the LPM provides an accurate estimation of G in the anti-resonances and in the transition regions in between these anti-resonances and theresonances (Figure 4.9a). However, a poor quality estimate is obtained for thepolynomial method in the resonance regions due to the limited modelling capacityof the polynomial model. The inspection of G[1, 2] reveals that the number offrequency bins that is present in the 3dB bandwidth of the resonances rangesfrom 1 to 11. The LPM requires at least 7 frequency bins in a resonance to ensurethat the approximation error remains sufficiently low [Scho 13]. The LRM-5(Figure 4.9b) improves the modelling of the resonances significantly to reach aperformance that is comparable to the LBTLS-5 estimator.

77


0.0 0.5 1.0

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Entire range

0.55 0.60 0.65

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Zoomed-in

(a) The Local Polynomial Method.

0.0 0.5 1.0

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Entire range

0.55 0.60 0.65

−20

−40

−60

Frequency [kHz]

R2 M

SEG

[1,2] [dB]

Zoomed-in

(b) The Local Rational Method.

Figure 4.9.: The R2MSE of the estimated G[1, 2] using the LPM (red +) and theLRM-1 (yellow +) is compared with LBTLS-5 (blue ×). Additionally,the reference standard deviation is shown (grey −).

To further evaluate the performance of the different methods over the wholefrequency range, the values of the following cost function are compared for thedifferent estimators (Table 4.1)

+cost =1

∑:=1

vecG( 9l : ) −G( 9l : ) C−1vecG vecG( 9l : ) −G( 9l : ) (4.41)

where is the total number of excited frequencies in the input signal, G is themean of G; taken over the Monte-Carlo runs and CvecG is the reference covariancematrix of the stacked G matrix. From Table 4.1, we conclude that the LPM is not

78


LPM LRM-1 LRM-5 LBTLS-1 LBTLS-5

+cost 2.7143 0.71 1.12 1.06 0.64=\ 5 4 4 4 4

Table 4.1.: Cost function comparison for the MIMO simulation example.

0.0 0.5 1.0

0

−20

−40

−60

Frequency [kHz]

STD

G[1,2] [dB]

Entire range

0.55 0.60 0.65

0

−20

−40

−60

Frequency [kHz]

STD

G[1,2] [dB]

Zoomed-in

Figure 4.10.: The STD of the estimated G[1, 2] using the LPM (red +) is comparedwith the LBTLS-5 estimator (blue ×). The reference standard devi-ation for one realisation is also depicted (grey −).

adapted to model resonant systems. A high value of the cost function is obtainedin that case (1000 times more) which results in a variability that is 30 times larger.The LRM-1 and LBTLS-5 have a comparable performance to model these resonantsystems. Application of the LBTLS-5 estimator results in the lowest value of +cost.Remark that iterating with the LRM increases the value of the cost function as hasbeen reported earlier in [Geer 16].

Another important feature of the LBTLS estimator is the capability to accuratelyestimate uncertainty bounds on the acquired G. To verify that the derivationof uncertainty bounds is indeed correct (Section 4.2), we compare the STandardDeviation (STD) obtained with the LPM and the LBTLS-5 estimator for one singleMonte-Carlo run (Figure 4.10). We observe that the estimated standard deviationoverlaps well in the regions where the LPM provides a reliable fit. In the resonanceregions, LBTLS-5 lowers the modelling error and thus results in an uncertaintybound estimation that is closer to the reference standard deviation.

79


Figure 4.11.: Dividing the QR decomposition of WJ into several QR decompos-itions for each output . 9 greatly reduces both the computationalcomplexity and memory usage.

4.5. Numerically efficient computation

One of the original aspirations when developing the LBTLS estimator was toensure that it can be applied to large-scale MIMO systems. However, we rapidlycame up against the limits of the current implementation of the estimator. Afterall, the estimated parameters are obtained by the calculation of the GSVD of thematrix pair (WJ, C) [Paig 86, Pint 98]. This GSVD algorithm starts with a QRfactorisation of WJ. The computation of the upper triangular matrix R, is themost important one for the estimation of the parameter vector ) since

WJ) =QR, ) ≈ 0 =⇒ R, ) ≈ 0 (4.42)

Unfortunately, this QR factorisation becomes computationally expensive whenevera large amount of inputs and outputs is considered.

We propose to explicitly take advantage of the structure of the matrix WJ to coun-teract the dimensional explosion (Figure 4.11). Instead of computing one singleQR decomposition of the whole matrix WJ, as done in a generic GSVD implement-ation, we propose to calculate a separate (and smaller) QR decomposition for eachoutput . 9 independently. Each upper triangular matrix R 9 belonging to output

80


. 9 can be divided into three parts. Each part describes a different projection.The numerator coefficients are projected in the top triangle ( ), the denominatorcoefficients in the bottom triangle ( ), and the mapping between these numeratorand denominator coefficients in the square (). Since the numerator coefficients ofthe different outputs are modelled independently, and can be used withoutfurther modifications for the construction of R, . Only for the denominatorpart, one additional QR decomposition needs to be performed on the stackedlower parts to compute the final projection Rden belonging to the denominatorcoefficients.

To analyse the computational complexity of the proposed method, the numberof floating point operations (flops) needed to compute R, is estimated. Thecomputational complexity of the QR decomposition is characterised 2<=2 flopsfor a <×=-matrix [Golu 12]. Straightforward application of the QR decompositionon WJ would therefore result in

2=H (2=ΔΩ +1)[(=+1) + [(= +1) =D + (= +1)] =H

]2=$

(=H =ΔΩ =

2D =

2H

)flops (4.43)

while our proposed method results in

2=H (2=ΔΩ +1)[(=+1) + (= +1) =D + (= +1)

]2=$

(=H =ΔΩ =

2D

)flops (4.44)

Analysing (4.43) and (4.44) shows that the proposed method effectively deflatesthe computational complexity of the QR decomposition from a cubic dependenceon =H to a linear one.

Remark that the dependency of the number of flops on the window width 2=ΔΩ +1holds an additional dependency on =D and =H . After all, =ΔΩ is chosen relativeto =\ and the DOF with (4.39). Since =\ = (=+1) + [(= +1) =D + (= +1)] =H , thismeans that the dependency of (4.43) and (4.44) on =D and =H increases with one.Therefore, the original computational complexity of the QR decomposition is notcubic with respect to =H but quartic. As a consequence, the proposed methodconverts this quartic dependency on =H to a quadratic one. The dependency on =Dbecomes cubic and is not affected by the method.

If no noise is present at the input (as in the output-error framework) then a secondstep can be applied to improve the numerical efficiency even further. In that case,the matrices Λ and Φ 9 of the column covariance matrix C, in (4.22)-(4.24) arezero. Consequently only a GSVD of the common denominator part is necessarysuch that the dimensional explosion can be further reduced. The projection of

81


the noiseless part of the matrix WJ onto its null space is already performed oncethe QR decomposition of WJ is calculated. Therefore the block structure of theresulting R, matrix is used to increase the efficiency of the computation withrespect to the number of outputs [Van 91]. The original model equations (4.14)are rewritten using the QR decomposition of WJ as follows

WJ) = W(Xnum Xmap

0 Xden +ΔXden

) ()num

)den

)= 0 (4.45)

Rnum and Rmap are, respectively, the numerator and mapping part of R, asshown in Figure 4.11. )num and )den represent the numerator and denominatorcoefficients. The input is not perturbed with noise in this case. Hence, onlythe Rden part that corresponds to the denominator coefficients is affected by theoutput noise through ΔRden. The GSVD is applied to the matrix pair (Rden+ΔRden,=H∑9=1|F 9 |−1Σ 9 ) only. This retrieves an estimate for the denominator coefficients )den.

Afterwards, the numerator coefficients )den are obtained by manipulating the firstblock row of (4.45) in the following manner

)num = −(Rnum)−1 Rmap )den (4.46)

Remark that computation of (4.46) can also be done for each output separately bymaking use of the block structure of R, .

When all the above mentioned measures are combined, a computationally efficientscheme results. This scheme allows to estimate MIMO systems with a largenumber of inputs and outputs, within a reasonable time frame.

Application to a large-scale MIMO system

The effectiveness of the improved numerical scheme is illustrated on a large-scaleMIMO system with 100 inputs and 100 outputs. The same dynamic behaviour asin (4.38) is used. Now, we set =? = 100 such that the estimators are challenged todescribe a complex lightly-damped system. The LPM (2nd order polynomial), theLRM-1 (= = = = 1) and LBTLS-5 estimator (= = = = 1) are applied to simulatedinput-output data sets (Figure 4.12). The DOF is chosen equal to 8 for everyoutput independently.

82


0.0 0.5 1.050

100

150

Frequency [kHz]

Am

pli

tud

e[d

B]

Entire range

0.8 0.85 0.9

150

135

120

Frequency [kHz]|G

[42,

17] |[

dB

]

Zoomed-in

(a) The Local Polynomial Method.

0.0 0.5 1.050

100

150

Frequency [kHz]

Am

pli

tud

e[d

B]

Entire range

0.8 0.85 0.9

150

135

120

Frequency [kHz]

|G[4

2,17

] |[d

B]

Zoomed-in

(b) Bootstrapped Total Least Squares.

Figure 4.12.: The error of the estimated G[42, 17] using the LPM (red •) and theLBTLS-5 estimator (blue •) is compared with the true FRF (grey −).Additionally, a zoomed-in view between 800Hz and 900Hz showsthe estimated G using the LPM (red ×) and the LBTLS-5 estimator(blue ×).

83


LPM LRM Efficient LBTLS Traditional LBTLS

Average compu-tation time

0.041s 1.84s 4.29s 1.5hours

Table 4.2.: Average computation time needed to estimate a local model at onesingle frequency for a MIMO system with 100 inputs and 100 outputs.In case of the LBTLS estimator and the LRM, the time for performingone single iteration is displayed.

To compare the performance of the non-parametric G estimates, we visualisethe results obtained by the LPM and the LBTLS-5 estimator for one arbitraryinput-output combination G[42, 17] . The LRM-1 had a similar performance asthe LBTLS-5 estimator. Comparable results were also obtained for all the otherinput-output combinations. Unlike the LBTLS-5 estimator, Figure 4.12 shows thatthe LPM cannot adequately model the resonant behaviour of the system. Dueto the high number of inputs and outputs, the size of the local window 2=ΔΩ +1is equal to 210. As a result, the system matrix G needs to be modelled over abroad frequency range which is challenging for the polynomial models. Onepossible solution would be to increase the order of the polynomials. This generallyimproves the modelling capabilities of the LPM. Unfortunately, increasing theorder negatively impacts the estimate. A larger order results in a wider localfrequency window in which the dynamic behaviour of G varied even more.

Comparing the computation time for each method on an Intel Xeon E5-2630 [email protected] processor-based Windows computer (Table 4.2), reveals that the LBTLSestimator requires an increased computation time when compared with the LPMand the LRM (Table 4.2). This increase is significantly reduced by implementingthe numerically efficient scheme that was outlined earlier in this section. In theend, the efficient BTLS implementation is only a factor 2.33 slower than the LRM.

84

4.6. Systems operating in feedback

4.6. Systems operating in feedback

Up to now, we always have assumed that the input U(:) of the system was noise-and distortion-free. This condition is rarely met in practice. Multiple systemsare mostly working in synergy to implement particular functionalities. In mostcontrol applications, the system is embedded in a feedback configuration, resultingin a dependency of the actual input of the system on both the actuator and thesystem dynamics. In these situations, the input is surely perturbed with noiseand nonlinear distortion. The suggested local modelling techniques are thennot applicable. A slight adaptation of the methods, however, enables one toapply these techniques for the identification of systems operating in feedback(Figure 4.13) [Pint 13].

The identification of MIMO systems operating in feedback requires the exactknowledge of the =A succesively applied reference signals '8 (:) (8 = 1 ... =A ). Thisrequirement is fulfilled in the vast majority of simulation/measurement scenarios.Instead of applying the local modelling techniques directly from the perturbedinput U(k) to the perturbed output Y(k), we use an intermediate modelling stepthat models Y(:) and U(:) as a function of R(:)(

Y(:)U(:)

)=

(GBLA

R→Y (Ω : )GBLA

R→U (Ω : )

)R(:) +

(LR→Y (Ω : )LR→U (Ω : )

)+

(NR→Y (:)NR→U (:)

)+

(DR→Y (:)DR→U (:)

)(4.47)

which can be rewritten using the stacked input-output vector Z(:) as follows

Z(:) =GBLAR→Z (Ω : ) R(:) +LR→Z (Ω : ) +NR→Z (:) +DR→Z (:) (4.48)

Since the reference signals are unperturbed by noise and nonlinear distortions,local modelling techniques can be applied to (4.48) without introducing a bias.Using the estimated properties it is then possible to derive the BLA from U to Y

GBLAU→Y (Ω : ) =GBLA

R→Y (Ω : ) (GBLAR→U)

† (Ω : ) (4.49)

where •† is the pseudo-inverse of the matrix [Ben 03]. Starting from the block-diagonal covariance matrix CvecGBLA

R→Z (Ω : ) it is furthermore possible to derive

uncertainty bounds on GBLAU→Y (Ω : ) [Pint 12, Appendix 7.F]

CvecGBLAU→Y (Ω : ) = 1CvecGBLA

R→Z (Ω : ) 1

with 1 =((GBLA

R→U)† (Ω : )

))⊗ [I=H −GBLA

U→Y (Ω : )](4.50)

85


Figure 4.13.: MIMO system operating in a feedback configuration.

Introducing the known reference signals in the local modelling techniques eleg-antly converts the closed-loop identification problem to the output-error frame-work. As shown in (4.49) and (4.50), only two post-processing steps are requiredto deduce from the augmented modelling in (4.47) a consistent estimate of theBLA and uncertainty bounds.

4.7. Measurements on the tailplane of a glider

The application of the proposed technique to a realistic measurement exampleis indispensable to illustrate the (dis)advantages of the proposed technique inan experimental setting. Therefore, the resonant behaviour of the tailplane of aglider is characterised using the previously described local modelling techniques.In this section, we describe the measurement setup and provide a measurementcomparison between the LBTLS, the LPM and the (iterative) LRM.

The measurement setup

The system under test is the tailplane of a glider that is connected to a wallvia its central mounting points (Figure 4.14a). The tailplane is excited by twomini-shakers (B&K 4810) and the responses are measured at 5 different locationsusing impedance heads (B&K 8001) and accelerometers (B&K 4371). Both thegeneration (HPE 1445A) and acquisition (HPE 1430A) of the different signals iscarried out by the VME eXtensions for Instrumentation (VXI) measurement system(Figure 4.14b). The output signal of the arbitrary waveform generators needs to be

86


(a) The tailplane of the glider.

(b) Measurement setup. Sensors 1 and 2 record forces, sensors 3, 4 and 5record accelerations.

Figure 4.14.: The VXI measurement setup makes it possible to identify the reson-ant behaviour of the tailplane of a glider.

amplified before application to the mini-shakers. Additionally, a power resistor(18Ω/5W) is connected in series to the shaker to reduce the inductive effects ofthe shaker on the amplifier’s output. The recorded signals are amplified usingcharge amplifiers and buffered before being applied to the acquisition channels.

Unlike the simulation examples, the output-error framework is invalid due tofeedback introduced by the dynamic interaction between the mini-shakers andthe glider’s tailplane. To cope with this issue, two impedance heads (1 and 2)are connected directly between the output of the mini-shakers and the excitationpoints of the tailplane. These impedance heads record the force signals *1 (:) and*2 (:) that are truly applied to the system. As output signals Y, the accelerationsat three different positions (3, 4 and 5) are recorded. The main advantage of the

87


VXI system is that all the different generation and acquisition modules share thesame internal clock. This ensures that all the signals are synchronised and use thesame fixed sampling frequency. This is needed to reliably convert the recordedsignals from the time domain to the frequency domain using the DFT.

Comparison between LBTLS, the LPM and the LRM

Using the described measurement setup, filtered Gaussian white noise with acutoff frequency of 100Hz is applied as reference signal to both mini-shakers.Remark that due to the aperiodic nature of the reference signals, the leakageterm L will also have to be estimated during the modelling stage. In this setup,both noise and nonlinear distortions disturb the measured signals U and Y. Wedeliberately apply enough force onto the tailplane of the glider to make sure thata fair amount of nonlinear distortion is present in the measurement data. Thisallows to verify how the proposed technique behaves in an experimental settingwhere the Signal-to-Noise and Distortion Ratio (SNDR) is lower than 20dB. Adata set was obtained with a total measurement time of 108s using a samplingfrequency of 305Hz. To generate a reference estimate, the LPM was applied to thecomplete data set using a 4th order polynomial for both G and L.

The main goal of the local modelling techniques is to generate a reliable estimatewhen the amount of available data is limited. Therefore, we restricted the lengthof the data set to one tenth of the total measurement time and subsequentlyapplied the LPM (4th order polynomial), the LRM (= = = = = = 1) and theLBTLS estimator (= = = = = = 1) on this shortened data record (Figure 4.15).The cost function defined in (4.26) is calculated to compare the obtained results.The outcome is that the LPM captures the dynamic behaviour less well than theLBTLS-5 estimator (Table 4.3). Again, the LRM-1 and LBTLS-5 have a comparableperformance in modelling the resonant behaviour of the system. To furtherevaluate the performance of the different techniques, we zoom in on the firstresonance and anti-resonance of G[4, 1] (Figure 4.16). This zoomed-in view revealsthat the LPM behaves poorly in estimating the resonance, while the LBTLS-5and the LRM-1 improve this estimation. In other regions, there is no significantdifference between the three methods.

88


0 25 50 75

−80

−60

−40

−20

0|G

[3,1] |

[dB

]

25 50 75 100

−80

−60

−40

−20

0

|G[3,2] |

[dB

]

0 25 50 75

−80

−60

−40

−20

0

|G[4,1] |

[dB

]

25 50 75 100

−80

−60

−40

−20

0

|G[4,2] |

[dB

]

0 25 50 75

−80

−60

−40

−20

0

|G[5,1] |

[dB

]

25 50 75 100

−80

−60

−40

−20

0

|G[5,2] |

[dB

]

Frequency [Hz]

Figure 4.15.: The non-parametric estimate G obtained with LBTLS-5 (blue ×) andthe LPM (red ×) is compared with the reference estimate (grey −).

LPM LRM-1 LRM-5 LBTLS-1 LBTLS-5

+cost 153 85 124 92 79=\ 5 4 4 4 4

Table 4.3.: Values of the cost function (4.26) for the different methods applied tothe measurement example.

89


6 8 10 12

0

−25

−50

Frequency [Hz]

|G[4,1] |[

dB

]Resonance

12 14 16 18

−30

−50

−70

Frequency [Hz]

|G[4,1] |[

dB

]

Anti-resonance

Figure 4.16.: Zoomed-in view of the first resonance and anti-resonance of G[4, 1] .The reference estimate (grey −) is compared with the non-parametricG[4, 1] obtained using the LBTLS-5 estimator (blue ×), the LPM(red ×) and the LRM-1 (yellow ×).

The main difference between the LPM and the LBTLS estimator is observed in theestimation of the uncertainty bounds (Figure 4.17)

- The LPM models the anti-resonances better in the frequency region between40Hz and 50Hz. The reason for this behaviour is that the LPM modelseach output independently with a different polynomial and thus is moreversatile in modelling these anti-resonances locally. The LBTLS estimatoron the other hand, shares a common-denominator part between all theinput-output combinations of G and L. Therefore, the uncertainty on theseparameters is common between the different input-output combinations.Hence, the elevated STD estimate in the anti-resonance regions.

- The uncertainty of the LBTLS estimator is significantly lower in the res-onance regions and in the higher frequency regions (> 70Hz). It differssignificantly from the one obtained with the LPM. This difference is a res-ult of the distinct model structure that models the rapid varying dynamicbehaviour better in the case of the LBTLS estimator.

90


0 25 50 75

−80

−60

−40

−20

0

STD

G[3,1][d

B]

25 50 75 100

−80

−60

−40

−20

0

|G[3,2] |

[dB

]

0 25 50 75

−80

−60

−40

−20

0

STD

G[4,1][d

B]

25 50 75 100

−80

−60

−40

−20

0

STD

G[4,2][d

B]

0 25 50 75

−80

−60

−40

−20

0

STD

G[5,1][d

B]

25 50 75 100

−80

−60

−40

−20

0

STD

G[5,2][d

B]

Frequency [Hz]

Figure 4.17.: When compared to the LPM, the LBTLS estimator reduces the ap-proximation error significantly in the vicinity of the resonances. Ittherefore results in a lower uncertainty. The standard deviation ofLBTLS (blue •) and the LPM (red •) are shown.

91


4.8. Summary

This chapter presents a local rational modelling technique for the accurate es-timation of the BLA and the corresponding uncertainty bounds. The techniqueuses the Bootstrapped Total Least Squares (BTLS) estimator to obtain an unbiasedestimate of the local common-denominator rational representation of the BLAGBLA and the leakage term L. Compared to existing local rational modellingtechniques, our technique provides a consistent estimate in each iteration Theiterative process is used to boost the weighting to obtain a nearly ML efficiency.This is done at the expense of an increase in computational complexity. Further-more, the curse of dimensionality is mitigated by taking explicit advantage ofthe common-denominator structure which opens the scope of applications tolarge-scale MIMO systems. The proposed technique has been thoroughly appliedto strongly resonant mechanical systems, both in simulation (on a 100×100 MIMOsystem of order 100) as in practice (on the tailplane of a glider).

92

5. Stability analysis ofVCO-based ΣΔ modulatorsusing the describing BLA

This chapter combines the local modelling techniques of chapter 4 with theconcepts of the Describing Function to suggest a combined approach forthe stability analysis of ΣΔ modulators excited by modulated signals. Theresults of this chapter are being prepared for publication.

The stability analysis is a vital part that needs to be taken care of during the designof ΣΔ modulators. Instability can generate spurious oscillations or can even resultin chaotic behaviour of the circuit. This disrupts proper functioning of the ΣΔmodulator. Designers want to have access to tools that properly predict in advancefor which signals and amplitude levels a potential unstable behaviour manifestsitself.

In this chapter, we develop a stability analysis technique for Continuous-Time (CT)ΣΔ modulators that incorporate a Voltage-Controlled Oscillator (VCO). This oscil-lator proves to be a key component to achieve an efficient implementation of themulti-level quantiser. The VCO-based quantisers differ from their voltage-basedcounterpart in the sense that they include an intermediate voltage-to-time conver-sion that is implemented by a ring oscillator. Our approach to the stability analysisof such circuits uses the Local Polynomial Method (LPM), which was introducedin chapter 4, in combination with the use of random-phase multisine excitations.It extends the original Best Linear Approximation (BLA) theory to enable theamplitude-dependent modelling that is inherent to the Describing Function (DF)theory. The outcome of the study is the so-called Describing Best Linear Approx-imation (DBLA), an amplitude-dependent BLA model, that combines the benefitsof both theories. While the DBLA can be derived for any nonlinear PISPO system,the DBLA of the internal quantiser is specifically used to predict the amplitudelevel for which the VCO-based ΣΔ modulator potentially becomes unstable.

93

5. Stability analysis of VCO-based ΣΔ modulators using the describing BLA

5.1. Stability assessment

Stability is an important property of any (electronic) device used in an (electronics)system. Its absence results in the creation of uncontrolled oscillations, circuitsaturation phenomena or even the destruction of the system. Unstable behaviouralways deteriorates the signal performance of the ΣΔ modulator and results in acollapse of the Signal-to-Noise Ratio (SNR) and the Signal-to-Noise and DistortionRatio (SNDR). The goal of the stability analysis for ΣΔ modulators is to answerthe following simple question: is the designed modulator stable when convertingsignals taken from a given class of inputs? Before we can answer this question, weneed to formally define what we mean by stability.

Depending on the stability definition, one can obtain a different answer to the sta-bility question of the ΣΔmodulator [Risb 95, Suar 03]. For example, the first-orderΣΔ modulator generates periodic limit cycles whenever a rational constant inputis applied [Schr 05]. This unwanted steady-state oscillation can be consideredto be unstable behaviour when taken from the user point of view. However, thesignal processing properties of the ΣΔ modulator remain primarily unaffectedin this case. To take these cases into account, the stability of a ΣΔ modulatoris defined as a condition on the quantiser input. To put it more clearly, the ΣΔmodulator becomes unstable if the quantiser input starts oscillating uncontrol-lably with a rapidly increasing amplitude [Risb 95]. These uncontrolled oscil-lations may become unbounded for a purely mathematical description of theΣΔ modulator. In practice, circuit saturation will limit the amplitude. In thepast, several rules of thumb have been proposed both from the analytical orthe behavioural simulation viewpoint to maximise the stable input amplituderange [Lee 87, Anas 89, Chao 90, Schr 93, Bour 03]. These rules mostly contain alist of instructions to be followed during the design of the Noise Transfer Func-tion (NTF). They are intended to avoid overloading of the quantiser as much aspossible.

Example Consider a third-order Discrete-Time (DT) ΣΔ modulator that hasthe following NTF

NTF(I) = (1− I−1)3 (5.1)

It operates at a sampling frequency of 1Hz with an Oversampling Ratio (OSR)equal to 32. Visualising the NTF of this modulator (blue) reveals that a highnoise amplification (almost 20dB) is achieved in the close vicinity of 5B/2. Thisamplification pushes the quantiser faster into overloading. As a consequence itbecomes more susceptible to instability and its stable input range is limited.

94

5.1. Stability assessment

One possible solution to enhance the stable input range is to lower the max-imal noise amplification. This can be obtained by adding poles to the NTF(red) [Schr 99]. The effect of this manipulation is best illustrated as follows. Wecalculate the influence of the quantiser’s gain : on the position of the polesof the closed-loop transfer function. The result is visualised on a root locusplot, shown below. The root locus plot shows that the closed-loop poles movefrom the stable region, which is inside the unit-circle in the I-plane, to theunstable region for a quantiser gain value that is equal to 0.5 for the NTF in(5.1). The NTF with reduced high-frequency amplification lowers this valueto 0.325. Since the quantiser’s gain decreases as a function of the input signalamplitude due to the presence of saturation in the circuit, a lower value meansthat more overloading can be accommodated with. Thus, an extended stableinput range is obtained for the NTF with the poles added.

−1 0 1

−1

0

1

k = 0

k =0.325

k = 0.5

Real part

Imag

inar

yp

art

Root locus

10−2 10−1

−60

−40

−20

0

20

Frequency [Hz]

|NT

F|[d

B]

Noise Transfer Function

The main culprit for instability is the saturation of the quantiser. Overloadingshould be avoided as much as possible to avoid instability. This can be obtainedby a careful design of the NTF as was demonstrated in the example above. Alinear root locus analysis determines the value of : for which instability oc-curs, but proves useless when it comes to predict how this gain changes as afunction of the input signal amplitude. This can be accomplished by the DFmodel (chapters 2 and 3) as explained in Section 1.4. Current research into thistopic [Arda 87, Lota 08, Romb 13] mainly focusses on constant inputs, single-toneand dual-tone sinusoids only. The goal of this chapter is to extend this analysis tomodulated signals. Due to the importance of the behaviour of the quantiser for thestability analysis, our main focus lies on modelling the amplitude-dependent gainof this quantiser. Before we dive into the explanation of our modelling approach,we start by giving an explanation of the operation of VCO-based quantisation.

95


5.2. VCO-based quantisation

Over the years, a lot of research has been devoted to increase the performanceof the ΣΔ modulators while keeping their power consumption to a minimum.These research efforts encompass improvements on both the system- and thecircuit-level [Schr 05]. One of the more recent evolutions is a paradigm shift inthe implementation of multi-level quantisers. The Complementary Metal OxideSemiconductor (CMOS) downscaling brings problems for the analogue circuitry.The analogue circuitry suffers from a lower intrinsic gain and a lower supplyvoltage. This complicates the implementation of multi-level quantisers more andmore. The limited headroom that is available to discriminate between consecutivevoltage levels resulted in a lower SNR and more complex circuits. On the brightside, this downscaling improves the performance of the digital circuitry as itincreases its speed and allows for a higher density.

Incorporation of a ring-based VCO in the quantiser of the continuous-time ΣΔmod-ulator alleviates the problems encountered due to the limited voltage headroom.It also narrows the boundary between the analogue and the digital world [Stra 08,Park 09a, Baba 17]. An efficient, high-speed implementation of such a VCO-basedquantiser is proposed in [Stra 08] (Figure 5.1). Instead of performing the quant-isation directly in the voltage or current domain using the voltage/current levelsof the analogue input signal Ein, the VCO-based quantiser uses a two-step archi-tecture. First, the input signal is converted to a frequency modulated high-speeddigital signal. It does so using a -stage voltage-controlled ring oscillator. Thefrequency of this oscillating signal becomes directly proportional to the magnitudeof Ein. To obtain this, the delay of each inverter stage is controlled. Next, theoutput of every inverter stage is connected with digital circuitry (two D flip-flopsand one XOR gate). The first D flip-flop samples the output signal, while thesecond one acts as a storage element of the previously sampled value. The XORgate verifies whether or not a transition from a logic 0→ 1 or 1→ 0 has happenedrelative to the previous sampling instant. The total number of inverter stages thatundergo a transition within one clock period )B depends on the delay occuringin each stage and is set by Ein (Figure 5.2). It therefore corresponds to a sampleddigital representation of the input voltage.

This highly digital alternative solution has the additional advantage that theseVCO-based quantisers automatically incorporate first-order noise shaping of thequantisation noise. This behaviour becomes apparent when we take a look atthe equivalent model of the VCO-based quantiser (Figure 5.3). The input of the

96

5.2. VCO-based quantisation

Figure 5.1.: The VCO-based quantiser samples every intermediate stage of thering oscillator independently to achieve multi-level quantisation.

Figure 5.2.: The VCO-based quantiser counts the total number of inverter stagesthat have transitioned during one clock period (indicated in red). Thehorizontal grey line in between two stages indicates which inverterstage is currently in transition.

97


Figure 5.3.: Model of one inverter stage of the VCO-based quantiser in Figure 5.1.

quantiser is first shaped by the nonlinear voltage-to-frequency transfer charac-teristic of the VCO. Next, it is converted to a phase signal (cycles) by the VCOthat acts as a phase integrator. The phase signal is consequently sampled andfed to a differentiator to obtain the frequency at which the VCO runs. Due to thefinite number of delay cells that are present in the VCO, phase quantisation occursduring the sampling operation. In the model of Figure 5.3, it is modelled by anadditive quantisation noise source @(=). Using the star operator •, as introducedin Appendix A, the equivalent Signal Transfer Function (STF) and NTF can beobtained if we assume linear operation of the circuit

. (I) = (1− I−1)[ VCO

B+in (B)

]+ (1− I−1) &(I) (5.2)

=

[︸︷︷︸STF(B)

VCO (1− 4−B)B )/B +in (B)]+ ︸︷︷︸

NTF(I)

(1− I−1) &(I) (5.3)

where +in (B) is the Laplace transform of Ein (C), and . (I) and &(I) are, respectively,the Z-transform of H(=) and @(=). From (5.3) it becomes apparent that the VCO-based quantisers automatically implement a first-order noise shaping, while theSTF corresponds to a constant gain factor for frequencies 5 << 5B = 1/)B.

The main problem that nowadays limits the performance of VCO-based quantisersis the presence of the nonlinear VCO transfer characteristic [Park 09a, Voel 14,Xing 15, Baba 16]. As a first step in the BLA-based stability analysis of VCO-basedΣΔmodulators as proposed here, we apply a multisine excitation to the VCO-basedquantiser. This signal is used to extract the generated amount of quantisationnoise and of nonlinear distortion. To do this in an accurate way, we have to get ridof the noise leakage. This is the subject of next section.

98

5.3. Noise leakage reduction using local modelling techniques

5.3. Noise leakage reduction using localmodelling techniques

The introduction of a time window is inescapable to accurately analyse the beha-viour of a ΣΔ modulator [Schr 05]. Remember that only output signals H(C) thatare exactly periodic within the observation time window result in a leakage-freeoutput spectrum . (:). We expect that if we apply periodic repetitions of a finitelength signal to a noise-free system, the output response is identical for everyrepetition of the signal. However, the random nature of the quantisation noiseproduced by a ΣΔ modulator results in an aperiodic output response even thougha periodic input signal is applied. This aperiodicness of the noise in turn resultsin noise leakage in the output spectrum . (:) (see Section 4.1). In most cases,this noise leakage can be safely ignored since it is small compared to the actualnoise contribution. For ΣΔ modulators, an accurate analysis of the in-band noisedensities requires to reduce the noise leakage [Schr 05, Appendix A]. Due to thehigh-frequency noise amplification, there can potentially be a large differencebetween the in-band and out-of-band noise densities (> 80dB) as is demonstratedin Figure 1.7a. The noise leakage of the amplified out-of-band noise can thendominate the in-band noise density. This results in a wrong determination of theactual in-band quantisation noise densities. A widely-used method to reduce theeffect of spectral leakage involves the multiplication of the time-domain outputsignal with a non-rectangular window before applying the Discrete Fourier Trans-form (DFT) transform [Harr 78]. These time windows are mathematical functionsthat are only defined over the total length of the observation time window, are zeroat the edges, are mostly symmetric, and reach a maximum right in the middle ofthe observation time window. They ensure that the noise leakage can be reducedto properly assess the noise shaping behaviour. Popular choices for such windowsare the Hann and the Hann2 window [Schr 05].

While time windowing results in a significant reduction of the noise leakage,it also introduces spectral widening of discrete spectral lines. It is convenient,both in simulation and in measurements, to use single-tone sinusoidal excitations.Ideally, the single-tone should occupy only one bin :exc in the output spectrum.However, when applying a window, additional ’artificial’ contributions pop up onthe bins surrounding :exc. In the case of a single-tone excitation, these windowingartefacts can be easily ignored as we know exactly and beforehand on which binsthey manifest themselves. However, in the case of a multisine excitation, thesurrounding bins might already be occupied by excitation lines. This means thata significant error is introduced when a window is applied.

99


Example Consider a first-order discrete-time ΣΔ modulator with the sametopology as the one used in Figure 1.3. The simulated modulator has a samplingfrequency 5B = 35.2MHz and a bandwidth = 275kHz. We record the responseto a single-tone and a flat random-phase multisine. We apply a Hann windowand obtain the following output spectra

1e5

0

−25

−50

−75

Ou

tpu

tsp

ectr

um

[dB

FS/N

BW

]

Single-tone

1e6

0

−25

−50

−75

Input

Ou

tpu

tsp

ectr

um

[dB

FS/N

BW

]

Multisine

Frequency [Hz]

These spectra reveal that the single-tone is still clearly visible while the win-dowing completely destroys the anticipated flattness of the multisine.

One solution to this problem is to increase the spectral resolution of the DFT.We then determine the time-domain response H(C) of the ΣΔ modulator to %

periods (repetitions) of the same multisine excitation. Converting these % periodssimultaneously to the frequency domain as a single time record results in aspectrum .% (?) (Figure 5.4). If the signal is perfectly periodic, .% (?) contains% − 1 zero spectral lines in between the originally neighbouring spectral lines.This is due to the increased spectral resolution. When taking % periods, adjacentspectral lines of the excitation are now spaced % bins apart. These additional binscan then be used to stash the windowing artefacts. As the spectrum of the Hannwindow consists of 3 spectral lines only, the Hann window requires at least 3periods to ensure that the excited bins of the multisine are not hampered by theirneighbours. If on top of that we want to characterise the in-band noise densities,then at least 4 periods are needed. For a Hann2 window these numbers evenincrease to respectively 5 and 6 periods.

From these examples it is clear that combining windowing with multisine excita-tions results in a significant simulation overhead. We want to reduce this overheadto a minimum. For this purpose, we apply the local modelling techniques which

100


Figure 5.4.: Top: steady-state response H(C) and the corresponding DFT spectrum. (:). ↑: excited bins, ↑: nonlinear distortion, ↑: noise, ↑: noise leakage.Bottom: the DFT spectrum .% (?) of % = 4 consecutive periods of thesteady-state response with an increased spectral resolution. It resultsin %−1 additional bins that pop up in between the original bins ofthe output spectrum. These are only perturbed by the noise and thenoise leakage.

are introduced earlier in chapter 4. These techniques require minimally twoperiods of the multisine excitation while being able to characterise the in-bandnoise densities and the in-band nonlinear distortion levels. This already halvesthe simulation time in comparison with the Hann window. The method we applyis called the ’fast method’ [Pint 11].

The fast method

The fast method starts from the same input-output DFT spectra of % periodsof the steady-state response U% (?) and Y% (?) to a multisine excitation as be-fore. It makes extensive use of the observation that, for Periodic-In Same Period-Out (PISPO) systems excited with periodic signals measured over an integernumber of periods (L (Ω : ) becomes zero), nonlinear distortion only manifestsitself at frequencies that are affine combinations of the original excited frequencies(Figure 5.4) [Pint 12]. Therefore, the additional bins in Y% (?) carry no periodic

101


contributions. They exclusively contain noise contributions N% (?) and noise leak-age contributions L%N (Ω ?). The idea used here is to apply the local modellingtechniques introduced earlier in Chapter 4. Since neither the STF nor the NTFcontain resonances within the signal bandwidth [Schr 05], we have chosen theLPM to model the local dynamic variations in relation to frequency. Remark thatif % is a non-integer number, or the input and output signals are aperiodic withinthe observation time, it is no longer possible to distinguish the contribution of thenoise from the nonlinear distortion.

Starting from the output Y% (?) and the noise- and distortion-free input U% (?),the fast method uses a two-step procedure to (Figure 5.5)

1. Obtain an estimate for the noise leakage L%N (Ω :%) and the noise covariancematrix C%

N (:%).

2. Retrieve an estimate for the BLA GBLA (Ω :%) and the covariance matrix ofthe nonlinear distortion C%

D (:%).

This procedure is repeated for every excited frequency Ω :% .

Step 1: Retrieve the noise information using a local noise model

Suppose that we work at the excited frequency bin :%. As explained earlier, theoutput spectrum Y% is only perturbed by the noise and the noise leakage at thebins around :%

Y% (:%+ An) = L%N (Ω :%+An ) +N% (:%+ An) (5.4)

where An belongs to the set Z \ =% | = ∈ Z. By exclusively using these additionalnon-excited bins, we can locally approximate the noise leakage with a polynomialmodel

L%N (Ω :%+An ) = L%N (Ω :% + XAn ) = L%N (Ω :%) +=!∑<=1ℓ:< (XAn )< (5.5)

where L%N (Ω :%) is the local noise leakage contribution, and ℓ:< are the unknownpolynomial model coefficients that model the dynamic variations of the noiseleakage within the local model. These coefficients are estimated, together withL%N (Ω :%), by substituting (5.5) in (5.4). The resulting equation is evaluated in thefirst =noise non-excited bins to the right and to the left of Y% (Ω :%) This results in anoverdetermined set of linear equations that is solved in linear least-squares sense.The covariance matrix C%

N (:%) is retrieved from the residuals of this estimationprocedure.

102


Figure 5.5.: The fast method is a two-step procedure. It first removes the noiseleakage and estimates the in-band noise variance (=noise = 5). Next, itretrieves the BLA and the in-band nonlinear variance (=ΔΩ = 3).↑: excited frequency for which the estimation is performed, ↑: nonlin-ear distortions, ↑: noise, ↑: noise leakage, −: estimated local model.

103


Step 2: Retrieve the BLA and the covariance matrix of the nonlinearities

The leakage present in the original spectrum is then removed using the resultsfrom step 1. The estimated noise leakage L%N (Ω :%) is subtracted from Y% (:%) atevery excited frequency bin

Y%corr (:%) = Y% (:%) − L%N (Ω :%) (5.6)

This results in the corrected spectrum Y%corr (:%) that is freed of the noise leakage if

the compensation is perfect. This time, we evaluate the corrected output spectrumat the excited frequency bins only

Y%corr (:%+Ae%) =GBLA (Ω :%+Ae%)U% (:%+Ae%) +N% (:%+Ae%) +D% (:%+Ae%) (5.7)

where Ae ∈ Z. Similarly to what we did in step 1, we can locally approximate theBLA with a local polynomial model

GBLA (Ω :%+Ae%) =GBLA (Ω :% + XAe%) = GBLA (Ω :%) +=∑<=1

g:< (XAe%)< (5.8)

where GBLA (Ω :%) is the local BLA estimate, and the coefficient matrices g:< arethe unknown polynomial model coefficients that model the dynamic variationsof the BLA within the local signal model. These local model coefficients andGBLA (Ω :%) are all estimated by substituting (5.8) in (5.7). The resulting equationis then evaluated in the closest =ΔΩ excited bins that lie to the right and to the leftof Y% (Ω :%). Again, this results in an overdetermined set of linear equations thatis then solved in linear least-squares sense. The residuals of this estimation pro-cedure are used to estimate the total covariance matrix C%

N+D (:%). The covariancematrix of the nonlinear distortions C%

D (:%) is calculated by subtracting the earlierobtained noise variance C%

N (:%) from the toal variance C%N+D (:%).

Application to a VCO-based quantiser

For illustration purposes, the fast method is applied to the VCO-based quantiserthat was shown in Figure 5.1. The results obtained with different windows (Hann,Hann2, Blackman) are compared. The quantiser is designed in Advanced DesignSystem (ADS) using a 0.18`m CMOS process with 1.8V supply. It consists ofa current-starved VCO with = 7 inverter stages. Every intermediate stage issampled by a dynamic edge-triggered D flip-flop. The sampling frequency is

104


105 106 107 108

−80

−100

−120

−14020dB/d

ec

PSD

YP

[10

log 1

0V

2 /H

z]Fast method

106 107 108

−80

−100

−120

−14020dB/d

ec

PSD

YP

[10

log 1

0V

2 /H

z]

Hann window

Frequency [Hz]

Figure 5.6.: Comparison between the Power Spectral Density (PSD) of the outputsignal obtained after application of the fast method using the LPM(left) and the Hann window (right). ×: excited frequencies, ×: in-bandnonlinear variance, ×: in-band noise variance, ×: in-band additionalbins. The out-of-band quantisation noise (−) is indeed subject tofirst-order noise shaping.

set to 1GHz. Care has been taken during the design of the VCO to ensure thatthe oscillation frequency does not exceed 500MHz. This condition needs to besatisfied to ensure that not all the inverter stages undergo a transition duringone sampling period. If this condition is not met, the digital output saturatesindependently from the input control voltage. The multisine that excites thesystem has a Root Mean Square (RMS) of 0.1VRMS. It has a base tone frequency50 of 250kHz and excites 40 tones with a total resulting bandwidth of 10MHz.A transient simulation is performed to retrieve four periods of the steady-stateresponse to the multisine excitation.

Application of the fast method using the LPM (= = =! = 2) and the Hann windowmethod on the simulated data reveals the differences that are present betweenboth methods even though they are fed with exactly the same data (Figure 5.6).The LPM provides an accurate estimate of the PSD of the output at the excitedfrequencies, together with the in-band noise variance and the in-band nonlinearvariance. The Hann window on the other hand generates unwanted contributionto the left and to the right of the excited frequencies due to the presence of a widermain lobe in its transfer function [Nutt 81]. Therefore, at least four periods areneeded to reveal the in-band noise with this window. Furthermore, the Hannwindow only provides information about the in-band noise and completely fails torecover the nonlinear distortion that is produced by the system. Remark that the

105


Periods % LPM Hann Hann2 Blackman

2 -132.85 - - -3 -136.65 - - -4 -138.20 -139.96 - -5 -138.95 -140.11 - -6 -140.22 -140.22 -141.73 -141.447 -141.15 -140.86 -142.06 -141.81

Table 5.1.: The average in-band noise PSD (10 log10 V2/Hz) obtained with the LPMand three commonly applied windowing techniques (Hann, Hann2

and Blackman) for different number of periods %.

in-band noise for this VCO-based quantiser should exhibit the same 20dB/decadeslope as the out-of-band noise. However, this is not the case. As it turns out, thereason for this is that the numerical accuracy of the simulator used is inadequate.A consequence of this limited accuracy is a flattening of the in-band noise PSD.Further lowering the transient simulation time step resolves this issue but wouldresult in an unreasonable simulation time.

To further evaluate the performance of the fast method when compared to thecommonly applied windowing techniques (Hann, Hann2 and Blackman) [Nutt 81],we determine the average in-band noise PSD for a different number of periodsof the steady-state response (Table 5.1). From Table 5.1, we conclude that theestimates for the average in-band noise PSD do not differ significantly for a givennumber of periods. What does change substantially, however, is the minimalamount of periods that is needed to acquire the in-band noise PSD. For example,the Hann2 and Blackman window require a simulation time that is three timeslarger when compared to the fast method.

Validity of the fast method for mixed-signal systems

Looking back on these results, it is remarkable that the fast method using the LPMcan be successfully applied to the VCO-based quantiser. After all, the quantisersare mixed-signal systems. They contain a sampling operation that converts thecontinuous-time signals to discrete-time sequences. If we want to model the beha-viour of such a mixed-signal system in general, the model should be able to mixcontributions of spectra taken in the B- and I-plane. Hence, it needs to properly

106

5.4. The Describing Best Linear Approximation

handle the not so trivial continuous-to-discrete-time transformation. Fortunately,the Impulse Invariant Transformation (IIT) enables the transformation of themixed-signal behaviour to an equivalent representation in a single domain usingthe following equivalence (Appendix A)

I = 4B)B or B =ln(I))B

(5.9)

as has been showcased earlier in (5.3). This equivalence discloses one majorobstacle in the parametric modelling of the STF and NTF: the model becomesnonlinear in the parameters. If one would aspire to estimate a parametric modelstarting from noisy input-output data that is valid over the whole frequency range,a nonlinear optimisation scheme is required. This involves the generation of goodinitial values to prevent local minima and requires an often tedious model orderselection procedure. The local modelling techniques, however, are less impactedby this nonlinear dependence. As long as the dynamic variations in the narrowlocal window are limited, polynomials are sufficient to model the dynamics. Dueto the IIT, polynomials in one variable, B or I, can be used for this purpose as ismade clear by the local series expansion around every excited frequency l:

I = 4B)B = 4 9l:)B+∞∑==0

)=B

=!(B− 9l: )= (5.10)

This reasoning explains why we could apply the LPM effortlessly to the VCO-basedquantiser.


In our search of an approach for the frequency- and amplitude-dependent linearapproximation of nonlinear PISPO systems, we gradually recognised the local mod-elling techniques as a potential candidate to satisfy the goals specified earlier insection 1.5. The requirement of no prior knowledge is automatically fulfilled sincethe local modelling techniques only require the knowledge of simulated/measuredinput-output spectra as input data. The previous section furthermore has shownthat the local context does also enable the modelling of mixed-signal systems.Finally, the user interaction is also kept to a minimum. The only parameters thatthe user needs to specify are the orders of the polynomials and the width of thelocal window. A convenient way to include the signal-dependent behaviour withinthe local modelling techniques is still to be found.

107


Figure 5.7.: The DBLA estimation uses a two-dimensional grid (Ω : ,f9 ) as a func-tion of the RMS value f and the generalised frequencyΩ. For every f9 ,a random-phase multisine with % periods is applied to the nonlinearPISPO system to acquire the input-output data.

To accomplish this inclusion, we propose to use an n-dimensional extension of theLPM, the so-called nD-LPM [Maas 17]. Originally, the nD-LPM was introduced forthe modelling of Linear Parameter-Varying (LPV) systems. It was meant to take theeffect of external parameter variations into account. One of the main assumptionsof the nD-LPM is that the LPV system varies smoothly as a function of the externalparameters. Relating this assumption to our problem setting, we assume here thatthe amplitude dependency of the BLA approximation of the nonlinear systemvaries smoothly as a function of the amplitude of the input signal. While thisassumption is not generally true for all nonlinear systems (e.g. chaotic systems),it is valid for the considered class of PISPO systems. Following the lines of thefast method introduced in Section 5.3, the proposed technique uses a two-stepprocedure to implement the nD-LPM. We took the liberty to name this techniquethe DBLA. As a preliminary, multiple periods % of the input-output time-domainresponse of the system under test to a random-phase multisine at different RMSlevels f9 are gathered by simulations/measurements (Figure 5.7). This creates atwo-dimensional grid (Ω: ,f9 ). A DBLA estimation is then performed at each gridpoint (Ω : ,f9 ) individually.

108


DBLA step 1: One-dimensional noise leakage removal

The whole reasoning starts from the fundamental equation (4.1). This time, it isslightly adapted to explicitly express the amplitude dependency of the differentsignals and transfer functions

Y% (:%,f9 ) = GDBLA (Ω :% ,f9 )U% (:%,f9 )+ L%# (Ω :%) (5.11)

+ N% (:%,f9 ) + D% (:%,f9 )

Since the noise leakage perturbation is only caused by the aperiodic nature ofthe noise perturbation when performing the DFT, it depends only on the fre-quency and does not vary with the amplitude of the signal. Therefore, the sameone-dimensional estimation procedure as described in Step 1 of Section 5.3 canbe applied to the additional bins that are present in Y% (?,f9 ) to remove thenoise leakage (Figure 5.8) and obtain an estimate of the noise covariance matrixC%

N (:%,f9 ) as before.

DBLA step 2: Two-dimensional DBLA estimation

The DBLA estimation starts from the corrected spectrum that is obtained at eachexcited frequencie Ω :% and corresponding frequency bin :% separately

Y%corr (:%,f9 ) = Y% (:%,f9 ) − LN (Ω :%) (5.12)

to derive GDBLA (Ω :% ,f9 ) and the total covariance matrix C%N+D (:%,f9 ) as a func-

tion of the frequency and the amplitude. Since the DBLA depends both on thefrequency and the amplitude, a two-dimensional Taylor expansion is used aroundevery grid point (Ω :% ,f9 ) (Figure 5.8)

GDBLA (Ω :%+Ae% ,f9+C ) =GDBLA (Ω :% + XAe% ,f9 + XC )

=

=Ω∑E=0

=f∑F=0" (:, 9)EF XEAe%X

FC

(5.13)

where Ae, C ∈ Z. XAe% and XC describe, respectively, the local frequency and amp-litude variations. " (:, 9)EF are the coefficients that model the two-dimensionallocal variations within the local model. The coefficient " (:, 9)00 corresponds toGDBLA (Ω :% ,f9 ). In general, the model orders =Ω and =f can be chosen small

109


σmin

σmax pmin

pmax

RMS σj Binsp

|YP(p,σ

j)|

Step 11D noise leakage removal

Excitation grid

Additional bins

σmin

σmax pmin

pmax

RMS σj Binsp

|YP(p,σ

j)|

Step 22D DBLA estimation

Figure 5.8.: The proposed DBLA estimation procedure consists of a one-dimensional noise leakage removal (step 1) followed by a two-dimensional DBLA estimation (step 2).

110


(≤ 4) since the variations within the local window remain limited if the grid iswell chosen in both the frequency and the amplitude direction. The unknowncoefficients " (:, 9)EF and the total variance C%

N+D (:%,f9 ) are retrieved by substitut-ing (5.13) in Y%

corr (:%,f9 ) and solving the resulting overdetermined set of linearequations in linear least-squares sense.

Example To demonstrate the modelling capability of the proposed technique,the DBLA of a multi-level quantiser (Figure 1.5) is determined and comparedwith the corresponding DF. This multi-level quantiser has 32 distinct levels(5 bits resolution) and can be explicitly described by a static staircaselikenonlinear function H = 5 (D) (Figure 1.6). The quantisation step is equal to1/16V. To determine the DBLA, the quantiser is excited at 40 distinct RMSlevels spaced logarithmically between −35dBVRMS and 10dBVRMS using thesame multisine. The multisine excitation contains 100 tones over a bandwidthof 100Hz with 50 = 1Hz and 5B = 1kHz. During the estimation, three steady-state periods of the simulated output are used. All the model orders (=! , =Ω and=f) are chosen to be equal to one.

−40−20

00

2550

75100

−15

−10

−5

0

5

σ [dBVRMS ] Frequency Ω [Hz]

|GD

BL

A(Ω

,σ)|

[dB

]

Note the dynamic variations in the DBLA esimate. These variations cannot beintuitively explained since a static nonlinear function does not exhibit dynamicbehaviour [Buss 75]. They are produced by the stochastic distortion contribu-tions to the DBLA. These contributions add an uncertainty to the estimate ofthe DBLA that appear to be ’dynamic’ variations around the main nonlineartrend. At the expense of an increased simulation or measurement time, thesevariations can be mitigated by averaging the results obtained from multipleexperiments with different phase realisations of the multisine excitation.

111


For random zero-mean inputs with a Gaussian distribution the Random-InputDescribing Function (RIDF) is calculated as [Gelb 68]

#A (f) =1

√2cf3

+∞∫−∞

5 (D) D exp(−D2

2f2

)dD (5.14)

Numerical integration of (5.14) yields the following RIDF that has a 1dB com-pression point equal to −4.1dBVRMS.

−40 −30 −20 −10 0 10

0

−5

−10

RIDF

GDBLA ±σGDBLA

σ [dBVRMS]

Lin

eari

sed

gain

[dB

]

The RIDF is compared with the 68% confidence interval of the DBLA. Thisinterval is obtained by averaging the DBLA (Ω,f) as a function of the frequency.For small values of f, the input signal only partly exceeds the quantisationstep. This results in a collapse of the linearised gain of the quantiser in thisregion. As could be expected, the DBLA exhibits the same average compressionbehaviour as the RIDF.

The DBLA of a VCO-based quantiser

Following the same lines of reasoning as in Section 5.3, the local modellingapproach can also be used here to estimate the DBLA of a mixed-signal system ina single frequency domain (I or B). Here we choose the Laplace variable B. Again,we explicitly assume that the two-dimensional dependency on the frequency andinput amplitude is smooth. This ensures that the partial derivatives with respectto Ω and f exist until the =th

Ωand =th

f order.

112


−30−25

−20−15

−10 02.5

57.5

10

−1.5

0

1.5

3

σ [dBVRMS] Frequency Ω [MHz]

|GD

BL

A(Ω

,σ)|

[dB

]

Figure 5.9.: The DBLA (=Ω = =f = =! = 1) of the continuous-time VCO-based quant-iser depicted in Figure 5.1.

−30−25

−20−15

−10 02.5

57.5

10

−80

−60

−40

σ [dBVRMS] Frequency Ω [MHz]

Var

ianc

e[d

BV

2 ]

Figure 5.10.: Estimated noise variance f2#(:,f9 ) (gray) and total variance

f2#+ (:,f9 ) (coloured) of the VCO-based quantiser.

113


To verify the validity of this reasoning, we apply the proposed technique to theearlier designed CMOS VCO-based quantiser of Section 5.3. The quantiser isagain excited by a multisine excitation that covers a bandwidth of 10MHz with abase tone frequency 50 equal to 250kHz resulting in 40 spectral lines. Transientsimulations are performed to retrieve four periods of the steady-state response.The quantiser is excited at 20 distinct RMS levels spaced logarithmically between−30dBVRMS and −10dBVRMS using the same multisine excitation. The resultingDBLA (Figure 5.9) shows that the continuous-time VCO-based quantiser has aconstant dynamic behaviour for low RMS values of the input. This behaviouris in correspondence with the approximate linear STF that was derived in (5.3).Nonlinear compression on the other hand is clearly visible with a 1dB compressionpoint of −14.5dBVRMS.

In addition to the DBLA, we also visualise the output noise variance f2#(:,f9 )

and the total output variance f2#+ (:,f9 ) in Figure 5.10. The noise contribution

remains fairly constant as a function of the input amplitude at an average variancelevel of −81dBVRMS. This noise variance level corresponds with the quantisationnoise power that is predicted by the linear system theory [Kim 09]. The totalvariance increases significantly with increasing RMS levels. Similar to what wasfound for the DBLA, the nonlinear distortion is also subject to compression forhigher input RMS levels [Wamb 13]. Note that for this specific example, it is im-possible to calculate the RIDF of the complete system due to the complexity of theunderlying transistor models and the unavailability of an explicit mathematicaldescription. One of the major advantages of the proposed technique is that theDBLA does not require such a mathematical description to be obtained. Onlysimulated/measured input-output is needed to estimate the DBLA, the outputnoise variance, and the output total variance simultaneously.

5.5. A DBLA-based stability analysis

The previous sections introduced the concept of the DBLA, the required two-stepestimation procedure, and the application on VCO-based mixed-signal quantisers.In this section, we describe how the DBLA of the quantiser is used to perform thestability analysis of the previous continuous-time ΣΔ modulator. The proposedanalysis makes the two following assumptions

1. The continuous-time ΣΔ modulator consists of a nonlinear quantiser sur-rounded by linear blocks as in Figure 5.11.

114


Figure 5.11.: The continuous-time ΣΔ modulator of Figure 1.5 is approximatedlinearly by the DBLA estimate of the quantiser.

2. The nonlinear quantiser is identified in open-loop using the method intro-duced in Section 5.4. This method uses random-phase multisine excitationswith a flat PSD. The embedding of the quantiser within a feedback loopdynamically shapes the PSD of the quantiser’s input and makes it no longercompletely flat. Theoretically, the estimated DBLA is then no longer valid.We assume that the effect of this dynamic shaping is limited. As a result, thequantiser can be replaced by its linearised DBLA estimate and the estimatednoise and distortion components, which are obtained with flat random-phasemultisine excitations.

The main goal of the proposed stability analysis is to obtain the output RMS levelf. as a function of the applied reference RMS level f'. When we combine thesewith the knowledge of f# and f#+ , the SNR and SNDR curve can be calculatedas a function of f'. The modulator overloading level of the ΣΔ modulator isobtained from an analysis of these curves [Schr 05].

Deriving the output RMS f. of the quantiser using the linear framework is astraightforward task. It suffises to first evaluate the linear transfer function '→.from the reference ' to the output. and next integrate the result over the completebandwidth of the random-phase multisine

f2. =

1

∑:=1|'→. (Ω: ,f* ) |2 (AA (Ω : ,f') (5.15)

where (AA is the auto-power spectrum of the input and '→. is the closed-loop linear transfer function from the reference to the output. Unfortunately,'→. (Ω : ,f* ) depends on DBLA which in turn depends nonlinearly on f* . Thedirect consequence of this dependency is that (5.15) cannot be evaluated directly.An iterative scheme, that is similar to Algorithm 1 in Section 3.3, is proposedto calculate the correct f. for a fixed f' (Algorithm 3). This algorithm takes

115


Algorithm 3 Iterative algorithm for the derivation of the correct f. using theDBLA estimate.

1. Initialise the iteration variable f* with the lowest RMS value used duringthe DBLA estimation. By doing so, we make sure that the starting DBLAcorresponds with the underlying linear system.

2. Using the current value of f* , find the grid RMS f9 that is closest to f* . Foreach frequency Ω : , evaluate the model (5.13) that is identified earlier at theclosest grid point (Ω : ,f9 ). Derive DBLA (Ω : ,f* ) for this gridpoint.

3. Similarly to (5.15), use the interpolated model DBLA (Ω : ,f* ) to retrieve anupdate of f*

f2*,new =

1

∑:=1|'→* (Ω : ,f* ) |2 (AA (Ω : ,f') (5.16)

4. Verify whether or not |f*,new−f* |/f* remains below a specified maximumrelative error. If it does, stop the iterative loop and go to step 5. If not, setf* = f*,new and reiterate starting from step 2.

5. Using the final estimated value for f*,new, we retrieve f. via (5.15). Forthe extraction of f# (Ω : ,f*,new) and f#+ (Ω : ,f*,new), a two-dimensionallocal interpolation similar to (5.13) is applied to the noise variance andtotal variance surface. An example of such surfaces is shown in Figure 5.10.After shaping these interpolated noise and total variance surfaces with theclosed-loop transfer function, the level of the noise and nonlinear distortionthat is present in the output can also be derived.

advantage of the estimated local parametric models in (5.13) for interpolationpurposes. After all, these models already describe the ’best’ (in least-squaressense) surface that describes the DBLA in the vicinity of every grid point (Ω : ,f9 ).In the remainder of this section, the accuracy of the proposed iterative schemeis assessed on two different continuous-time VCO-based ΣΔ modulators. Bothmodulators are simulated using CppSim, a C++ behavioural simulator that wasdeveloped for this purpose by Perrott et. al [Perr 97].

116


Application to a VCO-based voltage-to-phase quantiser

As a first example, a VCO-based voltage-to-phase quantiser is evaluated (Fig-ure 5.12) [Park 09a]. This type of modulator consists of a multi-stage voltage-controlled ring oscillator, a sampled quantiser, a phase detector, and a DAC. Thedifference with the original VCO-based quantiser in Section 5.3 is that the differ-entiator is not included. This transforms the quantiser’s STF from a constant inthe signal bandwidth to one that exhibits a dynamic integrative action 1/B. Thesampling frequency of the quantiser is set to 1GHz and 31 stages are present inthe ring oscillator which is equivalent to 5-bit quantisation. Additionally, thebehavioural model includes a static nonlinear polynomial dependency of the VCOwhich was derived by Park et al. from Spectre simulations [Park 09a].

As a first step, the open-loop DBLA is determined (=Ω = =f = =! = 2) from theinput D(C) of the VCO to the output H(=) of the phase detector using a multisineexcitation that contains 100 tones with a bandwidth of 10MHz and 50 equal to100kHz (Figure 5.13). Transient simulations calculate three periods of the steady-state response of the system for 40 different RMS levels of D(C) spread linearlybetween −60dBVRMS and −20dBVRMS. This large number of distinct RMS levelsis needed to ensure that the nonlinear contributions in the compression regioncan be estimated accurately. Otherwise the proposed iterative scheme is not ableto accurately predict the overloading level.

Analysing the resulting DBLA shows a dynamic behaviour that corresponds to1/B due to the voltage-to-phase conversion of the ring-based VCO (Figures 5.13and 5.14). Furthermore, the phase detector saturates from an input level f* equalto −35.9dBVRMS. This drastically degrades the quality of the estimated DBLA andresults in artificial resonances.

Applying the iterative scheme of Algorithm 3 to the voltage-to-phase quantiseryields the closed-loop RMS values f* and f. as a function of the applied referenceRMS value f'. During the analysis, we consider that the DAC is a linear blockwith a static gain in the signal bandwidth. For low reference RMS values, anaccurate fit is obtained between the iterative scheme (×) and simulations (−). Inthe literature, the modulator overloading level is generally defined as the RMSlevel for which the SNR shows a drop of at least 6dB when compared to the max-imal SNR level [Schr 05]. Simulation of the SNR curve for this voltage-to-phasequantiser reveals that the modulator overloading level is situated at −9.5dBVRMS.Around the modulator overloading level, the iterative scheme deviates most with

117


Figure 5.12.: The VCO-based voltage-to-phase quantiser implements a feedbackloop that consists of a ring-based VCO, a clocked quantiser, a phasedetector and a Digital-to-Analogue Convertor (DAC). No loopfilteris included.

−60−50

−40−30

−20 02.5

57.5

10

30

50

70

90

Input σU [dBVRMS ] Frequency Ω [MHz]

|GD

BLA(Ω

,σU)|

[dB

]

Figure 5.13.: The DBLA (=Ω = =f = =! = 2) of the open-loop VCO-based voltage-to-phase quantiser.

118


0.1 1 10

50

70

90

σU =−40.4dBVRMS

σU =−30.3dBVRMS

Frequency Ω [MHz]

|GD

BL

A(Ω

,σU)|

[dB

]

Figure 5.14.: Two cross sections of Figure 5.13 clearly show the linear (1/B) andnonlinear behaviour of the voltage-to-phase quantiser.

a maximum difference of around 2dB for f* (left plot in Figure 5.15). When ana-lysing f. (right plot in Figure 5.15), only a small discrepancy is visible betweenthe estimated and simulated results.

This example has shown that the signal-dependent DBLA in combination withAlgorithm 3 enables to predict the nonlinear dependence of f* and f. as afunction of f'. In the next example, the prediction capabilities for the SNR andSNDR curves is assessed.

Higher-order VCO-based Sigma-Delta modulator

The final example is a higher-order VCO-based ΣΔ modulator as is shown Fig-ure 5.16. The structure incorporates a dual-loop feedback, where the inner feed-back loop is included to accomplish excess loop delay compensation [Cher 99b].Excess loop delay is introduced by the non-ideal behaviour of the DACs and hasa destabilising effect. We consider in this example that the loopfilter is of 2nd-and 4th-order [Park 09b]. This results in, respectively, a 3rd- and 5th-order ΣΔmodulator. Remember that the VCO adds an additional integrator. The samplingfrequency is set to 950MHz. Again, 31 stages are present in the ring oscillatorthat has a nominal oscillation frequency of 250MHz. The ΣΔ modulator is excitedwith a flat random-phase multisine excitation. This multisine signal contains 100tones with a bandwidth of 9.5MHz with 50 = 95kHz. The open-loop DBLA of thequantiser (=Ω = =f = =! = 2) is obtained by using three periods of the steady-state

119


−30 −20 −9.5

−50

−40

−30

−20

Unstable

Reference σR [dBVRMS]

Inp

ut

σ U[d

BV

RM

S]

−9.5−7.5−35

−25

−30 −20 −9.5

−5

0

5

10

15

20


Ou

tpu

tσ Y

[dB

VR

MS]

−9.5 −7.514

15

Figure 5.15.: The closed-loop simulated input RMS f* and output RMS f. (−)is compared with results obtained using the proposed iterativescheme (×).

response at 50 different RMS levels spaced logarithmically between −40dBVRMS

and 0dBVRMS (Figure 5.17). The number of RMS levels is determined by examin-ing the uncertainty on the DBLA estimate. One has to ensure that the estimateduncertainty does not contain sudden outliers. These outliers are mostly caused bya coarse resolution that results in an approximation error.

To estimate the input f* as a function of the reference f', a combination ofthe resulting DBLA and Algorithm 3 is used. The comparison of the estimatedresults (×) and the transient simulations of the complete 3rd- and 5th-order ΣΔmodulators (-) lead to the following conclusions (Figure 5.18)

- In the case of the third order ΣΔ modulator, the estimated result largelypredicts the overall nonlinear trend.

- Simulations reveal that the fifth order ΣΔ modulator shows a more abrupttransition towards the unstable region due to more agressive filtering. TheDBLA is unable to predict this abrupt transition since it assumes explicitlythat the amplitude dependency is a smooth function that can be modelledby local polynomials.

To further evaluate the performance of the proposed method, we additionallyderive the SNR and SNDR curves for both ΣΔ modulators (Figure 5.19). Thebiggest advantage of the 5th order modulator is that the SNDR is significantlyincreased when compared to the 3rd order one. Analysis of the SNR curves revealsthat the linearised DBLA only provides a good prediction in the case of the 3rd

120


Figure 5.16.: The higher-order VCO-based ΣΔ modulator implements a dual-loopfeedback system [Park 09b]. It consists of a voltage-to-frequencyquantiser, a double feedback loop each with a different DAC (Return-to-Zero and Non-Return-to-Zero), some additional gain blocks, anda loopfilter (B).

−40−30

−20−10

0 02.5

57.5

1015

20

25

30

35

Input σU [dBVRMS ] Frequency Ω [MHz]

|GD

BLA(Ω

,σU)|

[dB

]

Figure 5.17.: The DBLA (=Ω = =f = =! = 2) of the open-loop VCO-based voltage-to-frequency quantiser used within the higher-order ΣΔ modulator.

121


−30

−20

−10

0

UnstableIn

pu

tσ U

[dB

VR

MS]

Third order

−20 −10 −3.08 0

−40

−30

−20

−10

0


Rel

ativ

eer

ror

[dB

]

−30

−20

−10

0

Unstable


Inp

ut

σ U[d

BV

RM

S]

Fifth order

−20 −10 −3.39 0

−40

−30

−20

−10

0


Rel

ativ

eer

ror

[dB

]

Figure 5.18.: The closed-loop simulated input RMS f* for both loopfilters (−) iscompared with results obtained from the proposed iterative scheme(×).

order modulator. Remark that in the unstable region, the estimated results forthe SNR curves become more variable due to the stochastic nature of the noiseestimate. Unfortunately, a comparison of the SNDR curves shows that there is asignificant difference between the estimated and simulated results. The simulatedSNDR curves start to degrade at lower RMS levels than anticipated with the DBLA.We believe this difference is related to the second assumption that is described onpage 115. Due to the feedback loop, the PSD of the quantiser’s input is shapeddynamically. The effect of this dynamic shaping on the behaviour of the nonlineardistortion is apparently significant and cannot be ignored. In the case of the5th order modulator, this effect even influences the SNR curve. In future work,we will investigate how the effect of this dynamic shaping can be included inthe estimation of the DBLA. The SNR and SNDR curves are subsequently usedto determine the modulator overloading level for both the simulations and theproposed iterative scheme (Table 5.2). Comparing these numbers shows that theestimated DBLA overloading levels are best determined using the estimated SNDRcurves.

122


−20 −10 −3.08 0

70

80

90

100


SNR

[dB

]Third order

−20 −10 −3.39 0

70

80

90

100


SNR

[dB

]

Fifth order

−20 −10 −3.08 0

70

80

90

100


SND

R[d

B]

−20 −10 −3.39 0

70

80

90

100


SND

R[d

B]

Figure 5.19.: Comparison between the simulated SNR and SNDR curves (−) andthe ones obtained with the proposed iterative scheme (×) and (×)for both loopfilter orders.

The major advantage of the proposed iterative scheme are time savings (Table 5.2).As with every modelling technique, an initial modelling time is required. Thisincludes the simulations of the open-loop quantiser and the DBLA estimation.This effort needs to be performed only once but results in a much more time-efficient prediction whenever a loopfilter is added. Time-wise, it is therefore morepractical to use the DBLA to predict the signal-dependency of ΣΔ modulatorsfor different loopfilter orders and structures. Once the designer is satisfied withthe performance of a specific loopfilter, more accurate simulations need to beperformed only once to verify the actual behaviour of the ΣΔ modulator withthe chosen loopfilter topology. Following this scheme allows for an overal timereduction by removing time-consuming simulations from the design flow.

123


Open-loopThird-order Fifth-order

quantiser

Ove

rloa

ding Simulations −3.08dBVRMS −3.39dBVRMS

DBLA - SNR −0.79dBVRMS −0.62dBVRMS

DBLA - SNDR −2.69dBVRMS −2.63dBVRMS

Max difference −2.29dB −2.01dB

Mod

eltime Simulations 50 × 2.3s

DBLA estimation + 43s

Total 158s

Simulation

s Simulation time 3.8s 3.9sDBLA interpolation 0.32sNo simulations × 50 ×50 ×50

Total 16s 190s 195s

Table 5.2.: This table provides an overview of the simulated/estimated modu-lator overloading levels and the time it took to obtain the results inFigures 5.17-5.19.

124

5.6. Summary

5.6. Summary

In this chapter, we develop a stability analysis technique for VCO-based ΣΔ

modulators excited by modulated signals.

In a first stage, we introduce the fast method. This method uses the LPM incombination with multisine excitations to estimate the BLA, the in-band noiseleakage, the in-band noise variance, and the in-band variance of the nonlineardistortion. Next, we apply the fast method to a VCO-based quantiser and comparethe outcome with the results obtained with time windowing techniques. Thiscomparison shows that the fast method reduces the simulation/measurement timewithout having to make concessions in terms of noise leakage reduction.

Building on the experience gained with the fast method, we propose a tech-nique for the estimation of the frequency- and amplitude-dependent DBLA ofan arbitrary nonlinear PISPO system. As a preliminary, the technique requiressimulated/measured input-output data of the system under test at different RMSlevels of a multisine excitation. The technique performs the following two stepssubsequently using the LPM

1. Removal of the frequency-dependent noise leakage.

2. Derivation of the DBLA of the nonlinear PISPO system, together with thecharacterisation of the noise and the nonlinear distortion.

In a final stage, we develop an iterative algorithm that uses the estimated DBLAof the internal VCO-based quantiser to perform a stability analysis of the ΣΔmodulator. The algorithm enables one to estimate the SNR and SNDR curves.From these curves it is then possible to predict the modulator overloading level.Throughout the chapter, both system-level and circuit-level examples are used toshowcase several aspects of the developed techniques.

125

6. Conclusions and future work

There is nothing like looking, if you wantto find something. You certainly usuallyfind something, if you look, but it is notalways quite the something you were after.

The Hobbit - J.R.R. Tolkien

The research that is described in this PhD thesis all started with an idea. This ideawas formed when I bumped into VCO-based ΣΔ modulators. As you can guess bynow, I wanted to use the knowledge about the Best Linear Approximation for theanalysis and design of these modulators. As with every research venture, it was notat all straightforward to obtain the eventual solution. Research is a long and twistyroad, that possibly contains bumps and detours. It translates an idea into findings,observations, solutions, and results. This chapter details the main conclusionsand the take-home messages. These are followed by possible directions for futureresearch to improve on and further develop the current findings.

127


6.1. Conclusions

The main goal of this thesis is the development of a technique that analysesthe stability of continuous-time VCO-based ΣΔ modulators that are excited bymodulated signals. The pursuit of this goal led to different research contributionswhich are summarised in the ’List of Publications’ on p. 145.

The first contribution is the development of an improved DF to describe saturationphenomena in chapter 3. We propose to describe the actual saturation by a modelthat fits a mathematical approximation to the measurements or simulations. Theadvantage of fitting is that no assumptions about the actual saturation behaviourneed to be made. Large errors are therefore avoided if these assumptions wouldbe violated. Using this mathematical approximation, an accurate sinusoidal-inputand random-input DF is obtained that enables time-efficient computation.

Next, we showed how the proposed DF is beneficial for the nonlinear analysisof circuits incorporating operational transconductance amplifiers. As it turnsout, saturation is the main phenomenon that dictates the nonlinear behaviour ofthese circuits. However, the DF theory cannot be straightforwardly applied for thenonlinear analysis of these circuits since they often consist of multiple saturatingelements. Therefore, two generally applicable iterative schemes were developedthat take these multiple nonlinear elements into account for Gaussian noise andsingle-tone sinusoids. The proposed DF in combination with these schemes wasthen applied successfully to a gm−C Tow-Thomas filter and a quadrature oscillator.In both cases, the nonlinear effects were accurately predicted while getting rid ofthe need for time-consuming simulations.

In chapter 4, we developed a local rational modelling technique to capture theresonant behaviour of strongly resonant systems adequately. These local mod-elling techniques enable one to efficiently estimate the BLA non-parametricallyfrom input-output data in the frequency domain. They also provide a separationof the steady-state response, the leakage contribution, the noise and the nonlin-ear distortion. Our developed technique uses the iterative Bootstrapped TotalLeast Squares (BTLS) estimator to consistently estimate a local rational model.Consistency is an important property since it ensures that both the estimate andthe corresponding uncertainty bound are correctly estimated. Several simulationexamples are used to highlight different aspects of the developed technique onMIMO systems. To illustrate the power of our method, we also devised a meas-urement campaign that resulted in the successful characterisation of the highlyresonant behaviour of the tailplane of a glider.

128

6.1. Conclusions

Furthermore, a numerically efficient scheme is proposed to apply the earlierdeveloped local modelling technique to large-scale MIMO systems. By takingexplicit advantage of the common-denominator parametrisation of the MIMOsystem, it is possible to mitigate the curse of dimensionality and compute the localBTLS estimate in a reasonable time. The effectiveness of this numerically efficientscheme is illustrated on a 100×100 MIMO system of order 100.

Building on the experience gained with the DF and the BLA theory, chapter 5details our efforts to accomplish the original idea and related goals described inSection 1.5. The resulting technique, which we call the Describing Best LinearApproximation, combines concepts from both theories and is largely based on thelocal modelling techniques introduced in chapter 4. The resulting model capturesboth the frequency- and the power-dependent behaviour of nonlinear PISPOsystems that are excited by random-phase multisines. The proposed techniquehas some distinct advantages:

- Unlike the DF theory, the model is retrieved from simulated/measured input-output data and hence does not require prior knowledge of the system understudy. This enables the application of the technique both to the system- andthe circuit-level.

- The local nature of the technique allows to seamlessly mix continuous-timesignals with discrete-time sequences. This is required for the analysis of thecontinuous-time VCO-based ΣΔ modulator.

- The frequency and power variations remain modest within a local window.Therefore, a low order model structure is sufficient to obtain a good model.This procedure is shown to require little to no user interaction.

- The technique additionally characterises the frequency- and power-dependentvariations of the noise and the nonlinear distortions. This further increasesthe power of this tool to assess the (non-)ideal performance of the VCO-basedΣΔ modulators.

Finally, an iterative scheme has been developed that performs a stability analysis.More specifically, the modulator overloading level is predicted starting from theDBLA of the nonlinear quantiser. This iterative scheme benefits greatly from theestimated local models that are reused for interpolation purposes. Again, severalexamples are used throughout chapter 5 to demonstrate the capabilities of thedeveloped techniques.

129


6.2. Future work

Research is never truly completed. Therefore, this section suggests several im-provements and extensions for the techniques developed during this PhD.

Further extension of local modelling techniques

The current local modelling techniques provide the choice between two modelstructures: the polynomial or the rational model. The former is preferred toconsistently model regions with limited dynamic frequency variation. The latterimproves the local estimation of resonances. A possible improvement combinesboth polynomial and rational models into one single model structure. This isfeasible when using improper rational functions. The degree of the numeratorpolynomial is then larger than the degree of the denominator polynomial. Thisapproach is based on the observation that these improper functions can alwaysbe rewritten as the sum of a polynomial and a proper rational function. Possibleextensions involve the

1. Application of the Local Bootstrapped Total Least Squares (LBTLS) tech-nique to consistently estimate these improper rational functions for large-scale Multiple-Input Multiple-Output (MIMO) systems.

2. Investigation of an efficient local model order selection procedure for theseMIMO systems.

Furthermore, pole-zero cancellations are to be avoided at all cost. As has beenshown in Section 4.3, they negatively impact the quality of the estimate. Onepossible solution to prevent these cancellations would be to improve the numericalconditioning of the BTLS estimation. Forsythe orthogonal polynomials can beused to this end [Rola 95].

130

6.2. Future work

Experiment design using adaptive sampling

The DBLA requires the acquisition of input-output data at a high number of differ-ent RMS levels to capture the behaviour of the system. Practically speaking, thisnumber is at least a few tens of levels. In the current implementation, we alwaysselected the RMS values on a linear or logarithmic grid. However, in practice weoften see that it is possible to leave some of these RMS values out without affectingthe quality of the estimate. This is especially true for RMS values that are located inthe linear region of the quantiser. The total simulation/measurement time wouldtherefore benefit from the use of an adaptive sampling scheme [Sack 89]. In ad-aptive sampling schemes, new samples are sequentially added to the sampling setin the regions where the accuracy of the model is too low. In this way, an optimalsample size can be reached iteratively. As a decision criterion, a whiteness test ofthe residuals of the local model estimates can be used to decide whether additionalsamples are needed to lower the model approximation error [Pint 12, Section 19.5].To make these adaptive sampling schemes efficient, an updating/recursive imple-mentation of the local modelling techniques needs to be developed. One possiblesolution would be to use an updating QR factorisation within the local modelsthat are affected by the addition of a sample.

DBLA estimation using non-stationary signals

Another approach to possibly speed up the simulation/measurement time usestime-varying estimation techniques [Pint 15]. Instead of applying different RMSvalues independently to the system under test, an alternative would be to varythe RMS value of the input multisine signal during the experiment (left plot inFigure 6.1). This transforms the input signal in a non-stationary process thatinherently contains leakage. In the case of a linearly increasing RMS value, thisleakage expresses itself as hyperbolic functions (called skirts) in the frequencydomain (right plot in Figure 6.1). With the help of the local modelling techniques,it is possible to model these skirts. The resulting skirt information can then beused to identify the magnitude-varying transfer function. Ideally, this transferfunction should correspond to the DBLA. Our hope is that future implementationof this technique will drastically decrease the simulation/measurement time ofthe DBLA.

131


−2

0

2

Time

Amplitude[V

]Time domain

−60

−40

−20

0

Frequency

Amplitude[dB]

Frequency domain

Figure 6.1.: Left plot: multiple periods of a random-phase multisine excitation (×)with a linearly increasing RMS value (−). Right plot: the time-varyingRMS creates skirts (×) in between the original excited frequencies (×).

Inclusion of additional variables within the DBLA estimation

During the stability analysis, we only include the frequency- and magnitude-dependency in the DBLA. It would be interesting to see how the DBLA varies as afunction of additional variables such as the bandwidth of the multisine excitation,the average value, etc. Also, the influence of other signal components on thelinearised DBLA can be studied. For example, we can include the magnitude andfrequency of a single-tone sinusoid superimposed on the multisine excitation. Itwould then be possible to verify the effect of possible limit-cycles on the signal-processing behaviour of the ΣΔ modulator.

132

A. Mixing continuous-time anddiscrete-time signals

Linear systems are intuitively described in the frequency domain by means oftheir Transfer Function (TF). This description is obtained starting from the Laplacetransform of the input/output signals for Continuous-Time (CT) signals and theZ-transform of these signals for Discrete-Time (DT) sequences. Unfortunately,we cannot simply choose one of these transformations and carelessly apply it tomixed-signal systems. The aim of this appendix is to provide the mathematicaltools that are needed for the rigorous analysis of mixed-signal systems. Thisappendix is based on [John 55, Oppe 99, Ortm 06].

The effect of sampling

Consider the CT signal H2 (C) that is sampled at a fixed time interval )B to retrievethe datasequence H(=) (Figure A.1)

H(=) = H2 (=)B) (A.1)

The Laplace transform .2 (B) of the CT signal H2 (C) is defined as

.2 (B) =+∞∫−∞

H2 (C) 4−BC dC (A.2)

while the Z-transform . (I) of the DT sequence is obtained as follows

. (I) =+∞∑==−∞

H(=) I−= (A.3)

133

A. Mixing continuous-time and discrete-time signals

Figure A.1.: The sampler and Digital-to-Analogue Convertor (DAC) implementthe transition from CT to DT and vice versa.

How are these two quantities .2 (B) and . (I) related to each other? To formulatean answer to this question, we take a look at the CT representation of the sampledsignal H(C) that can be obtained by multiplying H(C) with a Dirac comb Δ)B (C)

H(C) = H2 (C)Δ)B (C)

=

+∞∑==−∞

H2 (=)B) X(C −=)B)(A.4)

where X(C) is the Dirac-Delta function and • represents the sampling operation.Taking the Laplace transform of H(C) results in the following two equivalentexpressions for .(B)

.(B) =+∞∑==−∞

H(=)4−B)B (A.5)

=1)B

+∞∑==−∞

.2

(B+ 9=lB

)(A.6)

with lB = 2c/)B.

Combining the first equality (A.5) with the definition of the Z-transform (A.3)reveals the following relationship between the Laplace transform of the sampledsignal and the Z-transform of the corresponding DT sequence

.(B) = . (I) |I=4B)B or . (I) = .(B)B=ln(I)/)B (A.7)

134

This transformation I = 4B)B is called the impulse-invariant transformation andplays an important role in the CT to DT conversion [Gard 86].

The second equality (A.6) shows that the Laplace representation of a sampled-signal results in a periodic repetition of the original CT spectrum .2 (B). Thisphenomenon is directly related to the Nyquist-Shannon sampling theorem thatshould be fulfilled to avoid overlap in the spectrum .(B). A useful consequenceof (A.6) is that the • operator can be interchanged with a DT filter (I)

[. (B) (I) |I=4B)B ] = .(B) (I) |I=4B)B (A.8)

This property is used further on in this appendix to analyse ΣΔ modulators.

Digital-to-analogue conversion

Opposed to the sampler, the DAC conversion takes as an input a DT sequence andconverts it to a CT signal. There exist many different types of DACs [Ortm 06],each characterised by its own waveform IDAC (C) (Figure A.1). The most straightfor-ward one is the non-return-to-zero DAC that simply keeps the value of the inputsample constant during a whole sampling period )B. The functioning of the DACcan be understood as a repeated application of the DAC waveform for every inputsample. Mathematically, this boils down to a convolution in the time-domainthat corresponds with a multiplication in the frequency domain. Therefore, thebehaviour of the DAC can be naturally described by a filtering operation

- (B) = /DAC (B).(B) = /DAC (B) . (I) |I=4B)B (A.9)

where /DAC (B) is the Laplace transform of the DAC waveform. For example, thenon-return-to-zero DAC implements the following transfer function

/DAC (B) =1− 4−B) B

B(A.10)

135

A. Mixing continuous-time and discrete-time signals

Figure A.2.: The original CT ΣΔ modulator can be restructured to a modifiedrepresentation by relocating the loopfilter and sampler. The quantiseris replaced with the white noise approximation to retrieve a linearsystem representation.

136

The continuous-time ΣΔ modulator

In this section we want to derive the Signal Transfer Function (STF) and NoiseTransfer Function (NTF) of the CT ΣΔ modulator described in Figure 1.5. Duringour analysis we consider a single-loop feedback modulator as shown in Figure A.2.We assume that the quantiser can be modelled by a unity gain and a uniformlydistributed white quantisation noise source. To be able to analyse this systemusing the earlier introduced concepts, the loopfilter and sampler need to berelocated past the summation point to the input and feedback branch. Since (B)features a low pass characteristic, this modified representation reveals the implicitanti-aliasing present in the CT modulator.

Using (A.8) and the modified representation, it is possible to analyse the behaviourof the ΣΔ modulator in Figure A.2 in an elegant way. As a first step, the CT inputsignal is sampled, which results in the sampled spectrum

- (I) |I=4B)B = [ (B)* (B)] (A.11)

Furthermore, the feedback spectrum / (I) is obtained from . (I) using the follow-ing transformation

/ (I) |I=4B)B = [ (B) /DAC (B) . (I) |I=4B)B ]

= [ (B) /DAC (B)] . (I) |I=4B)B(A.12)

Using (A.12) it is possible to derive an equivalent DT filter eq(I) for the modulator

eq (I) = [ (B) /DAC (B)]B=ln(I)/)B (A.13)

The STF and NTF can then be derived using linear system theory

STF(B) = (B)

1+ eq (I)I=4B)B

(A.14)

NTF(I) =1

1+eq (I)(A.15)

137

B. Interpretation of theLBTLS cost function

In this appendix, we show a Local Bootstrapped Total Least Squares (LBTLS) costfunction interpretation under the assumption that the output noise is white. Thewhite noise assumption simplifies the output covariance as follows

C[ 9 , 9 ].(: + A) = f2

9 and C[8, 9 ].(: + A) = 0 (8 ≠ 9) (B.1)

where f29

is the total power of the white noise source that perturbs output 9 .

Since the cost function +LBTLS ()) is a quadratic funtion of the measurements [and _, it can be split into two parts

E+LBTLS ()) = E+LBTLS () ,[0,_0) +E+LBTLS () ,\. ) (B.2)

where E+LBTLS () ,[0,_0) depends entirely on the true input-output data [0 and_0. E+LBTLS () ,\. ) only contains the combined contributions of the output noiseand nonlinear distortions \. . From (B.2), it follows that the LBTLS estimatoris unbiased if E+LBTLS () ,\. ) is a )-independent constant. We now derive anexplicit expression for E+LBTLS () ,\. ) as a function of ) and V. . Consider forthis purpose the output-error framework. The original formulation of the costfunction (4.26) can be simplified to the following expression for E+LBTLS () ,\. )

E+LBTLS () ,\. ) =

=ΔΩ∑A=−=ΔΩ

=H∑9=1

=D∑8=1|F 9 (XA , )34=) |−1 |(XA ,)den) |2 I [ 9 , 9 ].

(: + A) | |)den | |

=ΔΩ∑;=−=ΔΩ

=H∑9=1|F 9 (X; , )34=) |−1 )denΣ 9 )den

(B.3)where )den is the subset of ) which contains the denominator coefficients only.)den is an estimate of )den that is obtained in the previous iteration. I. is the truecovariance matrix of the output noise. To derive (B.3), we assume that F 9 and Σ 9

139

B. Interpretation of the LBTLS cost function

are deterministic variables. Applying the white noise approximation (B.1) on Σ 9and F 9 results in

)denΣ 9 )den = f29 )

den (V

=) V = )den = f29 |(X; ,)den) |2 (B.4)

F 9 (X, )34=) = f29 |(X, )34=) |2 (B.5)

Substituting (B.4) and (B.5) in (B.3) results in

E+LBTLS () ,\. ) =

=ΔΩ∑A=−=ΔΩ

|(XA ,)den) |2

|(XA , )34=) |2

(=H∑9=1

I [ 9 , 9 ].(: + A)f29

)=ΔΩ∑

;=−=ΔΩ

|(X; ,)den) |2

|(X; , )34=) |2=H

| |)den | | =D (B.6)

Analysing (B.6) reveals that the LBTLS estimator uses (X, )den) to compensate forthe shaping introduced by the denominator polynomial (X,)den).

If we now assume that the output noise covariance truly originates from a whitenoise source (C[ 9 , 9 ]

.(: + A) = f2

9), then (B.6) transforms into

E+LBTLS () ,\. ) = | |)den | | =D (B.7)

Analysing the expression in (B.7), shows that E+LBTLS () ,\. ) in that case becomes)den-independent which corresponds to an unbiased estimator.

140

C. Vector Fitting applied toinput-output data

As an alternative to the Local Bootstrapped Total Least Squares (LBTLS) estimator,we want to incorporate prior knowledge about the pole locations in the localmodel estimation. The idea is to locally approximate BLA (Ω : ) and ! (Ω : ) withthe following model structure

BLA (Ω : + X) ≈=∑==0

1= X= +

=?∑==1

1=

Ω : + X− ?=

! (Ω : + X) ≈=!∑==0

2= X= +

=?∑==1

2=

Ω : + X− ?=

(C.1)

where 1=, 2= are the complex residues that belong to the complex pole ?=. Insteadof considering ?= as an unknown variable in the estimation, we here set the polelocation to a fixed value. The model then becomes entirely linear in the parameters.This decision, however, requires that the set of poles ?= is accurately known inadvance. Otherwise, an approximation error is unavoidably introduced.

In general, this pole information is not availabe a priori unless a profound know-ledge about the system is available through simulations or physical insight. Inorder to reliably extract the poles ?= without any prior knowledge, we basedourselves on the iterative pole relocation algorithm used within the Vector Fit-ting (VF) technique [Gust 99]. VF is a popular system identification tool thatreformulates the rational approximation of a transfer function 5 (B) as a linearleast squares problem. It iteratively improves this approximation by pole reloca-tion.

141

C. Vector Fitting applied to input-output data

In a nutshell, VF approximates 5 (B) with a rational model as follows [Gust 99]

5 (B) ≈=?∑==1

A=

B− ?=+ 3 + 4 B (C.2)

where B is the Laplace variable and 3, 4 are optional complex coefficients. VF usesa two-step procedure to obtain estimates for A=, 3, 4 and ?=. First, the polelocations ?= are derived by solving the following linear problem

f(B) 5 (B) = d(B) (C.3)

where

f(B) ==?∑==1

C=

B− @=+1

d(B) ==?∑==1

A=

B− @=+ 3 + 4 B

(C.4)

For a fixed set of initial poles @=, (C.4) is linear in its unknowns A=, C=, 3and 4. Equating (C.4) at a large set of different frequencies allows to generate anoverdetermined set of equations that can be solved with a linear least squaresestimator. As it turns out, the poles of the resulting rational approximation areexactly equal to the zeros of f(B) [Gust 98] and can be efficiently derived bycalculating the eigenvalues of the following matrix

?= = eigQ− IR) (C.5)

where Q is a diagonal matrix containing the initial poles @=, I is a column vectorof ones and R) is the row vector containing the residues C=. This pole estimatecan be further improved by iteratively substituting @= with the retrieved ?=and solving (C.5). Finally, the coefficients A=, 3 and 4 in (C.2) are calculatedby solving (C.2) with known pole locations ?= using a linear least squaresoptimisation.

Straightforward application of the traditional VF technique to our problem setting(Figure 4.3) results in a modelling error. This technique works directly with thefrequency response 5 (B) rather than with the input-output signals. Furthermore,the model structure (C.2) does not allow for the inclusion of the leakage contribu-tion !. It is thus not generally applicable for the signals under consideration here.To cope with the aforementioned issues, we adapt the VF technique such that?: can be locally derived from the measured input-output data. Starting from(4.1) and substituting (C.2) for both BLA (Ω : ) and ! (Ω : ) results in the following

142

model equations

. (: + A) =( =?∑==1

A=

Ω : + X− ?=+ 3 + 4 (Ω : + X)

)* (: + A)

+=?∑==1

A=

Ω : + X− ?=+ 3 + 4 (Ω : + X)

(C.6)

where all coefficients are again complex variables. To derive the pole locations?: , a similar linearised problem as in (C.3) is solved

f(Ω : + X). (: + A) = d(Ω : + X) (C.7)

where

f(Ω : + X) ==?∑==1

C=

Ω : + X− @=+1

d(Ω : + X) =( =?∑==1

A=

Ω : + X− ?=+ 3 + 4 (Ω : + X)

)* (: + A)

+=?∑==1

A=

Ω : + X− ?=+ 3 + 4 (Ω : + X)

(C.8)

Substituting (C.8) into (C.7) and using the frequencies that are present in a band-width ΔΩ around Ω : results in an overdetermined set of equations which canbe solved via linear least squares estimation. Afterwards, (C.5) can be used toretrieve an estimate of ?: . In a final stage the iteratively obtained pole locationsare substituted in (C.1) and a last linear least squares optimisation is performedto derive 1=, 1=, 2= and 2= with known poles.

143

List of Publications

Journal publications

D. Peumans, A. De Vestel, C. Busschots, Y. Rolain, R. Pintelon and G. Vandersteen.“Accurate estimation of the non-parametric FRF of lightly-damped mechanicalsystems using arbitrary excitations“. Mechanical Systems and Signal Processing, Vol.130, pp. 545-564, September 2019.

A. Cooman, P. Bronders, D. Peumans, G. Vandersteen and Y. Rolain. “DistortionContribution Analysis With the Best Linear Approximation“. IEEE Transactions onCircuits and Systems I: Regular Papers, Vol. 65, No. 12, pp. 4133-4146, December2018.

M. van Berkel, G. Vandersteen, H.J. Zwart, G.M.D. Hogeweij, J. Citrin, E. Westerhof,D. Peumans and M.R. de Baar. “Separation of transport in slow and fast time-scales using modulated heat pulse experiments (hysteresis in flux explained)“.Nuclear Fusion, Vol. 58, No. 10, p. 17, August 2018.

D. Peumans, C. Busschots, G. Vandersteen and R. Pintelon. “Improved FRFMeasurements of Lightly Damped Systems Using Local Rational Models“. IEEETransactions on Instrumentation and Measurement, Vol. 67, No. 7, pp. 1749-1759,July 2018.

D. Peumans and G. Vandersteen. “An Improved Describing Function With Ap-plications for OTA-Based Circuits”. IEEE Transactions on Circuits and Systems I:Regular Papers, Vol. 64, No. 7, pp. 1748-1757, July 2017.

145

List of Publications

Journal publication under review

R. Pintelon, D. Peumans, G. Vandersteen and J. Lataire. “Frequency ResponseFunction Measurements via Local Rational Modeling, Revisited“. Submitted toIEEE Transactions on Instrumentation and Measurement.

Conference publications

C. Busschots, A. Keymolen, H. Maes, D. Peumans, J. Pattyn, G. Vandersteen and J.Lataire. “Adaptive excitation signals for low frequency FOT“. 15th InternationalSymposium on Medical Measurements and Applications (MeMeA), Bari, Italy, 1-3June 2020.

D. Peumans, P. Bronders and G. Vandersteen. “Noise leakage suppression in VCO-based ΣΔ-modulators excited by modulated signals“. 16th International Conferenceon Synthesis, Modeling, Analysis and Simulation Methods and Applications to CircuitDesign (SMACD), Lausanne, Switzerland, 15-18 July 2019.

D. Peumans, A. Cooman and G. Vandersteen. “Analysis of Phase-Locked Loopsusing the Best Linear Approximation“. 13th International Conference on Syn-thesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design(SMACD), Lisbon, Portugal, 27-30 June 2016.

Conference posters

M. van Berkel, G. Vandersteen, H. J. Zwart, G. M. D. Hogeweij, J. Citrin, E. West-erhof, D. Peumans and M. R. de Baar. “Mathematical equivalence of non-localtransport models and broadened deposition profiles“. 61st Annual Meeting of theAPS Division of Plasma Physics, Florida, USA, 21-25 October 2019.

D. Peumans, A. De Vestel, C. Busschots, Y. Rolain, R. Pintelon and G. Vandersteen.“Exposing resonances using the Bootstrapped Total Least Squares estimator“. 27thERNSI Workshop in System Identifcation, Cambridge, U.K., 23-26 September 2018.

P. Bronders, D. Peumans, J. Lataire and G. Vandersteen. “Enhancing EnvelopeTracking Power Amplifiers’ Understanding using the wonderful properties ofLPV“. 27th ERNSI Workshop in System Identifcation, Cambridge, UK, 23-26 Septem-ber 2018.

146

D. Peumans, C. Busschots, G. Vandersteen and R. Pintelon. “Can BootstrappedTotal Least Squares improve the FRF estimate?“. 26th ERNSI Workshop in SystemIdentifcation, Lyon, France, 24-27 September 2017.

Scientific honors

Best paper award runner-up at the 16th International Conference on Synthesis,Modeling, Analysis and Simulation Methods and Applications to Circuit Design(SMACD 2019).

147

List of Abbreviations

AAF Anti-Alias Filter

ADC Analogue-to-Digital Converter

ADS Advanced Design System

AIC Akaike Information Criterion

BLA Best Linear Approximation

BTLS Bootstrapped Total Least Squares

CMOS Complementary Metal Oxide Semiconductor

CT Continuous-Time

DAC Digital-to-Analogue Convertor

DBLA Describing Best Linear Approximation

DF Describing Function

DFT Discrete Fourier Transform

DOF Degrees Of Freedom

DT Discrete-Time

GSVD Generalised Singular Value Decomposition

HB Harmonic Balance

IIT Impulse Invariant Transformation

LBTLS Local Bootstrapped Total Least Squares

149

List of Abbreviations

LMF Left Matrix Fraction

LPM Local Polynomial Method

LPV Linear Parameter-Varying

LRM Local Rational Method

LTI Linear Time-Invariant

MIMO Multiple-Input Multiple-Output

ML Maximum Likelihood

NTF Noise Transfer Function

OFDM Orthogonal Frequency Division Multiplexing

OSR Oversampling Ratio

OTA Operational Transconductance Amplifier

PDF Probability Density Function

PISPO Periodic-In Same Period-Out

PLL Phase-Locked Loop

PSD Power Spectral Density

RIDF Random-Input Describing Function

RMS Root Mean Square

SIDF Sinusoidal-Input Describing Function

SISO Single-Input Single-Output

SNDR Signal-to-Noise and Distortion Ratio

SNR Signal-to-Noise Ratio

STD STandard Deviation

STF Signal Transfer Function

TF Transfer Function

VCO Voltage-Controlled Oscillator

VF Vector Fitting

VXI VME eXtensions for Instrumentation

WGTLS Weighted Generalised Total Least Squares

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Analogue‐to‐digital conversion plays an essential role in the digital interconnected world we live in today. It provides the means to link our physical, analogue world to digital computers. In recent years, VCO‐based Sigma‐Delta modulators attracted growing interest as they effectively narrow the boundary between the analogue and digital world.

Stability is a vital aspect that needs to be analysed during the design phase of Sigma‐Delta modulators. Instability disrupts the proper functioning of the modulator and therefore degrades the envisioned performance. Designers want to have access to tools that accurately predict in advance for which signals and amplitude levels potential unstable behaviour manifests itself.

In this thesis, a linear approximation theory is developed that combines concepts from both the Describing Function and the Best Linear Approximation theory. The developed theory captures both the amplitude‐ and the frequency‐dependent behaviour of nonlinear systems that are excited by modulated signals. Application of this theory to VCO‐based Sigma‐Delta modulators results in an approach that allows to time‐efficiently predict the amplitude‐dependent stability behaviour.

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BLABASED DESIGN AND ANALYSIS OF VCOBASED …VCO-based Sigma-Delta Modulators SUPERVISOR Prof. Dr. ir. Gerd Vandersteen Vrije Universiteit Brussel, Belgium MEMBERS OF THE JURY Prof