Multicarrier Modulation for Broadband Return Channels · Multicarrier Modulation for Broadband Return Channels in ... quadrature amplitude modulation ... power PSD of signal prolate

Multicarrier Modulation for Broadband Return Channels in Cable TV Networks

Von der Fakultät Informatik, Elektrotechnik und Informationstechnik der Universität Stuttgart zur Erlangung der

Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung

Vorgelegt von

Stephan Pfletschinger

aus Abtsgmünd

Hauptberichter: Prof. Dr.-Ing. J. Speidel Mitberichter: Prof. Dr.-Ing. U. Reimers

Tag der mündlichen Prüfung: 27.02.2003

Institut für Nachrichtenübertragung der Universität Stuttgart

2003

Die vorliegende Arbeit ist während meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Institut für Nachrichtentechnik der Universität Stuttgart entstanden.

Bedanken möchte ich mich an erster Stelle bei meinem verehrten Lehrer, Herrn Prof. Dr.-Ing. Joachim Speidel, der mir nicht nur diese Arbeit ermöglichte, sondern mich auch durch seine ständige Gesprächsbereitschaft und seine daraus resultierenden, zahlrei-chen Anregungen unterstützte. Seine fachkundige Förderung, sein reges Interesse und seine ständige Gesprächsbereitschaft haben wesentlich zum Gelingen dieser Arbeit bei-getragen.

Herrn Prof. Dr.-Ing. Ulrich Reimers danke ich herzlich für die Übernahme des Mitbe-richts.

Bei allen ehemaligen Studien- und Diplomarbeitern bedanke ich mich für ihren Einsatz und die erfolgreiche Zusammenarbeit, wobei ich besonders die Arbeit von Herrn Ger-hard Münz hervorheben möchte.

Weiterhin danke ich allen Institutsmitarbeitern, die mich und meine Arbeit in vielfälti-ger Weise unterstützt haben.

3

Contents

Acronyms and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Kurzfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Cable TV Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 The Classical Coaxial CaTV Network and the Modern HFC Network . . . . . . 172.2 Simulation of an Existing CaTV Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Simulation Model with all Network Components . . . . . . . . . . . . . . . . . 202.2.2 Channel Models in Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Noise and Ingress in the Return Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Modelling and Simulation of Ingress Noise . . . . . . . . . . . . . . . . . . . . . 252.3.2 Broadband Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Narrowband Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.4 Impulse Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Conclusions for the Modulation Scheme in the Return Channel . . . . . . . . . . 322.5 A Laboratory Model of a Modern Interactive CaTV Network . . . . . . . . . . . . . 33

2.5.1 The Hardware Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.2 Creation of Software Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 Simulation of the Complete Network and Comparison with

Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.6 Existing Solutions for Return Channel Transmission . . . . . . . . . . . . . . . . . . 44

2.6.1 System Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6.2 Multiple Access in the Return Channel . . . . . . . . . . . . . . . . . . . . . . . . 462.6.3 Upstream Physical Layer Specification . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Principles of Multicarrier Modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 The Continuous-Time System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Elementary Impulses, Ambiguity Function and Orthogonality . . . . . . . 503.2.2 Rectangular Pulse Shaping and Guard Interval . . . . . . . . . . . . . . . . . 52

3.3 The Discrete-Time System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Filterbank Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.2 DFT-Based Implementation with Cyclic Prefix . . . . . . . . . . . . . . . . . . 573.3.3 Implementation of MCM with DFT and Polyphase Filterbank . . . . . . . 60

3.4 Multicarrier Offset QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.1 Continuous-Time System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.2 Pulse Shaping with Square-Root Raised Cosine Impulse. . . . . . . . . . 653.4.3 Discrete-Time System Model and Polyphase Implementation . . . . . . 66

5

6 Contents

3.5 Further Aspects of Multicarrier Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.1 Peak-to-Average Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.2 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.3 Equalisation for Multicarrier Systems with Pulse Shaping. . . . . . . . . . 71

4 Multicarrier Modulation with Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1 Optimisation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Localisation in Time and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1.2 Out-of-Band Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Pulse Shaping Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.1 Gaussian Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.2 Time-Limited Cosine Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.3 Hermite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.4 The Isotropic Orthogonal Transform Algorithm . . . . . . . . . . . . . . . . . . 784.2.5 Extended Lapped Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.6 Further Approaches and Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Optimised Impulses for Multicarrier Offset-QAM . . . . . . . . . . . . . . . . . . . . . . . 855.1 The Prolate Spheroidal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1.1 The Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . . . . . . . . 855.1.2 The Discrete Prolate Spheroidal Sequences. . . . . . . . . . . . . . . . . . . . 87

5.2 The Optimisation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.1 Orthogonality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2.3 Calculation of the Expansion Coefficients . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Simulation Results for Narrowband Interference . . . . . . . . . . . . . . . . . . . . . 96

6 Subcarrier Allocation and Bitloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1 Channel Capacity of a Single-User Channel . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1.1 The Waterfilling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1.2 Single-User Bitloading Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2 The Multiuser Waterfilling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 OFDM with Multiple Access: OFDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3.1 An Efficient Waterfilling Algorithm for OFDMA . . . . . . . . . . . . . . . . . 1076.3.2 Subcarrier Allocation with Bitrate and Power Constraints . . . . . . . . . 113

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.1 The Equivalent Lowpass Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.2 Hermite Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.3 Calculation of the Peak-to-Average Power Ratio. . . . . . . . . . . . . . . . . . . . . 124

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Acronyms and Symbols

Acronyms

ADSL asymmetric digital subscriber lineAM amplitude modulationATM asynchronous transfer modeAWGN additive white Gaussian noiseBER bit error ratioBFDM biorthogonal frequency division multiplexBK Breitbandkommunikation, broadband communicationCaTV cable televisionc.d.f. cumulative distribution functionCDMA code division multiple accessCNR channel gain to noise ratioCSI channel state informationCSMA/CD carrier sense multiple access with collision detectionDAB digital audio broadcastDAVIC digital audio-visual councilDCT discrete cosine transformDFT discrete Fourier transformDMT discrete multitoneDWMT discrete wavelet multitoneDOCSIS data over cable service interface specificationDPSS discrete prolate spheroidal sequencesDVB digital video broadcastDVB-C digital video broadcast, cableDVB-RCC digital video broadcast, return channel on cableDWMT discrete wavelet multitoneELT extended lapped transformESB Erweiterter Sonderkanalbereich, enhanced special frequency bandETSI European Telecommunication Standardisation InstituteFEC forward error correctionFDE frequency domain equaliserFDM frequency division multiplexFDMA frequency division multiple accessFIR finite impulse responseHE headendHFC hybrid fibre coaxICC International Conference on CommunicationsIDFT inverse discrete Fourier transformIEEE Institute of Electrical and Electronics Engineers

7

8 Acronyms and Symbols

IOTA isotropic orthogonal transform algorithmIP internet protocolISDN integrated services digital networkITU International Telecommunications UnionJSAC Journal on Selected Areas in CommunicationsLAN local area networkMAC media access controlMC multi-carrierMCM multicarrier modulationMCNS Multimedia Cable Network SystemMC-OQAM multicarrier offset quadrature amplitude modulationMCSIS multicarrier system with impulse shapingNE Netzebene, network layerOFDM orthogonal frequency division multiplexOFDMA orthogonal frequency division multiple accessOQAM offset quadrature amplitude modulationOSB Oberer Sonderbereich, upper special frequency bandPAPR peak-to-average power ratiop.d.f. probability density functionPR perfect reconstructionPSD power spectral densityPSTN public switched telephone networkPSWF prolate spheroidal wave functionQAM quadrature amplitude modulationQPSK quaternary phase shift keyingRSV revised singular vectorRV random variableSC single-carrierSER symbol error ratioSNR signal to noise ratioSTB set-top boxSVD singular value decompositionTDE time domain equaliserTDM time division multiplexTDMA time division multiple accessUMTS universal mobile telecommunications systemUSB Unterer Sonderbereich, lower special frequency bandVDSL very high bitrate digital subscriber lineVCR video cassette recorderVoD video on demandVoIP voice over IPVTC vehicular technology conferenceWLAN wireless LANWSSUS wide sense uncorrelated scattering

Acronyms and Symbols 9

Symbols

Fourier transform linear convolution

circular convolution

rounding towards the nearest smaller integer

attenuation coefficient, roll-off factor

phase coefficient

propagation coefficient

SNR gap

unit impulse

two-dimensional unit impulse

delta distribution

eigenvalues, multipliers of equivalent channel

noise power

signal power

PSD of signal

prolate spheroidal wave function

FDE coefficients

carrier frequency spacing

Nyquist frequency

wave parameters

A subcarrier allocation matrixnumber of bits per QAM-symbol on subchannel for user ambiguity function

⊗

x

x[ ]+ x for x 0>0 else

=

α

β

γ a jβ+=

Γ

δ n[ ] 1 for n 0=0 for n 0≠

=

δ n m,[ ] δ n[ ] δ m[ ]⋅=

δ t( )

λn

σr2

σs2

Φx ω( ) x t( )

ψk t( )

ψµ

∆ω

ωN

ai bi,

bu ν,ν u

A t ω,( )


channel capacity

integer division

, time dispersion, frequency dispersion

expectation operator

energy per QAM-symbol on subchannel

centre frequency

impulse response of receive filter

Fourier transform

inverse Fourier transform

c.d.f. of RV

impulse response of transmit filter

Hilbert transform

transfer function

, continuous-time, discrete-time impulse response

Laplace-transform of current

imaginary unit

base 2 logarithm

modulo operator

upsampling factor

set of integer numbers

set of non negative integer numbers

number of subcarriers

normal distribution with mean and variance

noise sample function

elementary impulse of the second kind

Q-function

C

divN n( ) nN----=

Dt Dω

Ε x t( )

Eν ν

fc

f t( )

F x t( ) x t( )e jωt– td∞–

∞

∫=

F 1– X ω( ) 12π------ X ω( )ejωt ωd

∞–

∞

∫=

FX x( ) X

g t( )

H x t( )

H ω( )

h t( ) h n[ ]

I p( )

j 1–=

ld x( ) log2 x( )=

modN n( ) n N divN n( )⋅–=

M

N

N0

N

N µ σ2,( ) µ σ2

n t( )

pν µ, t( )

Q x( ) 12π

---------- et2

2---–

tdx

∞

∫12---erfc x

2-------

= =

Acronyms and Symbols 11

rectangular impulse of duration 1

, , elementary impulse

set of real numbers

set of positive real numbersS scattering matrix

scattering parameters

unit step function

sinc function

symbol period, duration of OFDM-symbol

sampling period

guard time

, channel gain to noise ratio

voltage and its Laplace-, Fourier-transform

twiddle factor

complex conjugate

lower case boldface fonts or underscore for vectors

X upper case boldface fonts for matrices

matrix transpose

Hilbert transform of

, continuous-time signal, discrete-time signal

, real part, imaginary part

discrete prolate spheroidal sequence

set of integer numbers

characteristic impedance

double-sided z-transform

rect t( ) 1 for t 1 2⁄<0 else

=

rν µ, t( ) rd t( ) rν µ, k[ ]

R

R+

Si j,

s n[ ] 1 for n 0≥0 for n 0<

=

si x( ) x( )sin x⁄ for x 0≠1 for x 0=

=

T

TA

TG

T ω( ) Tu ν,

u t( ) U p( ) U ω( ), ,

w j2πM------–

exp=

x*

x λ,

XT

x t( ) H x t( ) = x t( )

x t( ) x n[ ]

x' Re x = x'' Im x =

vk n[ ]

Z

ZL

Z g n[ ] g n[ ]z n–

n ∞–=

+∞

∑=


Abstract

Cable TV networks are currently evolving into interactive multimedia networks. For the transmission on the return path, which suffers from reflections and ingress noise, both robust and bandwidth efficient modulation schemes are required. In this thesis, multi-carrier modulation schemes are investigated and improvements of standard multicarrier systems are proposed in order to cope better with the characteristic channel impairments of the return path. A multicarrier system with optimum impulse shaping is derived which not only increases the spectral efficiency but also reduces the susceptibility to nar-rowband noise. The CaTV return channel is a multiuser channel over which a high number of users access one central station. Multicarrier modulation offers the possibility of adapting the access and modulation scheme to the channel conditions in such a way that the optimum solution given by the multiuser waterfilling theorem of information theory can be closely approximated. Two novel computational efficient subcarrier alloca-tion algorithms are presented in detail. Although originally intended for the use in CaTV networks, these algorithms are not limited to this application, and are also well-suited for wireless multiple-access communication systems.

Kurzfassung

Kabelfernsehnetze entwickeln sich immer mehr zu interaktiven Multimedianetzen. Für die Übertragung im Rückkanal, die durch Reflexionen und Störungseinkopplungen beeinträchtigt wird, müssen sowohl robuste als auch bandbreiteneffiziente Modulati-onsverfahren eingesetzt werden. In dieser Arbeit werden Mehrträgermodulationsver-fahren untersucht und Verbesserungen von bestehenden Verfahren vorgeschlagen, welche die Modulation besser an die Eigenschaften des Rückkanals anpassen. Es wird ein Mehrträgerverfahren mit optimaler Impulsformung hergeleitet, welches neben der Erhöhung der spektralen Effizienz auch die Empfindlichkeit gegenüber Schmalbandstö-rern verringert. Der Kabelfernsehrückkanal ist ein Vielfachzugriffskanal, über den viele Teilnehmer auf eine zentrale Instanz zugreifen. Mehrträgerverfahren bieten die Möglich-keit, das Zugriffs- und Modulationsverfahren so an den Kanal anzupassen, dass die aus der Informationstheorie bekannte optimale „multiuser-waterfilling“-Lösung approxi-miert wird. Zwei neuartige Trägeraufteilungsalgorithmen, die eine effiziente Implemen-tierung ermöglichen, werden im Detail vorgestellt. Obwohl ursprünglich für den Einsatz in Kabelfernsehnetzen konzipiert, sind diese Algorithmen nicht auf diese Anwendung beschränkt, sondern eignen sich auch hervorragend für drahtlose Vielfachzugriffsver-fahren.

13

1 Introduction

With the advent of digital multimedia services, the demand for high bitrates on access networks has increased significantly. The large increment in data rate that is delivered to and from residential customers is driven by the extension of the internet and the transi-tion from analogue to digital TV. In order to meet the customers’ demand for high bitrates, efficient and economic solutions have to be realised. In contrast to wide-area networks, cost is crucial in access networks. In wide-area networks, state-of-the-art transmission systems with optical fibre can be deployed because due to their high utili-sation the cost of implementation amortises rapidly. In contrast, in access networks utili-sation is low and thus cost effective solutions have to be found. Most network providers chose therefore to built upon existing infrastructures. Nowadays, two communication networks exist that reach nearly every home: the public switched telephone network (PSTN) and the cable television (CaTV) network [1-3]. While the first has always been a fully bidirectional network, the latter is basically a distribution network which is evolv-ing into a fully interactive multimedia network [4-9]. A third network that reaches even more homes than the PSTN, but has not been considered a communication network until recently, is the power supply network. Recent investigations aim at using the power line as an access network [10-14]. While it has been shown that bidirectional digital transmis-sion is possible, power line communication systems are unlikely to reach competitive bitrates for a high number of users. Other possibilities for last mile access include wire-less access systems, satellite links and optical wireless systems.

The most extended broadband access technology for residential customers in Germany is ADSL (asymmetric digital subscriber line), which builds upon the existing twisted pair telephone line and delivers bitrates of several Mbit/s. Cablemodems can be consid-ered a competing technology as they offer comparable bitrates and similar pricing. They use as transmission medium the CaTV network which has to be upgraded to provide a return path for bidirectional transmission. The bandwidth allocation for upstream and downstream is highly asymmetrical. This is mostly due to compatibility issues: CaTV plants in service have allocated almost all available bandwidth to broadcast services. They have been upgraded by extending the frequency range and adding a relatively nar-row frequency band for the upstream at lower frequencies. Although this allocation was chosen due to compatibility necessities, it fits well to the expected type of traffic which is generated by interactive services like webbrowsing and video on demand (VoD) and hence is highly asymmetric.

Although the demand for bandwidth has been ever increasing, a key figure can be iden-tified: a digital video stream in today’s resolution and quality has about 4 Mbit/s. Although we cannot imagine future applications or services, we may state that video delivers probably the highest bitrate a human can ʺconsumeʺ and it seems reasonable to consider this bitrate as sufficient for most future applications, including real-time games.

15

16 Introduction

Access via the CaTV network facilitates the convergence between the TV and the compu-ter networking world. As the complete TV signal is fed into the cablemodem in addition to the signal for internet access, with appropriate processing a seamless integration of TV and internet should be possible, e.g. watch TV in a browser window or open a browser window on the TV screen. While the CaTV networks impose no restrictions to a possible convergence between TV and internet services, current terminal equipment is not ready to handle both services in a seamless way, and thus the approach actually taken by network providers is contrary to convergence: for interactive TV a set-top box (STB) is used while for internet access a cablemodem is connected to the same cable. Both devices use the same technology to connect to the CaTV network and may thus merge into one device in the future.

Moreover, CaTV networks can offer a combination of broadcast and interactive services. Mobile network operators, who are currently planning the introduction of 3rd generation mobile communications, are beginning to realise that purely interactive infrastructure will be too costly for services which are requested by a great number of subscribers and that the solution for future personal communication systems will consist in an intelligent combination of broadcast and interactive services and the corresponding technical infra-structure [15]. This also applies for residential customers, making bidirectional CaTV networks a proper transmission medium for future personal communication services.

Although modern CaTV plants dispose of bandwidths of nearly 1 GHz, the bandwidth for the return channel is limited to only 30 – 60 MHz, which has to be shared by all users in the same network segment. Thus, bandwidth is a scarce resource in the upstream channel and transmission schemes with high spectral efficiency have to be employed. In this thesis, spectral efficient multicarrier modulation schemes are developed which extend the existing techniques in various aspects.

In the next chapter we describe the characteristics of the CaTV return channel. Therefore we provide simulation models for the channel and develop a mathematical model to describe and simulate the noise and interference, which is a major impairment on the return channel. A laboratory model of a modern hybrid-fibre coax (HFC) plant is described and corresponding software models are derived.

Chapter 3 introduces the basics of multicarrier modulation (MCM) and lays the ground for the following chapters which deal with pulse shaping in multicarrier systems.

Different approaches for pulse shaping and their motivations are presented in chapter 4 in a common framework and in chapter 5 a new impulse shape for multicarrier offset-QAM is developed.

In chapter 6, the extension of multicarrier modulation to a multiuser environment is con-sidered. Two algorithms are developed which allocate the subcarriers to the users in an optimal way.

2 The Cable TV Network

2.1 The Classical Coaxial CaTV Network and the Modern HFC Network

In Germany the public CaTV network is organised in different hierarchical layers as illustrated in Fig. 2.1 [1-3]. The whole network, beginning from the point of production and ending with the TV receiver in the subscriber’s home, can be decomposed into the distribution network including the network layers NE 1 and NE 2, the public access net-work NE 3 and the private in-home or community access network NE 4, where NE stands for ʺNetzebeneʺ, i.e. network layer. Sometimes, a fifth layer is defined informally as NE 5, referring to the in-home wiring from the TV wall outlet to the TV set or VCR which is often installed in a non-professional manner and thus constitutes the weakest link in the distribution network [16].

In the following, when speaking of the CaTV access network we refer exclusively to the layer NE 3. This part of the network is the key element in the transition from a dumb analogue distribution network to an analogue/digital bidirectional multimedia access network which offers services like fast Internet access, VoD, cable telephony, video con-ferencing and possibly other broadband services [4-9]. Not only high bandwidth and a return channel for interactive communication are crucial for the successful operation of such a multimedia network, but also the requirements on reliability increase dramati-cally. In a simple TV distribution network, in many cases the only fault detection ʺsys-temʺ is the user who calls the cable operator in case of failure. In a multimedia network which offers telephony among other bidirectional services, high reliability must be guar-anteed by a network management system which continuously monitors the components of the network and isolates faulty branches [17].

productionof TV andradio signals

distribution networktransmission of TVsignals from place ofproduction to satelliteor wide area networkand from there tocable headends

access networkdistribution ofsignals to homes

private networkin-house network

studio TV switchingpoint or studio

headend publicaccesspoint

TV walloutlet

NE 1 NE 2 NE 3 NE 4

Fig. 2.1 Definition of the network layers in the German cable TV network.

17


The classical German CaTV network, which was operated first by the Deutsche Bundes-post and then by the Deutsche Telekom, has a tree-and-branch architecture and is com-posed of coaxial cables with different attenuations, line and distribution amplifiers and passive drops, as shown in Fig. 2.2. As the frequency range extends to 450 MHz, the net-work was named BK-450 (BK, Breitbandkommunikation, broadband communication). In this network, up to 23 amplifiers are cascaded, which allows to connect several thou-sand subscribers to one headend (HE). The trunk lines A and B constitute the active part while the distribution line C and the drop cable D make up the passive part of the net-work.

As shown in Fig. 2.3, the frequency range spans from 47 to 450 MHz which accommo-dates 28 7-MHz channels and 18 8-MHz channels among analogue FM radio and (for-merly) digital radio. Twelve of the 18 8-MHz channels are typically used for digital TV according to the DVB-C (digital video broadcast - cable) standard [18, 19]. The BK-450 is still the prevailing access network for CaTV in Germany, but with the sale of the network to private investors it is actually being upgraded to offer a broader downstream fre-quency range and a return channel for interactive services.

In the year 2000, about 270 million homes worldwide were connected to CaTV. In Ger-many the number was about 21 million, which is more than 50% of all homes. In the US, even 65% of the homes have cable, while in Europe the percentage is little more than 25%, but the penetration rate is increasing rapidly [20].

Headend

PrivateNetwork

Drop Cable D

AmplifierCabinet

Trunk Line A

Trunk Amplifier

Bridge Amplifier

Distribution Amplifier

Trunk Line B

PublicAccessPoint

Distribution Line C

max. 409 m

max. 409 m

max. 280 m

max. 20 m

Fig. 2.2 All-coax CaTV network: BK-450.

The Classical Coaxial CaTV Network and the Modern HFC Network 19

In order to transform the BK-450 to an interactive multimedia network, two upgrading steps have to be undertaken: (1) the extension of the downstream frequency range to typically 862 MHz to accommodate more TV channels and bandwidth for other services and (2) the provision of a return channel which is generally located in the frequency band from 5 to 35 MHz. As this range seems rather small, many suppliers offer return channel components for the range 5 – 65 MHz. In this case, the three TV channels K2 – K4 in the VHF I band have to be relocated to higher frequencies. The upgrading of the existing BK-450 requires significant investments due to several reasons. First, the coaxial cables used nowadays have high attenuation at high frequencies and therefore new amplifiers with strong predistortion are required. Second, amplifiers with high linearity in the broad frequency band from 47 to 862 MHz are not easy to manufacture and due to linearity requirements the number of cascaded amplifiers is limited. In order to provide a return channel, all active components must be furnished with frequency splitters and return amplifiers. The reliable operation of the return channel is challenging because of ingress and interference problems. Noise is coupled into the network at the subscribers’ premises where, due to poor shielding and missing line terminations, the in-house wir-ings act like antennas. This noise accumulates in the tree-and-branch structure and adds up in the headend. This effect is often referred to as noise funneling [21]. Because of this ingress and the scarce bandwidth in the return path, the number of subscribers is critical. For this reason the number of users in a bidirectional CaTV network is usually reduced to a few hundred.

Modern CaTV networks are implemented as HFC networks [22-25] as depicted in Fig. 2.4. The broadcast signals are distributed via optical fibres to the fibre nodes, where they are converted into electrical signals and distributed via the coaxial part of the net-work. The coaxial part only contains about 6 to 8 cascaded amplifiers, alleviating the lin-earity requirements on the amplifiers. Also, as one fibre node only serves some hundred homes instead of thousands as in the BK-450, the ingress problem is less severe and the requirements on the return path amplifiers are less stringent.

The comparison of the topologies of the traditional BK-450 with the modern HFC raises the question of how to upgrade existing networks in an economically reasonable man-ner. A cost-effective solution was presented in [26, 27] where an upgrade of all amplifiers with frequency splitters and return path amplifiers was proposed. This is probably the

47 6887.5 108

111 174 230 300302 446355

f / MHz

returnchannel

3 8 channels 10 channels10 channels 18 channels

7 MHz 7 MHz 7 MHz channel spacing 8 MHz

digital / analogue TV

7 MHz

VHF I

TV

USB

analogue TV

VHF III ESB (Hyperband)OSB

FM

radio

Fig. 2.3 Channel allocation in the BK-450.


cheapest solution as it takes advantage of all of the existing infrastructure. The drawback of this approach is the high number of subscribers that share the same return channel, resulting in small data rates per user and ingress problems. These problems are inherent to the tree-and-branch structure of the BK-450 and can only be solved by reducing the network size. An interesting approach of virtually reducing the node size by switching off idle subscribers at the public access point was proposed by Alcatel [28, 29]. Another solution is to exploit only a part of the network that is close to the subscriber and feed one of the last amplifiers in the cascade by optical fibre, thus adding a optical overlay network [30]. This approach breaks up the network into smaller segments and thus solves the ingress and bandwidth problem. Although much more costly, this procedure promises to be more future-proof and is the preferred way chosen by most network operators.

2.2 Simulation of an Existing CaTV Network

2.2.1 Simulation Model with all Network Components

For the investigation about the upgrading possibilities of an existing CaTV network, a public access network which is in operation in a suburb of Stuttgart has been modelled with the simulation tool Ptolemy [31] in [32, 27]. The network corresponds to the BK-450 guidelines and its structure is as illustrated in Fig. 2.2. It covers an area of 14 km2 and passes 13798 homes, of which 10212 are connected. The simulated part of the network, consisting of one trunk line (A line) with all its branches connects 3370 subscribers. The simulation model is shown in Fig. 2.5 where the source is connected to the public access

HEfibrenode

bidirectionalamplifier

drop

publicaccesspoint

Fig. 2.4 HFC network with return path amplifiers.

Simulation of an Existing CaTV Network 21

point and the sink which saves the arriving impulses is connected to the beginning of the trunk line, thus simulating the return channel. The downstream channel to an arbitrary subscriber can be simulated as well with the same model by exchanging the impulse source and the sink. The simulation method for calculating the transfer function is based on the scattering parameters of the network components and is performed in frequency domain. The exact procedure will be explained in detail in section 2.5.2. The scattering parameters of most components have been measured [27] and where this was not possi-ble, specified data has been used to derive the S-parameters.

The simulation models have been used to elaborate a proposal for the incorporation of return channel components into a BK-450 network in a cost-effective way. Details about the required upgrades in the amplifier cabinets as well as a cost estimate of different upgrade strategies have been worked out in [27]. Both a passive and an active solution for the return path have been followed. Fig. 2.6 shows the resulting transfer functions for two subscribers which are located at the beginning and the end of the same C-line when active return path amplifiers are used. Due to the high number of reflections that occur in the branches of the network, the transfer functions have many ripples.

2.2.2 Channel Models in Time Domain

In order to perform efficient simulations, all signals and transfer functions have been simulated in the equivalent lowpass channel, as detailed in the appendix. The simulation with S-parameters yields the transfer function at the discrete frequencies

,

where is the frequency spacing, is the centre frequency of the band-pass signal, is the sampling frequency and is the number of samples in the sim-ulation. Thus the sampling time and the number of samples are already fixed. In [33] many transfer functions for different network configurations have been generated. As these simulations of the whole network are time consuming, the resulting transfer functions have been saved for future use. For subsequent simulations in time domain, however, the desired sampling time may differ from that used in the S-parameter simulation and therefore either the transfer function or the impulse response has to be interpolated. These two possibilities are described in the following subsections.

Interpolation in Frequency Domain

First, the continuous-frequency function is obtained by interpolation, either lin-ear or with cubic splines [34], from the frequency samples , then resampling with the desired sampling time gives

,

f1 n[ ] fcfA12

-------– n ∆f1⋅+= n 0 … NA1 1–, ,=

∆f1 fA1 NA1⁄= fcfA1 NA1

TA1 1 fA1⁄= NA1

H 2πf( )H1 n[ ]

TA2 1 fA2⁄=

H2 n[ ] H 2πn∆f2( )= n 0 … NA2 1–, ,=


Fig. 2.5 Simulation model of a part of an existing BK-450 network. At the top right the public access point with the connected subscriber is located. The headend (which is the sink for the simulation of the return channel) is located at the left.

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Noise and Ingress in the Return Channel 23

Now, the impulse response can be calculated with an inverse discrete Fourier transform (IDFT) from .

Interpolation in Time Domain

With this method, first the impulse response is determined by IDFT; then the con-tinuous-time function is obtained by ideal-lowpass filtering with the cutoff frequency

:

and the impulse response is obtained by sampling this function with the desired sampling time :

Fig. 2.7 shows the impulse responses that have been calculated with interpolation from a given transfer function as an example.

2.3 Noise and Ingress in the Return Channel

With the transfer function or the impulse response, we can describe the linear distortions in the return channel. Besides these impairments, the influence on the return path trans-

Fig. 2.6 Transfer functions of a near-HE and a far-HE subscriber connected to the same C-line.

Hf()

5 10 15 20 25 30 35 40 45−10 dB

−8 dB

−6 dB

−4 dB

−2 dB

0 dB

2 dB

4 dB

far−HE subscribernear−HE subscriber

f MHz⁄

h2 n[ ]H2 n[ ]

h1 n[ ]

fA1 2⁄

h t( ) TA1 h1 n[ ]δ t nTA1–( ) 1

TA1--------- si πt

TA1---------

n∑⋅ h1 n[ ] si π

TA1--------- t nTA1–( )

n∑= =

h2 n[ ]TA2

h2 n[ ] h nTA2( ) h1 m[ ] si π nTA2TA1--------- m–

m∑= =


mission caused by interference and noise plays a dominant role [21, 35-37]. The ingress noise mainly couples into the network at the subscribers’ premises, i.e. in the layers NE 4 and the in-home wiring. While German CaTV networks in the NE 3 are equipped with high-quality components, the in-house networks are often implemented with low-cost devices which are installed by amateurs. Due to poor shieldings, loose contacts, mis-matches, etc. noise and interference couple into the network. As this happens in all branches of the network, these influences accumulate while they propagate through the tree-and-branch network and sum up in the HE. This noise funneling effect can turn a CaTV plant into a hostile environment for return path transmission.

In a first approach, the noise can be divided into internal and external noise. Noise that is generated in the network itself stems from thermal noise in the amplifiers, from noisy switched power supplies and from connected TV sets. The external noise sources are manifold. They may originate from electrical household appliances, computers, car igni-tions, amateur and short wave radio, corona noise from high-voltage lines and couple into the network mainly at the users’ wirings. While the internal noise sources can be quantified relatively easy, the external sources are hardly predictable. Furthermore, these influences are strongly time-varying, both in the range of milliseconds as well as in the course of hours and even days, and they are different from one region to another. Details about these dependencies can be found in [38, 39].

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05

0.1

0.15

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05

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0

0.05

0.1

0.15

0 2 4 6 8 10 12 14 16 18 20−0.1

−0.05

0

0.05

0.1

0.15

Fig. 2.7 Complex envelope of the impulse response of a far-HE subscriber, obtained by IDFT and interpolation from the transfer function. (a) Interpolation in frequency domain, (b) interpolation in time domain.

t/µs

(a) (b)

Re

ht()

Imh

t()

t/µs

t/µs

Re

ht()

Imh

t()

t/µs


For the modelling of the noise it is advantageous not to consider the sources but to base the classification on the statistical properties. Roughly, we may distinguish three types of noise:

• broadband noise which emanates mainly from internal noise sources like thermal noise of amplifier components, power supplies and TV sets, but also from a variety of external sources which couple into the network. As this broadband noise is composed of the sum of a great number of independent noise sources, it may be assumed Gaus-sian according to the central limit theorem. Because nearly all components have a fre-quency dependent transfer function, this noise will not be white.

• narrowband interference, like e.g. shortwave or amateur radio. These mainly external interferers affect only a limited frequency range and are relatively stationary. The affected frequency range however depends on local circumstances and may vary from time to time.

• impulse noise which is also caused by external influences. This type of ingress is gen-erated by lightning, electrical machines, car ignitions, etc. These perturbations occur very irregularly and last typically for a very short time, but disturb a broad frequency range.

2.3.1 Modelling and Simulation of Ingress Noise

For the simulation of the described noise phenomena, the time domain representation of the channel is preferred. As illustrated in Fig. 2.8, we represent the linear distortions with the impulse response and model the ingress as additive noise with the sample function , which is the sum of three sample functions. As already mentioned above, this decomposition into broadband, narrowband and impulse noise is independent from the origin of the noise and is not the only possibility to pro-ceed.

h t( )n t( ) nb t( ) nn t( ) ni t( )+ +=

h(t)x(t) y(t)

n(t) = nb(t) + nn(t) + ni(t)

Fig. 2.8 Channel model in time domain with ingress noise.


2.3.2 Broadband Noise

This type of noise can be modelled as coloured Gaussian noise. In Fig. 2.9, the results of ample meas-urements conducted by R.P.C. Wol-ters and described in [38] are illustrated. The measurements have been performed at different loca-tions and in different networks, but in all cases the power spectral den-sity (PSD) shows a similar behav-iour, especially the decreasing for higher frequencies. The protruding peaks in the PSD curves can be attributed to narrowband interfer-ers which are treated separately in the following subsection. The meas-urements have been conducted in two major European cities in the Netherlands and in Belgium. Net-work A is an HFC network with 100 000 connections which was built in the early eighties, all cables are underground and the amplifiers are located in cabinets at street level. Network B is a tree-and-branch net-work with 70 000 connections, which dates back to the late sixties. Cables and amplifiers are mostly mounted on the façades of houses. As one might expect, the noise level in the older network is significantly higher.

For the simulation in the time domain, a sample function with the desired statistical properties has to be generated. As the simulation is carried out in the equivalent lowpass system, the sample function will be complex-valued. Given that and are

distributed random processes, the sample function can be written as

, (2.1)

where has to be specified according to the desired PSD. Routines for generating the random sequences and are available in practically all programming environ-ments, so there is nothing more to be said about how to obtain them. The function

constitutes a complex, white Gaussian random process, provided that real and imaginary part are statistically independent. The probability density function (p.d.f.) of is also Gaussian, as linear filtering of a Gaussian process yields again a Gaussian process [40, 41]. Please note that this is not true in general for other p.d.f.s. In [42], the filter was designed to match the PSD of Fig. 2.9. The transfer function of

Fig. 2.9 Measured noise PSD in the HE (reproduced from [38]) (a) network A, city centre (b) network A, residential quarter (c) trunk line of network A, d) network B

f MHz⁄

pow

er s

pect

ral d

ensi

ty

nb t( ) rr t( ) ri t( )N 0 σ2,( )

nb t( ) rr t( ) jri t( )+( ) hb t( )=

hb t( )rr t( ) ri t( )

rr t( ) jri t( )+

nb t( )

hb t( )


this filter and the estimated PSD of the simulated broadband noise are illustrated in Fig. 2.10.

2.3.3 Narrowband Noise

In the noise spectrum determined in [38] and shown in Fig. 2.9, among the broadband noise, some narrow-band interferers stick out. These are mostly caused by shortwave radio stations and couple into the CaTV network at poorly shielded junctions or at the sub-scribers’ premises. As this noise mainly consists of AM radio, these influences are easy to model: like illustrated in Fig. 2.11, white (Gaussian) noise is fil-tered with a lowpass filter and subsequently modu-lated to the appropriate carrier frequency. Note that because simulation takes place in the equivalent lowpass system, the bandpass centre frequency has to be subtracted

0 10 20 30 40 50 60-70 dB

-60 dB

-50 dB

-40 dB

-30 dB

-20 dB

0 10 20 30 40 50 60-90 dB

-80 dB

-70 dB

-60 dB

-50 dB

-40 dB

f MHz⁄

10 lg

Hb

ω()

f MHz⁄

PSD

of n

b(t)

Fig. 2.10 (a) Magnitude transfer function of the filter , (b) Estimated PSD of the sample function .

hb t( )nb t( )

(a)

(b)

hLP(t)r(t) ns(t)

Fig. 2.11 Block diagram for the simulation of narrowband noise. The filter is a narrowband lowpass.

hLP t( )

ej ω1 ωc–( )t

ωc


from the interferer frequency . For the filter, a lowpass with a cutoff frequency of 30 kHz was used, which yields a very sharp peak in the PSD curve as shown in Fig. 2.12.

2.3.4 Impulse Noise

Beside the broadband and narrowband noise which are approximately stationary over long time periods, we have to deal with short-time, wideband impulsive interference of possibly significant noise power. These disturbances are typically caused by automobile ignitions, electrical machines, arc welding, lightning strikes, etc. These noise sources are by nature difficult to describe in a mathematical framework and to represent properly in a simulation model.

ω1

5 10 15 20 25 30 35 40 45 50 55-140 dB

-120 dB

-100 dB

-80 dB

-60 dB

-40 dB

-20 dB

0 dB

17.9 17.92 17.94 17.96 17.98 18 18.02 18.04 18.06 18.08 18.1

-140 dB

-120 dB

-100 dB

-80 dB

-60 dB

-40 dB

f MHz⁄

PSD

of n

n(t)

Fig. 2.12 Estimated PSD of a narrowband interferer.


One approach to identify the impulse noise and describe it with a simple mathematical model was presented by Li et al. [43] who con-ducted extensive measurements on rural HFC network and recorded the waveforms of impulse noise with an automated measuring sys-tem. The return channel range of the considered plant was 5 to 42 MHz, as is common in the U.S. The cable plant consisted of several fibre nodes, each with about 500 connected homes. The data was gathered from individual fibre nodes as well as from a combina-tion of 31 nodes. Measurements were carried out at various times of day and no user data was present on the return channel, i.e. all the gathered noise data was solely due to noise and interference.

With the automated measuring equipment, a set of 75 impulses was gathered, and out of this set two representative waveforms were derived. First the time delay between the impulses was determined with the cross correlation function. Then, the impulses were aligned in time and placed into the columns of a matrix. By application of singular value decomposition (SVD), two dominating waveforms have been extracted. These two vec-tors, called Revised Singular Vectors (RSV), are used to represent the whole set of meas-ured impulses. In Fig. 2.13, the Fourier transforms of these RSVs and are shown. The impulse noise waveforms are approximated by

(2.2)

where and are dependent random variables (RV). The p.d.f. of is given in [43, 42] and for holds:

(2.3)

Fig. 2.13 Transfer functions of the characteristic impulses.

f MHz⁄H

12,

f()

0 5 10 15 20 25 30 35 40 45−25 dB

−20 dB

−15 dB

−10 dB

−5 dB

0 dB

5 dB

10 dBH

1H

2

h1 t( ) h2 t( )

ai t( ) x1 h1 t( ) x2 h2 t( )⋅+⋅=

x1 x2 x1x2

x2 1.9895 x– 12 0.1688 x1– 0.0147–=


Fig. 2.14 shows the block diagram for the genera-tion of a sample function for the impulse noise. A source generates a train of delta impulses

which are weighted with the random varia-bles and . For each delta impulse a new value of is generated and is calculated according to (2.3). The two impulse trains are fed into the FIR filters with the characteristic impulse responses and , and summation of the filter outputs finally yields the sample function

.

The impulse source emits delta impulses each , where is an exponentially distrib-uted random variable with the p.d.f.

(2.4)

The mean distance between two impulses is for this p.d.f. . A random generator for the exponential distribution is not available in most programming environments, but it can be derived easily from a uniform random generator: provided that is a RV uni-formly distributed in [0,1), with the cumulative distribution function (c.d.f.)

(2.5)

Then the RV

(2.6)

is exponentially distributed with the p.d.f. (2.4). This can be shown easily with the c.d.f.:

for :

for q.e.d.

The resulting c.d.f. corresponds to the p.d.f. (2.4).

Fig. 2.15 shows a sample function and the corresponding PSD for a chosen mean interarrival time of . The impulses vanish within few microseconds. The noise

ni(t)

h1(t)

h2(t)

x2(t)

x1(t)d(t)

Fig. 2.14 Block diagram for the generation of impulse noise.

ni t( )

d t( )x1 x2

x1 x2

h1 t( ) h2 t( )

ni t( )

Ti Ti

pT ti( )0 for ti 0<

λeλti–

for ti 0≥

=

T 1 λ⁄=

X

FX x( ) P X x≤[ ]0 for x 0≤x for 0 x 1≤ ≤1 for 1 x≤

= =

T 1λ--- ln X( )–=

FT t( ) P T t≤[ ]=

t 0≥ FT t( ) P 1λ--- ln X( ) t≤– P X e λt–≥[ ] 1 P X e λt–<[ ]– 1 e λt––= = = =

t 0< FT t( ) 0=

ni t( )T 10 µs=


spectrum shows clearly that the impulse noise affects the whole frequency range of the return channel.

The powers of the noise components have to be adapted for the simulation of a specific CaTV network. As the mathematical models of the noise components presented above are based on different studies, their power levels cannot be compared quantitatively. In the simulation model [42], the power of each component (broadband, narrowband, and impulse noise) can be adjusted individually and hence be adapted to a real-world CaTV plant.

Fig. 2.15 Impulse noise: (a) sample function (b) estimated PSD.

f MHz⁄

t µ⁄ s

impu

lse

nois

e sa

mpl

e fu

nctio

n

PSD

of n

i(t)

(a)

(b)

0 20 40 60 80 100 120 140 160 180 200−0.3

−0.2

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0

0.1

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−80 dB

−70 dB

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−50 dB

t µ⁄ s

Re

n it()

Imn i

t()


2.4 Conclusions for the Modulation Scheme in the Return Channel

Based on the knowledge about the channel characteristics regarding linear distortions and the noise profile, some important conclusions for the modulation format to be used on the return channel can be drawn. The following reasonings will be detailed in later chapters. First it is important to note that the CaTV channel is an essentially time invari-ant channel, like most cable channels are. This constitutes a substantial difference to mobile channels and indicates that most of the concepts considered there may not be appropriate. E. g. the channel estimation for a cable channel reduces to a rather simple task: during initialisation of the system, the channel parameters are estimated and are relatively stable during transmission, so that slow channel tracking will be sufficient.

We have observed that due to reflections the transfer function has many ripples. This means for a single carrier (SC) system that powerful equalisation has to be used in order to achieve high data rates. As is apparent from Fig. 2.9, the noise also varies strongly over frequency, which makes it difficult to equalise the channel without fomenting the noise at the receiver side. Therefore, a transmission system with one single carrier for the entire return channel and hence time division multiple access (TDMA) seems very diffi-cult, as was also concluded in [21]. A system with several carriers and combined fre-quency/time division multiple access has been elaborated by standardisation working groups [44-46]. While this approach mitigates these problems to a great extent, other drawbacks still persist. We will treat some details in section 2.6. Recently, a noise cancel-lation technique has been presented which aims at reducing the degradation due to nar-rowband interference [47].

A modulation scheme which can cope well with this channel characteristic is multicar-rier modulation with adaptive constellation size. A multicarrier scheme partitions the broadband channel into a great number of flat subchannels, which renders the complex channel equalisation into a simple one-tap equalisation per subchannel. Because of the frequency dependence of the transfer function and the noise, the subchannels have dif-ferent signal-to-noise ratios (SNR). This can be taken into account by applying different constellation sizes on the subchannels. For low SNR, a robust scheme like QPSK is appropriate while for high SNR higher order QAM schemes are more suited. Hence, multicarrier schemes can be well adapted to channels with frequency-dependent SNR. A necessary condition is that neither the transfer function nor the noise power vary over time. This is fairly true for the broadband and the narrowband noise, but not for the impulse noise as visible in Fig. 2.15. As these influences are not localised in the time or frequency domain, the modulation scheme alone can offer no remedy, but channel cod-ing is the proper antidote. With forward error correction (FEC), the interference due to the impulse noise can be combated.

A Laboratory Model of a Modern Interactive CaTV Network 33

2.5 A Laboratory Model of a Modern Interactive CaTV Network

2.5.1 The Hardware Setup

At the Institute of Telecommunications, a small CaTV plant was set up in order to allow measurements of the network components, carry out transmission experiments and ver-ify simulation methods. The hardware setup can thus be considered a complement for theoretical investigations and computer simulations. The laboratory plant was set up as a model to resemble a typical state-of-the-art coaxial CaTV access network with return channel. The model therefore represents the coaxial part of an HFC network or a small all-coax network. Modern amplifiers have been used which dispose of a forward fre-quency range of 47 to 862 MHz and a return range of either 5 to 35 MHz or 5 to 65 MHz, selectable by plugable modules [48]. The smaller range of 5 to 35 MHz was selected for the laboratory model because the forward path should carry a standard BK-450 signal whose spectrum starts at 47 MHz.

In order to have the possibility of testing the multiuser access in the return channel, two cable lines have been implemented for the time being, with the option to add more lines if need be. The plant counts eight amplifiers, of which six are connected in cascade. This is typically the maximum for modern HFC plant, in contrast to the classical BK-450 where up to 23 amplifiers can be cascaded. The limited number of amplifiers is due to linearity requirements in the downstream and the reduced number of subscribers per network segment. The typical cable lengths between two amplifiers are ca. 200 – 400 m. As in the laboratory model thinner (and thus cheaper and more flexible) cables with higher attenuation have been used, the cable lengths could be shortened to 50 – 100 m, maintaining the typical attenuation between the amplifiers. Fig. 2.16 shows a photo of the hardware setup, and Fig. 2.17 shows the block diagram of the plant. For easy access, the amplifiers are mounted on a wooden panel on a mobile rack.

In order to create simulation models, the scattering parameters of all amplifiers have been measured with a network analyser [49] and stored on a hard disk for later process-ing.

2.5.2 Creation of Software Models

Description of a Two-port with Scattering Parameters

As scattering parameters (S-parameters) can be obtained directly from measurements with a network analyser, they are the preferred parameters for system characterisation at radio frequencies. These parameters are also well suited for simulation models [27]. In the following, we describe the steps from the S-parameter measurements to the simula-tion of the system.


Instead of input and output voltages as defined in Fig. 2.18, a two-port can also be described with arriving (incident) and departing (reflected) wave parameters , which are defined as [50, 51]:

, (2.7a)

, (2.7b)

For termination with the characteristic impedance at input and output, this simplifies to

, , , (2.8)

Fig. 2.16 Test bed of a CaTV network.

AMP33 AMP25

AMP25 AMP25 AMP25 AMP33

AMP25 AMP25

1 25 6 7 8

3 4

50 m

1

2

3

4 5 6 7

50 m

50 m

50 m 50 m100 m

50 m

Headend

89/93 dBµV@ 47/450 MHz

100/105 dBµV@ 47/450 MHz

100/105 dBµV

@ 47/450 M

Hz

user 1

user 2

A

B

C

Fig. 2.17 Block diagram of the laboratory network with voltage levels for the downstream.

ai bi

a1U1 ZL I1⋅+

2 ZL

----------------------------= a2U2 ZL I2⋅+

2 ZL

----------------------------=

b1U1 ZL I1⋅–

2 ZL

---------------------------= b2U2 ZL I2⋅–

2 ZL

---------------------------=

a1U1

ZL

----------= a2 0= b1 0= b2U2

ZL

----------=


The departing and arriving waves are related via the S-matrix:

(2.9)

The transfer function of the two-port is given as

(2.10)

Analytical Model and Measurement Results for the Coaxial Lines

A coaxial cable with length can be described as a symmetrical, reciprocal two-port [51] with the S-matrix

(2.11)

Measurements in the frequency range up to 900 MHz showed that the coaxial cables are in fact well matched to 75Ω, so that the reflections are negligible. The transmission coeffi-cient is frequency dependent and can be closely approximated for a coaxial transmis-sion line by

(2.12)

The four parameters , , , can be determined by measuring the S-parameters and the length of the cable and subsequent parameter fitting with (2.11).

All network components, i.e. amplifiers and cables, have been measured individually with a vector network analyser which is capable of determining the S-matrix at 2001 dis-crete frequencies in one measurement step. In the return channel, the frequency range was chosen from 585.9375 kHz to 49.4140625 MHz with 2001 equidistant frequency sam-ples. For the forward channel, the range from 10.546875 MHz to 889.453125 MHz was used.

The attenuation coefficient and the phase coefficient can be determined from the set of measured S-parameters:

, (2.13)

SU0 U1

ZL

ZL

I1 I2

U2

a1 a2

b1 b2

Fig. 2.18 Two-port with wave parameters.

b1

b2 S11 S12

S21 S22 a1

a2

=

H ω( )U2 ω( )U1 ω( )---------------- S21 ω( )= =

l

S 0 γl–( )expγl–( )exp 0

=

γ

γ α jβ+= A0 f A1f A2+ +( ) j A0 f B0f+( )+≈

A0 A1 A2 B0

α β

α fi( ) 1l--- S12 fi( )ln–=

β fi( ) 1l--- S12 fi( )( )arg–=

i∀ 1 … 2001, ,=


Inserting this into (2.12) yields an over determined linear equation system for the real-valued coefficients , , :

(2.14)

This equation system provides 2001 equations for three variables. Alternatively, the unknown variables in (2.14) can be considered as coefficients of a second-order polyno-mial in , and an algorithm for polynomial curve fitting (available in Matlab) can be applied. This leaves us with the coefficient which can be found by solving the over determined equation system

(2.15)

This equation system is solved with an algorithm which minimises the squared error. Now all four coefficients are determined and together with the cable length, the S-matrix is given for all frequencies according to (2.11) and (2.12). In Fig. 2.19 where the measured and the approximated -parameter are drawn, the two curves are hardly distiguisha-ble. Thus, the description of the coaxial cables with the four parameters , , , closely matches the real transfer function.

Measurement of the Amplifiers

The S-matrices of all eight amplifiers have been measured for the return and forward path. As can be seen from Fig. 2.17, two different types are in use (AMP25 and AMP33), one is equipped with two output ports. The frequency responses are nearly identical, they only differ in the gain factor, which has been adjusted during installation in order to achieve input levels of about 94 dBµV in the downstream.

A0 A1 A2

α f1( ) A0 f1 A1f1 A2+ +=

α f2( ) A0 f2 A1f2 A2+ +=

α f2001( ) A0 f2001 A1f2001 A2+ +=

…

fB0

β f1( ) A0 f1 B0f1+=

β f2( ) A0 f2 B0f2+=

β f2001( ) A0 f2001 B0f2001+=

…

S12A0 A1 A2 B0


0 5 10 15 20 25 30 35 40 45 50−3 dB

−2.5 dB

−2 dB

−1.5 dB

−1 dB

−0.5 dB

0 dBmeasuredcalculated

0 5 10 15 20 25 30 35 40 45 50−25 π

−20 π

−15 π

−10 π

−5 π

0measuredcalculated

Fig. 2.19 Measured and analytical transfer function for line 6.

f MHz⁄

S 12ar

cS 12

()

f MHz⁄


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1|S

11(f)|

|S22

(f)|

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.8 MHz

2.1 MHz

3.6 MHz

34.2 MHz

37.8 MHz

S11

S22

Fig. 2.20 Reflection factor at the amplifier input and output. (a) Complex locus curves, (b) magnitudes of as a function of frequency.Sii

f MHz⁄

Re Sii

ImS ii

(a)

(b)

f


In the return path the gains are adjusted to compensate for the cable loss, i.e. to obtain a unity gain system. In Fig. 2.20 the reflection factors for the input and output of an amplifier are drawn. The S-parameter locus in Fig. 2.20 (a) exhib-its a strong impedance mismatch, but Fig. 2.20 (b) reveals that this occurs only in the transition region of the splitter filter and in the passband the amplifier is well matched to the 75Ω cable. Fig. 2.21 shows the S-parameters and , i.e. the magnitude of the transfer function for the up- and downstream. The stop-band attenuation of the amplifiers is about 60 dB and 70 dB, respectively, thus mutual interference between for-ward and return channel is kept to a minimum. The gain in the return channel is ca. 2 dB which accounts for the attenuation of the cables. The lower band edge of the first TV channel in the forward path is located at 47 MHz, which means that the filter transition band consumes 12 MHz of valuable bandwidth.

Especially for the measurements in the return channel, it was important to adjust the voltage level of the network analyser to a value within the linear range of the amplifiers. In order to check the linearity, some simple meas-urements with a sinusoidal signal at various voltage levels and frequencies have been carried out. For input volt-ages above 100 mV (100 dBµV), the amplifiers reach saturation as is visible in Fig. 2.22. Such high voltage levels thus have to be avoided in the return channel, otherwise strong distortion due to nonlinear effects would result.

0 10 20 30 40 50−80 dB

−70 dB

−60 dB

−50 dB

−40 dB

−30 dB

−20 dB

−10 dB

0 dB

10 dB

|S12

||S

21|

Fig. 2.21 Transfer functions for forward ( ) and return ( ) path of amplifier 6.

S21 S12

f MHz⁄

S ijf()

S12 S21

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

160

5 MHz15 MHz25 MHz35 MHz

Fig. 2.22 Input-output characteristics of the amplifiers at different frequencies.

u1 mV⁄

u 2m

V⁄


Concatenation of Two-ports and Simulation with S-Parameters

There are various possibilities to determine the S-matrix of the entire network from the S-matrices of the components: (1) the S-parameters can be transformed into transmission parameters; (2) the matrix can be obtained by successive multiport reduction; or (3) with a calculation by simulation of all propagating waves. The first possibility is not followed further as it involves much calculations that give no further insight. The successive mult-iport reduction combines two concatenated two-ports to one until only one two-port is left, which represents the whole network. The S-matrix of the concatenation of the two-ports with S-matrices and is given by

(2.16)

A more descriptive way to obtain the transfer function of the whole network is the simu-lation of the wave parameters , , where the two-ports are also characterised by their S-parameters [27]. Fig. 2.23 shows the simulation model of two two-ports connected together. The input-output transfer function of the combined two-port shall be deter-mined by simulation as described in the following. Like in Fig. 2.18 we assume matched termination at input and output, which signifies for the wave parameters that the input

at the right is fed with zeros while the output at the left is left open and ignored. The S-parameters depend on frequency and after measuring, for each parameter a set of values is available. In order not to complicate the notation without need, in the subse-quent derivations we consider all parameters at one frequency. Generalisation to parame-ter vectors is straightforward as the calculation is identical for all frequency samples.

With this simulation method, the input signal is set to one in the first step and zero in the following, these steps account for the reflections in the network. Thus the input sequence in Fig. 2.23 is the unit impulse . In Fig. 2.24 (c), the Simulink [52] model of the two-port is illustrated in detail: the block mainly consists of adders and multipliers. The out-put is delayed by one cycle because otherwise the circuit would contain an algebrai-cal loop [52]. Strictly speaking, the signals in Fig. 2.23 do not represent the wave parameters itself as defined in (2.7a,b), but are a sequence which is a decomposition into

S 1( ) S 2( )

S 11 S22

1( ) S112( )⋅–

-------------------------------- S111( ) det S 1( ) S11

2( )⋅– S121( ) S12

2( )⋅

S211( ) S21

2( )⋅ S222( ) det S 2( ) S22

1( )⋅–

⋅=

ai bi

a2 b1

S(1) S(2)

S

a1(1) a1

(2)a1

δ[n]x1

y1

b1(1) b1

(2)

b1

a2(1) a2

(2)

x2

b2(1) b2

(2)

z-1

Fig. 2.23 Simulation of concatenated two-ports based on wave parameters.

δ n[ ]

b1


the direct and the reflected parts. The index n represents the simulation steps. For the signals in Fig. 2.23

(2.17)

(2.18)

(2.19)

hold. Eq. (2.19) in (2.17) gives

This is a first-order difference equation which can be solved by the z-transform1:

(2.20)

has a pole at . For stability reasons, it must hold:

(2.21)

Inserting (2.20) into the z-transform of (2.18) gives the output signal

(2.22)

The inverse z-transform yields:

(2.23)

where stands for the discrete step function. We obtain the transfer function by sum-mation of all sequence elements, i.e. the summation of the direct and the reflected parts:

(2.24)

In order that this series converges, (2.21) must be fulfilled. The result is in accordance with (2.16). Eq. (2.24) gives the value of the transfer function for one frequency sample. For the calculation of all frequency samples, the scalar signals in Fig. 2.23 have to be replaced by vectors.

Thus, we have shown that the simulation method with propagating waves yields the same results as the calculation by successive multiport reduction. The advantage of the

1. Note that the z-domain in this case has nothing to do with frequency domain, nor is n related in any way to a time domain index.

x1 n[ ] S211( ) a1 n[ ] S22

1( ) x2 n[ ]⋅+⋅=

y1 n[ ] S212( ) x1 n[ ]⋅=

x2 n[ ] S112( ) x1 n 1–[ ]⋅=

x1 n[ ] S211( ) a1 n[ ] S22

1( ) S112( ) x1 n 1–[ ]⋅ ⋅+⋅=

X1 z( )S21

1( )

1 S221( ) S11

2( ) z 1–⋅ ⋅–--------------------------------------------=

X1 z( ) z0 S221( ) S11

2( )⋅=

S221( ) S11

2( )⋅ 1<

Y1 z( ) S211( ) S21

2( ) zz S22

1( ) S112( )⋅–

-------------------------------⋅ ⋅=

y1 n[ ] s n[ ] S211( ) S21

2( ) S221( ) S11

2( )⋅( )n⋅ ⋅ ⋅=

s n[ ]

H S21 y1 n[ ]n 0=

∞

∑ S211( ) S21

2( ) S221( ) S11

2( )⋅( )n

n 0=

∞

∑⋅ ⋅S21

1( ) S212( )⋅

1 S221( ) S11

2( )⋅–--------------------------------= = = =


simulation is that the simulation model can be set up in the same way as the block dia-gram of the network and thus resembles the topology of the real network.

2.5.3 Simulation of the Complete Network and Comparison with Measurement

With the measured S-parameters of all components, a Simulink model of the complete network can be set up. The model can be made very descriptive by hierarchically com-bining the network elements to functional blocks. In the lowest layer, in Fig. 2.24 (c), an amplifier or a cable is built up with elementary blocks (adders, multipliers, delay ele-ments); these are combined to network branches which in turn are used by the top level description (see Fig. 2.24 (a)) to form the complete network. To calculate the transfer function, at the input a unit impulse source has to be connected and at the output the arriving values have to be summed up. In the Simulink model there is no source or sink specified, instead connections to Matlab are defined. This allows to use the same model for simulations of transfer functions between arbitrary connection points. If the source and sink had to be defined in the block diagram (as it is the case in e.g. Ptolemy [31]), we would need separate simulation models for the forward and the return channel.

Besides the simulation of the components, the S-matrix of the network as a whole has been measured. Note that this is normally not possible for a network in service. In Fig. 2.25 (a) the measured frequency responses for a subscriber at connection point C in Fig. 2.17 are shown. The measured and the simulated frequency response for the return channel are compared in Fig. 2.25 (b). The curves differ for most frequencies less than 1 dB. There are various effects that explain the discrepancy between the two curves: (1) inaccuracies of the measurement and the analytical approximation of the cables accumu-late, (2) in the simulation, not all reflections are considered as the sum in (2.24) is trun-cated, and (3) as can be seen in Fig. 2.22, the gain factor not only depends on frequency but also on the input voltage level. The last effect is by far the least significant because the return path amplifiers are all adjusted to provide the same input level at the next amplifier’s input. Thus, this effect is not considered in the simulation, which would not be possible for a simulation in frequency domain anyway. In time domain it is princi-pally possible to consider nonlinearities, but this is only feasible at reasonable computa-tional expense if the nonlinearity can be described by an input-output characteristic. If this is not the case, nonlinear differential equations would come into place, boosting up the system complexity. In a correctly adjusted CaTV plant, nonlinear effects play a minor role and therefore simulations based on S-parameters yield results in compliance with real-world measurements.


Fig. 2.24 (a) Complete Simulink model for the laboratory network; (b) network branch to subscriber B; (c) block for one amplifier or for one cable.

z1

Unit Delay C

z1

Unit Delay B

VC

To Workspace C

VB

To Workspace B

VA

To Workspace A

a1

a2

a3

b1

b2

b3

Three-port amplifier

a1

a2

b2

b1

Reflection factor C

a1

a2

b2

b1

Reflection factor B

a1

a2

b2

b1

Reflection factor A

[T,UC]

From Workspace C

[T,UB]

From Workspace B

[T,UA]

From Workspace A

a1

a2

b2

b1

Branch C

a1

a2

b2

b1

Branch B

a1

a2

b2

b1

Branch A

2b1

1b2

a1

a2

b2

b1

Transmission line 3

a1

a2

b2

b1

Transmission line 2

a1

a2

b2

b1

Amplifier 4a1

a2

b2

b1

Amplifier 3

2a2

1a1

2b1

1b2

z1

Unit Delay

OutputReflectionFactor

InputReflection

Factor

Forward TransmissionFactor

Backward TransmissionFactor

2a2

1a1

(a)

(b)

(c)

S21

S22S11

S12


2.6 Existing Solutions for Return Channel Transmission

Data transmission in the return channel of CaTV networks started in the early nineties with several field trials [53] to test technical feasibility and customer reactions. While everybody was dreaming of optical fibres to everybody’s home, it became soon apparent that this solution would be by far too costly; neither were the huge data rates, that fibre can deliver, really requested. So it was clear that networking solutions which offer high data rates at reasonable cost must be found. CaTV networks appeared as proper candi-dates because they offer high bandwidth in the downstream at high SNR and have a high penetration rate in most countries, especially in the USA. What was missing (and is still missing nowadays in most parts of Germany and Europe) is the return path, prefer-ably on the same physical network, to provide full interactive services. Different upgrad-ing approaches have been followed, but in the end it boiled down to locate the return channel at frequencies below the broadcast TV spectrum (see Fig. 2.3) and to add digital channels for the downstream at higher frequencies. In the early years of interactivity, the envisioned applications were VoD, video conferencing, home banking, and homeshop-ping. With the rapid emergence of internet, the interest of cable operators and their cus-tomers has shifted toward fast internet access, which already subsumes some of the aforementioned applications. One of these, home shopping is nowadays offered with surprising success on plain old TV which offers not the slightest touch of interactivity. This shows that it is nearly impossible to foresee future applications and that commer-cial success not always depends on technical brilliance.

Return channel transmission was driven by two different forces which still have its impact in today’s equipment: (1) computer networking, where the goal is to connect the user’s computers to a wide area network as the internet, (2) interactive TV with the aim

0 5 10 15 20 25 30 35−7 dB

−6 dB

−5 dB

−4 dB

−3 dB

−2 dB

−1 dB

0 dB

1 dB

2 dB

measuredsimulated

0 100 200 300 400 500 600 700 800 900−60 dB−50 dB−40 dB−30 dB−20 dB−10 dB

0 dB10 dB20 dB30 dB40 dB50 dB

|S12

||S

21|

Fig. 2.25 (a) Transfer functions of forward ( ) and return ( ) channel to connection point C in Fig. 2.17; (b) measured and simulated return channel transfer function. For further details, see [49].

S21 S12

f MHz⁄f MHz⁄

S 12f()

S 21f()

(a) (b)

S 12f()

Existing Solutions for Return Channel Transmission 45

to provide a return channel to offer interactive services. Two terms have evolved for the device that connects the subscriber’s apparatus to the network: cable modem for the com-puter and set-top box (STB) for the TV set. Although both devices apply the same trans-mission scheme for access to the return channel, there is still little convergence between both worlds, despite of the many evocations of the industry.

Commercial interest arose with the boom of the internet. Cable modems which provide fast access to the internet, leaving aside other applications for the moment, were devel-oped [e.g. 54-56]. Soon it was recognised that interoperability was crucial because pro-prietary solution would impede fast market penetration. Time to market was a crucial issue because alternative technologies were evolving or were already on the market [4, 57-59]. Several working groups emerged to harmonise the efforts of many manufactur-ers and cable operators and many proposals for physical layer design [37, 60-67] and medium access [68-80] were advocated. In the US, the IEEE 802.14 working group started with a rather general approach which included ATM and real-time services. Ret-rospective, this task turned out to be too challenging, and due to slow progress, the industry forum MCNS (Multimedia Cable Network System) emerged which hammered out the DOCSIS (data over cable service interface specification) specification, which is now the prevailing standard for cable modems. In Europe the starting point was differ-ent. The DVB working group took up much of the work done in DAVIC (digital audio visual council), focusing on digital audio and video transmission and interactivity. Nev-ertheless, all the necessary functions for computer networking have been included in the DVB-RCC standard [44]. Both standards have been adopted by the ITU-T as J.112 Annex A and J.112 Annex B [81- 83].

Although starting from rather disparate points of view, with DOCSIS and DVB-RCC two standards have been created that have a lot in common. The basic transmission schemes are conceptually the same, although in detail they are as different as they could be, and hence are completely incompatible. In the following, the basic transmission schemes and the access mode are shortly reviewed. While we focus more on the DVB-RCC specifica-tion, the basic ideas apply as well for DOCSIS. For a more detailed description, please refer to [24, 84-89], or to the standards itself [44, 45]. A very good introduction to cable modems with reference to DOCSIS gives [90]. Extensive information related to CaTV and cable modems can be found in the web [91].

2.6.1 System Overview

A cable modem or STB is described in DVB in a more general context, as indicated in Fig. 2.26. Keeping in mind the vicinity of broadcast services, a broadcast delivery medium is defined in addition to the interaction network. This leads to a generic system definition which allows e.g. to combine a forward channel via satellite with a wireline ISDN return channel. For the CaTV network, both the broadcast and the interaction channel reside on the same physical network, separated in frequency.


The forward interaction path can be realised as downstream channel with 1 or 2 MHz bandwidth or it can be embedded into the MPEG-2 transport stream of a DVB cable channel. Thus, for the forward path the physical layer is already defined and the addressing of the individual subscriber stations can be accomplished easily via a unique address assigned to the station. Thus, each station filters out the packets which carry its address. Of course, the data in the packets has to be encrypted to ensure that only the destined subscriber can read them.

2.6.2 Multiple Access in the Return Channel

As the return channel is a shared medium which all subscribers wish to access, a media access control (MAC) protocol has to control and manage the access in order to avoid collisions and to ensure efficient operation. Simple decentralised access schemes known from LANs like CSMA/CD cannot be used due to high attenuations between the sub-scribers, thus these are not able to detect collisions. The only instance capable of detect-ing collisions is the HE. Hence, the access control must be managed exclusively by the HE. This allows to implement a more sophisticated access scheme than simple collision based protocols which are known to perform very poor under heavy load. Bandwidth is a scarce resource in the return channel, so heavy network load is to be expected. Further-more, each subscriber who cannot access at a given moment signifies lost revenues for the cable operator. In DVB three major access modes have been defined, accounting for services with different requirements:

1. Contention based Access: users send data with the risk of collisions which are solved by a contention resolution protocol under the control of the headend

2. Fixed rate Access (Reserved slots with fixed rate reservation): the user gets a reserva-tion of one or more time slots in each frame. This mode is best suited for real-time services like voice, audio, etc.

3. Reservation Access (Reserved slots with dynamic reservation): the user announces his demand for transmission capacity, the HE provides a finite amount of slots. For more details, see [e.g. 92, 93].

broadcastdeliverymedia

interactionnetwork

broadcastinterfacemodule

broadcastnetworkadapter

interactivenetworkadapter

interactiveinterfacemodule

ne

two

rkin

terfa

ce

un

it

headend cablemodem or STB

broadcastchannel

interactionchannels

returninteractionpath

forwardinteractionpath

wideareanetwork

enduser

Fig. 2.26 Reference model of an interactive system according to DVB.

Existing Solutions for Return Channel Transmission 47

With these access modes, it is possible to mix real-time and best effort services and thus services that require constant bitrate like voice or video can be supported without the detour of VoIP (voice over IP). Besides these access modes there are some special modes which are used during initialisation of a subscriber station. During the so called ranging procedure, channel parameters are estimated and the station is synchronised to the time basis of the HE and its power is adjusted. After the basic parameters have been set, the station listens to a configuration channel which contains information about the time slots for contention based access.

Both standards wisely leave open which resources should be assigned to which user. They define the mechanism but not the strategy which leaves it to the manufacturers to distinguish their product from others. Furthermore, these strategies heavily depend on the traffic type on the CaTV plant which in turn depends on future services.

2.6.3 Upstream Physical Layer Specification

The upstream frequency band is divided into channels with a bandwidth of 200 kHz to 4 MHz, and each channel can be shared by several users. Thus, for the multiple access a combined FDMA/TDMA scheme is used. The modulation format is either QPSK, 16-QAM or 64-QAM1 with square-root raised cosine filtering and a rolloff factor of or . The FEC consists of a Reed-Solomon-Encoder which in DOCSIS is very flexible and allows various code word lengths and strengths of protection. As shown in Fig. 2.27 for DVB, the upstream channel is organised in slots which contain ATM cells along with bytes for synchronisation, addressing and FEC.

1. Only for DOCSIS 2.0

α 0.3=α 0.25=

unique word payload area (ATM cell) Reed-Solomon-parity guard band

4 byte

5 byte 8 byte 40 byte

53 byte 6 byte 1 byte

ATMheader

MACaddress

information payload

Fig. 2.27 Upstream slot format in DVB.


3 Principles of Multicarrier Modulation

The key idea of multicarrier modulation (MCM) is to partition a high-rate data stream into a large number of low-rate data streams and transmit them in parallel over equidis-tantly spaced subchannels. As the occupied bandwidth of a signal is in principle recipro-cal to its symbol rate, the sum of all low-rate data streams occupies the same bandwidth as the high-rate signal. The main point of MCM is that the low-rate (and hence narrow-band) signals are much less susceptible to channel impairments and thus reconstruction of the subchannel signals at the receiver side is simplified to a great extent. However, all subchannel signals are transmitted in parallel and thus have to be processed simultane-ously at the receiver.

The subject of MCM has been treated in few books [94-96], and many articles cover the subject in quite some detail [e.g. 97-99].

3.1 Historical Overview

The roots of MCM reach back into the late fifties and sixties when the first multicarrier systems were developed for military high frequency transmission. The idea of synchro-nously modulating several narrowband signals onto equidistant carriers with overlap-ping spectra was realised as early as 1957 in the Collins Kineplex system [100], which used PSK modulation to transmit 3000 bit/s over 20 subcarriers with overlapping spec-tra. This early system already made use of a guard time to prevent intersymbol interfer-ence. In the sixties, it was followed by the Kathryn modem, another military high frequency system. The Kathryn modem also used a guard time and PSK modulation and already made use of DFT processing to modulate the data onto 34 subcarriers [101, 102]. It was already recognised that MCM is a proper solution for the multipath fading chan-nel. In 1966, Chang [103] proposed a technique which is based on bandlimited orthogo-nal signals to provide high spectral efficiency without interference. This method was refined [104] and the performance was investigated shortly afterwards by Saltzberg [105], and a corresponding patent was issued [106]. Still another military system using MCM, the ANDEFT modem, was introduced by Porter [107]. In 1968, the TADIM modem was presented [108], which was implemented mainly with digital signal processing and used a DFT processor to modulate 16 subcarriers. The idea of using the DFT instead of oscillator banks for modulation became widely known with the classical paper of Weinstein and Ebert [109]. The intersymbol and interchannel interferences were mitigated by smooth transitions of the impulse and a guard space between two consecu-tive symbols. This idea was advanced by Peled and Ruiz [110] who introduced the cyclic extension instead of an empty guard space. Hirosaki [111] extended Chang’s idea of bandlimited base functions and combined it with Weinstein’s DFT processing to the so-called digital orthogonally multiplexed QAM. A complexity comparison between his

49


multicarrier system and a conventional digital single-carrier system showed a considera-ble advantage for the MCM system in terms of number of multiplications. Later on, he presented an application [112] based on this theory. In 1985, Cimini investigated and suggested the use of OFDM for transmission over mobile channels [113]. Multicarrier systems had still not found their way into applications, but with the ascent of digital sig-nal processing and the substantial progress in semiconductor technology, Bingham [97] concluded in the early nineties, that the time had come for MCM. Kalet [114] and Kastu-ria [115] provided in-depth investigations of the potential of MCM, and then the race was on. Further investigations followed [116-119], and then OFDM found its way into the first major application: Digital Audio Broadcast [120]. OFDM was also chosen for digital terrestrial TV (DVB-T) [19], showing clear advantages in comparison with the North American single carrier system. It was in discussion for third generation mobile communication (UMTS) and is a hot candidate for the fourth generation. Another important field of application is the transmission over twisted-pair telephone lines, namely the subscriber line. Under the name DMT, multicarrier transmission has become a world standard for ADSL and VDSL transmission systems [121-124].

The nomenclature for multicarrier systems can be quite confusing. The most common terms are OFDM for wireless applications and DMT for wireline systems. The term DMT insinuates the use of adaptive modulation and baseband processing. In combination with pulse shaping it is sometimes referred to as discrete wavelet multitone (DWMT).

3.2 The Continuous-Time System Model

3.2.1 Elementary Impulses, Ambiguity Function and Orthogonality

In a multicarrier system, first the incoming bitstream is divided into bitstreams with lower rate, as indicated in Fig. 3.1. These are mapped into QAM symbols, yielding the sequences , . In a multiple access system, these sequences may stem from different sources, which does not affect the following treatment as long as the sequences are synchronised. The impulse modulator translates the discrete-time sequence into the continuous-time function

, (3.1)

The complex lowpass output signal of the MC transmitter is hence given by

(3.2)

b n[ ] N

Xν k[ ] ν 0 … N 1–, ,=

Xν k[ ]

xν t( ) T Xν k[ ] δ t kT–( )⋅k ∞–=

∞

∑= ν 0 … N 1–, ,=

s t( ) T ejωνt

Xν k[ ] g t kT–( )⋅k ∞–=

∞

∑ν 0=

N 1–

∑=

The Continuous-Time System Model 51

where is the impulse response of the impulse shaping filter. The carrier frequencies are equidistant and thus integer multiples of the carrier spacing :

, (3.3)

The aim of the receiver is to recover the sequences without distortion. As the given system is linear and time-invariant, we can restrict the analysis to the evaluation of the response to one unit impulse on an arbitrary subchannel. If this reveals no interference, the superposition theorem guarantees that the system is free of interference for any input signal. We can distinguish two basic types of interference:

1. Intersymbol interference (ISI): a symbol sent at time instant has impact on previous or subsequent samples on the same subchannel.

2. Interchannel interference (ICI) is the result of crosstalk between different subchannels at the sampling times .

Without loss of generality we assume that the system in Fig. 3.1 has zero delay, i.e. the filters , are not causal. We assume that a unit impulse is sent on subcarrier ν:

(3.4)

The received signal is free of interference if .

To gain further insight into the nature of ISI and ICI, we take a closer look at the received signals before they are being sampled. An input signal (3.4) yields the transmitter output

(3.5)

which is also the input signal of the receiver as we assume an ideal channel. The elemen-tary impulse is defined as the response to the input signal (3.4) [125, 126] as

(3.6)

g t( )ων ∆ω

ων ν ∆ω⋅= ν 0 … N 1–, ,=

Xν k[ ]

Fig. 3.1 Continuous-time system model of a multicarrier transmission system.

XN-1[k] YN-1[k]

Yµ[k]

Y0[k]

Xν[k]b[n]

X0[k]

g(t) f(t)

f(t)

f(t)

g(t)

g(t)

xN-1(t) yN-1(t)

T

yµ(t)s(t)

y0(t)

xν(t)

x0(t)δ[k]Tδ(t)

δ[k]Tδ(t)

δ[k]Tδ(t)

S S

P P

impulsemodulator

modulation demodulation receiverfilter

sampling demapping& decision

impulseshaping

mapping

……

… … …

… … …

kT

kT

g t( ) f t( ) δ k[ ]

Xi k[ ] δ ν i–[ ] δ k[ ]⋅=

Yµ k[ ] δ ν µ–[ ] δ k[ ]⋅=

s t( ) T g t( )ejν∆ωt=

rν µ, t( ) yµ t( )

rν µ, t( ) T g t( )ej ν µ–( )∆ωt( ) f t( ) T g τ( )ej ν µ–( )∆ωτf t τ–( ) τd∞–

∞

∫= =


Obviously, only depends on the difference . Thus, the definition simpli-fies to

(3.7)

The introduction of the elementary impulses allows us to write the condition for zero interference in a very compact form:

(3.8)

This condition is referred to as extended Nyquist criterion as it not only forces zero inter-symbol interference but also zero interchannel interference. It is also called orthogonality condition1 or criterion for perfect reconstruction (PR). In frequency domain, the elementary impulses are defined as

(3.9)

Eq. (3.8) leads to the condition for perfect reconstruction in frequency domain:

(3.10)

A similar concept for defining perfect reconstruction and visualising the interference characteristics of the base functions and is the ambiguity function [127] which is defined as

(3.11)

This leads to the PR condition

(3.12)

Both (3.8) and (3.12) lead to the condition on and

(3.13)

3.2.2 Rectangular Pulse Shaping and Guard Interval

Traditional OFDM systems apply rectangular pulse shaping. Without guard interval holds and with the carrier spacing

(3.14)

1. In the OFDM literature, orthogonality is often loosely explained by a diagram of the subcarrier spectra. Although a nice diagram, it does not give an exact definition for zero interference.

rν µ, t( ) d ν µ–=

rd t( ) T g t( )ejd∆ωt( ) f t( )=

rd kT( ) δ d k,[ ]=

rν µ, t( ) Rν µ, ω( ) T G ω ν µ–( )∆ω–( ) F ω( )⋅ ⋅=

1T--- Rν µ, ω k2π

T------–

k ∞–=

∞

∑ δ ν µ–[ ]=

g t( ) f t( )

A t ω,( ) T g τ( )f t– τ–( )e jωτ– τd∞–

∞

∫=

A kT d∆ω,( ) δ d k,[ ]=

g t( ) f t( )

T g τ( )f kT τ–( ) ej d ∆ω τ τd∞–

∞

∫ δ d k,[ ]=

g t( ) f t( ) 1 T⁄( ) rect t T⁄( )= =

∆ω 2πT

------=

The Continuous-Time System Model 53

we obtain the elementary impulses

(3.15)

These functions are shown in Fig. 3.2 (a) and help to understand the characteristics of ISI and ICI. We clearly see that at the sampling instants no interference is present. This is not the case between the sampling instants where significant crosstalk coming from the adjacent channels occurs. The oscillation frequency of the crosstalk increases with the distance d between transmitter and receiver subchannel index, whereas the amplitude decreases proportional to . Due to this slow decline, many adjacent subchannels contribute to the crosstalk between the sampling instants. As long as this crosstalk is zero at the sampling instants, no interference occurs there. However, it can-not be ignored for other time instances, as becomes clear from Fig. 3.2 (b) where the eye-diagram of the real part of is plotted for a 16-QAM transmission on carriers. Although the orthogonality condition (3.8) is fulfilled, the horizontal eye open-ing tends to zero, producing decision errors even for very little sampling jitter or channel influences.

The orthogonality condition can also be visualised with a two-dimensional time-fre-quency plane of the ambiguity function. This function is plotted in Fig. 3.3 and is given as

r0 t( ) 1 t T⁄– for t T≤0 elsewhere

=

rd t( )t( ) 1–( )dsgn

j2πd----------------------------- 1 jd2π

T------t

exp– for t T≤

0 elsewhere

for d 0≠,

=

t kT=

1 d⁄

yµ t( ) N 256=

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 r0

Rer1

Imr1)

Rer2

Imr2

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−6

−4

−2

0

2

4

6

Fig. 3.2 (a) Elementary impulses of an OFDM system with rectangular pulse shaping (b) Eye diagram of real part of for 16-QAM, subcarriers, .

rd t( )yµ t( ) N 256= µ 127=

t T⁄t T⁄

Re

y µt()

(a) (b)

r0

Rer1

Imr1)

Rer2

Imr2


(3.16)

The ambiguity function is limited in time direction but extends infinitely in frequency direction and reveals strong sidelobes near the sampling grid where the orthogonality conditions (3.12) demands zero values. It is clear that such a system would be highly sen-sitive to time or frequency dispersion.

An effective solution to this problem is the introduction of a guard interval by choosing different impulse lengths at transmitter and receiver. The duration of the guard interval is

(3.17)

where and denote the length of the impulse response of the transmitter and the receiver filter and , respectively. In contrast to (3.14), the carrier spacing is now

(3.18)

A t ω,( )

ejω t T 2⁄+( ) e jωT 2⁄––jωT

------------------------------------------------- for T t 0 ω 0≠,< <–

ejωT 2⁄ e jω t T 2⁄–( )––jωT

------------------------------------------------ for 0 t T ω 0≠,< <

1 t T⁄– for T t T ω 0=,< <–0 elsewhere

=

−3−2

−10

12

3

−2

−1

0

1

20

0.2

0.4

0.6

0.8

1

Fig. 3.3 Ambiguity function for rectangular impulses.

ω ∆ω⁄t T⁄

At

ω,(

)

TG T TU–=

T TUg t( ) f t( )

∆ω 2πTU------=

The Discrete-Time System Model 55

which leads to the elementary impulses

(3.19)

As can be seen from Fig. 3.4, there are now flat regions around the sampling instants which impede any interference. This holds also for a channel with an impulse

response shorter than . In Fig. 3.4 (b), the horizontal eye opening is now as long as the guard interval. During this period, no information can be transmitted and thus the spec-tral efficiency is reduced by the factor .

3.3 The Discrete-Time System Model

3.3.1 Filterbank Implementation

While the continuous-time model is valuable for the theoretical system analysis, the implementation of MCM systems is exclusively done with digital signal processing ever since this technique became available. We obtain the discrete-time system model depicted in Fig. 3.5 by sampling the continuous-time model in Fig. 3.1 with the sampling

r0 t( )

t– T TG 2⁄–+T TG–

-------------------------------------- for TG2

------ t TTG2

------–≤ ≤

1 for t TG 2⁄<

0 for t T TG 2⁄–>

=

rd t( )t( ) 1–( )dsgn

j2πd----------------------------- j t( )dπ

TGTU------sgn

jd2πTU------t

exp–exp for

TG2

------ t TTG2

------–≤ ≤

0 elsewhere

d 0≠,

=

t kT=TG

TG T⁄

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 r0

Rer1

Imr1)

Rer2

Imr2

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−6

−4

−2

0

2

4

6

Fig. 3.4 (a) Elementary impulses ; (b) eye diagram of real part of for 16-QAM and 256 subcarriers, both with guard interval .

rd t( ) yµ t( )TG TU 8⁄=

t T⁄t T⁄

Re

y µt()

(a) (b)

r0

Rer1

Imr1)

Rer2

Imr2


period . The impulse modulators are replaced by upsamplers which insert zeros between each incoming symbol . Thus

(3.20)

holds and for the complex carriers in Fig. 3.1 the sampling yields with (3.3) and (3.14)

where (3.21)

Since we have made no assumptions about the filters and , the sampling theo-rem is not fulfilled generally and thus the discrete-time system model does not exactly represent the analogue model. Nevertheless, it is a reasonable approximation and the concepts like the elementary impulses, the ambiguity function and the orthogonality cri-terion can be applied in the same way.

A guard time can be introduced into the model of Fig. 3.5 in the same way as before, but there are some subtle differences between this approach and the realisation using DFT processing. Since for the further discussion no guard interval is required in the model according to Fig. 3.5, we omit these details which can be found in [128].

The discrete-time model of Fig. 3.5 can be further simplified by incorporating the modu-lation stage into the pulse shaping filters, resulting in the block diagram of Fig. 3.6, where the impulse responses , are given as

(3.22)

and , are often referred to as prototype filters.

The equivalence of both structures can be shown easily by calculating in the trans-mitter and in the receiver. Note, that although the receiver structures are equiva-lent regarding the output signals , this is not true for the signals .

TA M 1–Xν k[ ]

T M TA⋅=

ejωνnTA e

j2πM------νn

w νn–= = w ej2πM------–

=

g t( ) h t( )

XN-1[k] YN-1[k]

Yµ[k]

Y0[k]

Xν[k]

X0[k]

g[n] f [n]

f [n]

f [n]

g[n]

g[n]

yN-1[n]

yµ[n]s[n]

y0[n]

up-sampling

modulation demodulation receiverfilter

down-sampling

impulseshaping

…… ……

…… ……

M M

M M

M M

w-0n w0n

w-νn wµn

w-(N-1)n w(N-1)n

discretechannel

Fig. 3.5 Discrete-time system model for MCM.

gν n[ ] fµ˜ n[ ]

gν n[ ] g n[ ] w νn–⋅=

fµ˜ n[ ] f n[ ] w µn–⋅=

g n[ ] f n[ ]

sν n[ ]Yµ k[ ]

Yµ k[ ] yµ n[ ]


The structure in Fig. 3.6 is called a transmultiplexer [129] and is a rather general system model. By choice of the filters , it can represent an MCM, TDMA or CDMA system. E.g. for , it represents a TDMA system. If the impulse responses are orthogonal, i.e. , then the transmultiplexer is a CDMA system. Transmultiplexers are strongly related to filterbanks which are applied for e.g. signal analysis or speech and image coding. For filterbanks, a highly developed theory exists [130-138] which can be applied to MCM [139-140].

3.3.2 DFT-Based Implementation with Cyclic Prefix

In practically realised systems is chosen, i.e. the upsampling factor equals the number of subchannels. Such a system is called maximal decimated or critically sam-pled, because then the sampling frequency just equals the band-width of all modulated carriers. In order to avoid spectral overlap with adjacent bands, some outer subchannels are not used. Note, that not modulating some subcarriers is another form of oversampling, which is commonly used in most applications. E.g. in DVB-T, out of subcarriers, only 6817 are modulated.

Without a guard interval, the filters in Fig. 3.5 are chosen as

(3.23)

The transmitter output signal can then be written as

(3.24)

Here we recognise the expression of the discrete Fourier transform and its inverse:

DFT: IDFT: (3.25)

gν n[ ]gν n[ ] δ n ν–[ ]=

gν n[ ] gµ n[ ]⋅n 0=M 1–

∑ δ ν µ–[ ]=

XN-1[k]YN-1[k]

Y1[k]

Y0[k]

X1[k]

X0[k]

yN-1[n]

y1[n]s[n]

s1[n]

s0[n]

sN-1[n]

y0[n]

… …

… …M M

M M

MM

discretechannel

synthesis filterbank analysis filterbank

Fig. 3.6 System model of a transmultiplexer.

M N=

ωA 2π TA⁄ M ∆ω⋅= =

N 8192=

g n[ ] f n[ ] 1 N⁄ for n 0…N 1–=0 elsewhere

= =

s n[ ] 1N

-------- Xν divN n( )[ ] w νn–⋅ν 0=

N 1–

∑=

Xm1N

-------- xn wnm⋅n 0=

N 1–

∑= xn1N

-------- Xm w nm–⋅m 0=

N 1–

∑= m n, 0 … N 1–, ,=


Thus, we identify the input signals as a block with argument k and define the blockwise IDFT as

, (3.26)

which allows us to express (3.24) as

(3.27)

At this point, the guard interval is inserted by copying the last samples of each block to its beginning as indicated in Fig. 3.7, thus extending a block of data samples to

samples. Because the guard interval is inserted at the beginning of each OFDM block, it is referred to as cyclic prefix.

At this point, it is convenient to introduce the matrix notation which takes advantage of the blockwise independence of the transmitter signal and avoids the tedious div and mod notations.

The receiver input signal according to Fig. 3.8 is given by

(3.28)

X0 k[ ] … XN 1– k[ ], ,

xi k[ ] 1N

-------- Xν k[ ] w iν–⋅ν 0=

N 1–

∑= i 0 1 … N 1–, , ,=

s n[ ] xmodN n( ) divN n( )[ ]=

GN

NS N G+=

IDFT DFTdiscretechannel

X0 Y0x0

s[m]

xN-G

xN-1

X1 Y1

XN-1 YN-1

… …

…

…

…

Fig. 3.7 Insertion and removal of the cyclic prefix in transmitter and receiver.

X0 y0x0

x[n] s[m] y[n]

Y0

X1 y1x1 Y1

XN-1 yN-1xN-1 YN-1

h[m]IDFT DFTCP CP… …… …

S S

P P

HF-1 F

Fig. 3.8 DFT-based OFDM system with cyclic prefix.

y n[ ] s n[ ] h n[ ] s m[ ] h n m–[ ]⋅m ∞–=∞∑= =


We assume that the channel can be described by a finite length impulse response with samples . From (3.28) we conclude that for independence of the received blocks it must hold

(3.29)

Thus, the CP must be at least as long as the length of the channel impulse response. If this condition is satisfied we can adopt a vector notation for the block data:

, , , (3.30)

The receiver input signal after removal of the CP can be expressed as

(3.31)

or

(3.32)

This directly relates the signals x and y and transforms the effect of the CP insertion and removal into the channel matrix H. The matrix equation (3.32) represents the circular convolution which corresponds to multiplication in the frequency domain of the DFT [141]. In scalar notation, this is expressed as

h n[ ]Lc 1+ h 0[ ] … h Lc[ ], ,

Lc G≤

x

x0

x1

xN 1–

=

…

s

xN G–

xN 1–

x0

xN 1–

=

……

y

y0

yG 1–

yG

yN G 1–+

=

……

y

y0

y1

yN 1– yG

yG 1+

yN G 1–+

= =

… …

y

h G[ ] h G 1–[ ] … h 0[ ] 0 … 00 h G[ ] h G 1–[ ] … h 0[ ] … 0… …0 0 … h G[ ] h G 1–[ ] … h 0[ ]

s⋅=

y0

y1

yG 1–

yG

yG 1+

yN 1–

h 0[ ] 0 … 0 h G[ ] … h 2[ ] h 1[ ]h 1[ ] h 0[ ] 0 … 0 h G[ ] … h 2[ ]… … … … … … … …

h G 1–[ ] … … h 0[ ] 0 … 0 h G[ ]h G[ ] h G 1–[ ] … … h 0[ ] 0 … 0

0 h G[ ] h G 1–[ ] … … h 0[ ] 0 …… … … … … … … …0 … … h G[ ] … … h 1[ ] h 0[ ]

x0

x1

xG 1–

xG

xG 1+

xN 1–

=

H

……

……


(3.33)

Thus, the CP transforms the linear convolution (3.28) into a circular convolution. The matrix H is a circulant matrix [142] whose eigenvalues and eigenvectors are given by

, , (3.34)

This can be easily verified by checking the equation . We recognise that the eigenvectors of the channel matrix are the columns of the inverse DFT matrix, hence

(3.35)

As a consequence, we can diagonalise the channel matrix by multiplying with the IDFT and its inverse, the DFT matrix F:

, where (3.36)

With and we can write

, , (3.37)

We can now describe the input-output relation of the whole transmission system of Fig. 3.8 with a single diagonal matrix. The result shows that due to the CP the parallel subchannels are independent and perfect reconstruction can be realised by a simple one-tap equaliser with tap weights .

Following from (3.34) we can interpret the eigenvalues of the channel matrix as the DFT of the channel impulse response . If we define

(3.38)

as the discrete-time Fourier transform of , we see that the eigenvalues are just the values of the channel transfer function at the frequencies :

(3.39)

3.3.3 Implementation of MCM with DFT and Polyphase Filterbank

The filterbank implementations in Figs. 3.5 and 3.6 are well-suited for system analysis, but for the implementation, more effective structures exist. As well as for standard OFDM, for MCM systems with pulse shaping, the implementational cost can be reduced

y n[ ] x n[ ] h n[ ]⊗ x m[ ] h modN n m–( )[ ]⋅m 0=

N 1–

∑= =

λµ ϑµ

λµ h n[ ]wµn

n 0=

N 1–

∑= ϑµ1N

-------- w 0– w µ– w 2µ– … w N 1–( )µ– T

= µ 0 … N 1–, ,=

Hϑµ λµϑµ=

F 1– ϑ0 ϑ1 … ϑN 1–, , ,( )=

Λ FHF 1–= Λ diag λµ( )=

X X0 k[ ] … XN 1– k[ ], ,( )T= Y Y0 k[ ] … YN 1– k[ ], ,( )T=

x F 1– X= y Hx HF 1– X= = Y Fy FHF 1– X ΛX= = =

1 λµ⁄

λµh n[ ]

H ω( ) h n[ ] ejωnTA–

⋅n 0=

N 1–

∑=

h n[ ]ωµ µ ∆ω⋅=

λµ H ωµ( )=


significantly by help of the DFT. In addition to the structure for rectangular , for general filters a polyphase filterbank has to be added. The transmitter is depicted in Fig. 3.9. In order to show the equivalence to the above circuits, we have to calculate the output signal and compare it to the output signal of Fig. 3.5.

For the following, we assume that the impulse response has length and is symmetrical to :

(3.40)

The polyphase filterbank in Fig. 3.9 consists of M FIR filters with impulse responses

, , (3.41)

This decomposition of into M filters is called polyphase decomposition of type 1 [130, 131] and reads in the other direction:

(3.42)

The IDFT output signal is

(3.43)

which yields after filtering

(3.44)

The output signal after serial-parallel conversion, symbolised by the commutator, is

g n[ ]

g n[ ] LM 1+LM 2⁄

g n[ ] 0 for n 0< n LM>,,=g n[ ] g LM n–[ ]=

…

x0[k]

gi[k] = g[kM+i]

s0[k]

s[n]x1[k] s1[k]

xM-1[k] sM-1[k]

M-pointIDFT

X0[k]

X1[k]

XN-1[k]

0

0

0 0

1

1

N-1

N

M-1 M-1

…

…Fig. 3.9 MCM transmitter with IDFT and polyphase filterbank.

gi k[ ] g kM i+[ ]= i 0 … M 1–, , ∈ k Z∈

g n[ ]

g n[ ] gmod M n( ) divM n( )[ ]=

xi k[ ] 1M

--------- Xν k[ ] w νi–⋅ν 0=

N 1–

∑=

si k[ ] Mgi k[ ] xi k[ ] g k l–( )M i+[ ] Xν l[ ] w νi–⋅ν 0=

N 1–

∑l ∞–=

∞

∑= =


from which follows with and the periodicity of

(3.45)

which is equivalent to the output signal of Fig. 3.5 and thus it is shown, that for filters that comply with (3.40) the circuit in Fig. 3.9 is equivalent to the structures in Figs. 3.5and 3.6.

The output signal of the receiver in Fig. 3.5 is

(3.46)

where we already inserted the polyphase decomposition of the input signal with

, , (3.47)

By defining the polyphase filterbank as

(3.48)

the receiver structure in Fig. 3.10 yields the same output signal as Fig. 3.5.

s n[ ] s mod M n( ) div M n( )[ ] g M div M n( ) lM– mod M n( )+[ ] Xν l[ ] wν modM n( )⋅–

⋅ν 0=

N 1–

∑l ∞–=

∞

∑= =

mod M n( ) n M div M n( )–= w νi–

s n[ ] g n lM–[ ] Xν l[ ] w νn–⋅ν 0=

N 1–

∑l ∞–=

∞

∑=

Yµ k[ ] wµi si l[ ] g k l–( )M i–[ ]⋅l ∞–=

∞

∑i 0=

M 1–

∑=

s n[ ] si l[ ]=

n lM i+= i 0 … M 1–, , ∈ l⇒ modM n( )= i divM n( )=

fi k[ ] g kM i–[ ]=

…

y0[k]

s[n]y1[k]

yM-1[k]

M-pointDFT

Y0[k]

Y1[k]

YN-1[k]

…

Fig. 3.10 MCM receiver with DFT and polyphase filterbank.

Multicarrier Offset QAM 63

3.4 Multicarrier Offset QAM

3.4.1 Continuous-Time System Model

Multicarrier offset QAM (MC-OQAM), as illustrated in Fig. 3.11, can be considered as a straightforward extension of single carrier OQAM or as a variation of OFDM. The incoming QAM symbols arrive at symbol rate and are decomposed into their real and imaginary parts. For even indices ν the real part is filtered with , while the imaginary part is filtered with which corresponds to a delay of half a symbol period. For odd ν, the real part is delayed by . In other words, the real and the imag-inary part are delayed alternately by half a symbol period.

To determine the elementary impulses for the MC-OQAM system, we have to calculate the response to the input signal (3.4), just as for the MCM system without offset. Because the filters process the real and imaginary part of their input separately, (3.6), (3.7) do not hold for this case.

The output signal of the transmitter can be written without distinguishing two cases for even and odd ν with the modulo operator as

(3.49)

With this input signal, the output after the receiver filters is the elementary impulse:

(3.50)

Xν k[ ] 1 T⁄g t( )

g t T 2⁄–( )T 2⁄

yµ t( )

δ[k]

δ[k]

δ[k]XN-1[k] YN-1[k]

Y 1[k]

Y0[k]

X1[k]

X0[k]

Tδ(t)

Tδ(t)

Tδ(t)

g(t) g(t)

g(t) g(t)

g(t) g(t)

g(t-T/2) g(t+T/2)

g(t-T/2) g(t+T/2)

g(t-T/2) g(t+T/2) yN-1(t)

T

y1(t)s(t)

y0(t)

… …

……

ψ0

ψ1

ψN-1

FDE

Fig. 3.11 Multicarrier Offset-QAM transmission system: real part and imaginary part of the filter input are processed separately. The upper impulse response holds for the real part, while the lower impulse response is for the imaginary part. The frequency domain equaliser (FDE) is placed in front of the receiver filters.

s t( ) T g t mod 2 ν( )T2---–

ejν∆ωt= S ω( ) T G ω ν∆ω–( )ej mod 2 ν( )ωT 2⁄–

=

rν µ, t( ) T g t mod 2 ν( )T2---–

ν µ–( )∆ωt( )cos⋅

g t mod 2 µ( )T2---+

=

jT+ g t mod 2 ν( )T2---–

ν µ–( )∆ωt( )sin⋅

g t mod 2 µ 1+( )T2---+


which is in frequency domain

(3.51)

Because the imaginary part is processed separately, it is not guaranteed by the orthogo-nality condition based on that no interference originating from the imaginary part of the input signal occurs. Therefore, the elementary impulse of the second kind has to be defined as the response to a unit imaginary impulse:

if (3.52)

This leads to the orthogonality condition

(3.53a)

in addition to

(3.53b)

Both terms lead to the same conditions on . With it must hold for

• d even ( even, even or odd, odd):

(3.54)

• d odd ( even, odd or odd, even):

(3.55)

Rν µ, ω( ) T2---G ω( )e

jωT2

------- mod 2 ν( )–1 e

jωT2

-------+

G ω ν µ–( )∆ω–( )ej ν µ–( )∆ω mod 2 ν( ) T

2---⋅ ⋅

1–( )µ 1 ejωT

2-------

–

G ω ν µ–( )∆ω+( )ej– ν µ–( )∆ω mod 2 ν( ) T

2---⋅ ⋅

+

=

rν µ, t( )

pν µ, t( ) yµ t( )= Xi k[ ] j δ ν i–[ ] δ k[ ]⋅⋅=

pν µ, kT( ) j δ ν µ– k,[ ]⋅=

rν µ, kT( ) δ ν µ– k,[ ]=

g t( ) d ν µ–=

ν µ ν µ

T g τ( )g kT τ–( ) d∆ωτ( )cos τd∞–

∞

∫ δ d k,[ ]=

T g τ T 2⁄–( )g kT τ– T 2⁄+( ) d∆ωτ( )cos τd∞–

∞

∫ δ d k,[ ]=

g τ T 2⁄–( )g kT τ–( ) d∆ωτ( )sin τd∞–

∞

∫ 0=

g τ( )g kT τ– T 2⁄+( ) d∆ωτ( )sin τd∞–

∞

∫ 0=

ν µ ν µ

g τ( )g kT τ– T 2⁄+( ) d∆ωτ( )cos τd∞–

∞

∫ 0=

g τ T 2⁄–( )g kT τ–( ) d∆ωτ( )cos τd∞–

∞

∫ 0=

g τ T 2⁄–( )g kT τ– T 2⁄+( ) d∆ωτ( )sin τd∞–

∞

∫ 0=

g τ( )g kT τ–( ) d∆ωτ( )sin τd∞–

∞

∫ 0=


These conditions simplify significantly for the carrier spacing (3.14) and even impulse response, i.e. to the only condition:

, for d even (3.56)

For odd d, the orthogonality conditions are already fulfilled.

3.4.2 Pulse Shaping with Square-Root Raised Cosine Impulse

A well-known impulse which is widely used in single carrier systems, also fulfils the orthogonality criterion (3.56) for MC-OQAM systems: the square-root raised cosine impulse1 which is defined with the roll-off factor as

, (3.57)

, (3.58)

For this impulse the elementary impulses can be calculated with (3.50) resulting in

(3.59)

1. Recently, another impulse which yields a greater eye opening has been proposed [143].

g t( ) g t–( )=

T g t kT2

------+ g t kT

2------–

d∆ωt( )cos td∞–

∞

∫ δ d k,[ ]=

α

g t( ) 11 4αt T⁄( )2–-------------------------------- 1 α–

T------------si π 1 α–( ) t

T---

4απT------- π 1 α+( ) t

T---

cos+ = 0 α 1≤ ≤

G ω( )

1 for ωωN------- 1 α–≤

π4---

ω 1 α–( )ωN–αωN

-------------------------------------- cos for 1 α– ω

ωN------- 1 α+≤ ≤

0 for 1 α+ ωωN-------≤

= ωNπT---=

r0 0, t( ) 1 α+2

-------------si 1 α+( )πtT-----

1 α–2

------------si 1 α–( )πtT-----

+=

α2--- 1 α–( )πt

T-----

cos si π 1 2αtT

---------–

si π 1 2αt

T---------+

+

⋅ ⋅+

2α2tπT

----------- 1 α–( )πtT-----

sin 1 2παt T⁄( )cos+1 2αt T⁄( )2–

-------------------------------------------⋅ ⋅+

r1 0, t( ) T2

4πα---------- πt

T-----

απ t T⁄ 1 2⁄–( )( )cosT

2α-------

2t T

2---–

2–

--------------------------------------------------- j απt T⁄( )cosT

2α-------

2t2–

------------------------------⋅+

sin–=

rν µ, t( ) 0 ν µ– 2≥,=


This is easiest calculated by using the frequency domain definition (3.51). For , no spectral overlap between two adjacent subchannels occurs and thus the elementary impulses vanish. Fig. 3.12 shows the elementary impulses for two different roll-off fac-tors. For the impulse is most concentrated in time domain and the bandwidth of a modulated subcarrier is the double of the carrier spacing . The limiting case leads to rectangular spectra without any spectral overlap.

3.4.3 Discrete-Time System Model and Polyphase Implementation

Like in section 3.3, we can derive a discrete-time representation for MC-OQAM from Fig. 3.11 by sampling all continuous-time functions with the sampling period

. This results in the digital system model in Fig. 3.13. In view of the imple-mentation, we adopted a causal notation for the receiver filters. The elementary impulses for this model can be calculated as

(3.60)

The orthogonality condition for causal filters is

, , (3.61)

where is the delay introduced by the causal transmitter and receiver filters.

While this model is well-suited for theoretical analysis, for practical implementation or for simulation faster solutions exist. The counterpart of Fig. 3.9, the fast implementation for MC-OQAM, is depicted in Fig. 3.14 which contains two IDFT processors with

ν µ– 2≥

α 1=∆ω α 0=

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 r0,0

Rer1,0

Imr

1,0)

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1 r0,0

Rer1,0

Imr

1,0)

Fig. 3.12 Elementary impulses for pulse shaping with square-root raised cosine impulses. (a) , (b) .α 1= α 0.2=

(a)t T⁄

(b)t T⁄

TA T M⁄=

rν µ, n[ ] g n m– mod2 ν( )M2-----– 2π

M------ ν µ–( ) n m–( )

g m M2-----– mod2 µ 1+( )M

2-----–cos

m∑=

jg n m– mod2 ν( )M2-----– 2π

M------ ν µ–( ) n m–( )

g m M2-----– mod2 µ( )M

2-----–sin+

rν µ, kM[ ] δ ν µ– k dg–,[ ]= k Z∈ ν µ, 0 … N 1–, , ∈

dg N∈


attached polyphase filterbanks1. In order to show the equivalence of both circuits, we again compute the output signal for arbitrary input sequences.

With the auxiliary variables

, (3.62)

we can write the output signal of the transmitter in Fig. 3.13 as

1. Another fast implementation for digital MC-OQAM can be found in [144].

Aν k[ ]Re Xν k[ ] ν evenjIm Xν k[ ] ν odd

= Bν k[ ]jIm Xν k[ ] ν evenRe Xν k[ ] ν odd

=

XN-1[k] YN-1[k]

Y 1[k]

Y0[k]

X1[k]

X0[k] g[n] g[n-M]

g[n] g[n-M]

g[n] g[n-M]

g[n-M/2] g[n-M/2]

g[n-M/2] g[n-M/2]

g[n-M/2] g[n-M/2] yN-1[n]

y1[n]s[n]

y0[n]

… …

……

ψ0

ψ1

ψN-1

M M

M M

M M

w-0n w0n

w-1n w1n

w-(N-1)n w(N-1)n

Fig. 3.13 Digital MC-OQAM.…

…

a0[k]

b0[k]

d0[k]

e0[k]

s[n]u[n]

a1[k]

b1[k]

d1[k]

e1[k]

aM-1[k]

bM-1[k]

dM-1[k]

eM-1[k]

M-pointIDFT

M-pointIDFT

A0[k]

B0[k]

A1[k]

-B1[k]

A2[k]

B2[k]

AN-1[k]

-BN-1[k]

0

0

0

0

0

0

0

0

1

1

2

2

1

1

N-1

N-1

N

N

M-1

M-1

M-1

M-1

……

……

Fig. 3.14 Implementation of MC-OQAM transmitter with IDFT and polyphase filterbank.


(3.63)

The auxiliary variables , serve as input signals to the DFT processors. The output of the polyphase filterbank is

, (3.64)

The signal after the delay element is

(3.65)

which leads for both cases to the output signal (3.63), and thus the equivalence of both structures is proved.

For the receiver, it is especially important where to attach the one-tap equaliser coeffi-cients since the receiver filter decomposes its input signal into real and imaginary part and processes them separately. Therefore the FDE (frequency domain equalisation) stage must be connected before the receiver filters and cannot be attached behind as it is possi-ble for MCM without offset.

We denote the real and the imaginary part of the input signal as and , respec-tively, the equaliser coefficients are decomposed accordingly:

, (3.66)

The output signal of the receiver in Fig. 3.13 for branch µ is given by

(3.67)

In the fast implementation of the receiver in Fig. 3.15, the polyphase filterbank is defined as in (3.48). The output signals of the DFT processors are combined to yield the desired output :

s n[ ] w νn– Aν k[ ]g n kM–[ ] Bν k[ ]g n M2-----– kM–+

k∑

ν 0=

N 1–

∑=

Aν k[ ] Bν k[ ]

di k[ ] gi k l–[ ] Aν l[ ]w νi–

ν 0=

N 1–

∑l

∑= ei k[ ] gi k l–[ ] 1–( )νBν l[ ]w νi–

ν 0=

N 1–

∑l

∑=

u n[ ]ei M 2⁄+ k 1–[ ] for i 0 … M

2----- 1–, ,=

ei M 2⁄– k[ ] for i M2----- … M 1–, ,=

=

s' n[ ] s'' n[ ]

s n[ ] s' n[ ] js'' n[ ]+= ψµ ψ'µ jψ''µ+=

Yµ k[ ] ψ'µ s' m[ ] 2πM------µmcos ψ'µ s'' m[ ] 2π

M------µmsin ψ''µs' m[ ] 2π

M------µmsin ψ''µs'' m[ ] 2π

M------µmcos–+ +

m∑=

g kM M2-----– mod2 µ 1+( )M

2-----– m–⋅

j ψ'µs' m[ ] 2πM------µmsin– ψ'µs'' m[ ] 2π

M------µmcos ψ''µs' m[ ] 2π

M------µmcos ψ''µs'' m[ ] 2π

M------µmsin+ + +

+

g kM M2-----– mod2 µ( )M

2-----– m–⋅

Yµ k[ ]


(3.68)

The DFT outputs are given by

(3.69)

which yields with (3.68) the same output signal (3.67) as the filterbank system in Fig. 3.13.

Yµ k[ ]Re Aµ k[ ] j Im Bµ k[ ] + µ, even

Re Bµ k[ ] – j Im Aµ k[ ] + µ, odd

=

Aµ k[ ] ψµ wµms m[ ]g kM M– m–[ ]m∑=

Bµ k[ ] 1–( )µψµ wµms m[ ]g kM M 2⁄– m–[ ]m∑=

Fig. 3.15 Implementation of MC-OQAM receiver with DFT and polyphase filterbank.

…

…

a0[k]

b0[k]

u[n]

s[n]a1[k]

b1[k]

aM-1[k]

bM-1[k]

M-pointDFT

M-pointDFT

A0[k]

B0[k]

A1[k]

B1[k]

AN-1[k]

BN-1[k]

…

…

ψ0

ψ0

ψ1

ψ1

ψN-1

ψN-1


3.5 Further Aspects of Multicarrier Modulation

3.5.1 Peak-to-Average Power Ratio

The high peak-to-average power ratio (PAPR) or crest-factor is considered a major prob-lem of OFDM for practical implementation, because a high PAPR requires a wide linear range of the power amplifier.

The PAPR for the system in Fig. 3.1 is calculated in the appendix as

(3.70)

where it has been assumed that on all subchannels the same -QAM square constella-tion is used, the sequences are uncorrelated and the transmitter filter has unit signal energy.

The second term in (3.70) is the PAPR of which ranges between 1 for 4-QAM and nearly 3 for large constellations. The most important conclusion we can draw out of (3.70) is that the PAPR is proportional to the number of subcarriers, while the depend-ence on the pulse shaping filter is rather low, taking into account that the values of

for are close to zero anyway. This means that for MCM with pulse shaping on the one hand the peak power problem is still present, but on the other hand the same algorithms [145-154] for mitigating it can be employed.

3.5.2 Spectral Properties

For the calculation of the PSD, the system model in Fig. 3.1 is best suited. Assuming that the samples of the sequence are uncorrelated and the mean energy per symbol is

, its autocorrelation function is . This gives the PSD of the modulated subcarrier ν:

(3.71)

Thus, the PSD of the output signal is given by

(3.72)

The output power spectrum is thus governed by the pulse shaping filter . The PSD of standard OFDM (see Fig. 3.8) has a ripple which is increasing with the length of the guard interval [128]. The MC-OQAM system as described in section 3.4.2 gives exactly a constant PSD [155]. This is the ideal PSD from an information theoretical point of view,

γ max s t( ) 2

Ε s t( ) 2 ------------------------------- N

3 Mq 1–( )2

Mq 1–------------------------------- T g kT( )

k ∞–=

∞

∑ 2

⋅ ⋅= =

MqXν k[ ] g t( )

Xν k[ ]

Ng t( )

g kT( ) k 0≠

Xν k[ ]EX ϕX k[ ] EXδ k[ ]=

Φsνω( ) TEX G ω ων–( ) 2=

Φs ω( ) TEX G ω ων–( ) 2

ν 0=

N 1–

∑=

g t( )

Further Aspects of Multicarrier Modulation 71

provided the channel gain to noise ratio is constant, too. Further details concerning this point shall be discussed in chapter 6.

3.5.3 Equalisation for Multicarrier Systems with Pulse Shaping

In MCM systems with pulse shaping, the impulse response of the transmit filter is gener-ally longer than the symbol period . This inhibits the use of a guard interval like explained in section 3.3.2 and thus the decomposition of the channel into independent flat subchannels does not hold any more. Hence, the simple one-tap equaliser is not the optimum solution for such an MCM system. The optimal equaliser structure for MCM with pulse shaping and a non flat channel remains an open issue where further research is required. This is also the case for a fading channel [156]. For the time-invariant CaTV return channel, initial simulation results indicate that with a high number of subcarriers, the one-tap equaliser performs nearly as well in MCM with pulse shaping as in OFDM systems with guard interval [157].

For DMT systems, equaliser structures have been developed which apply in addition to the FDE an additional time domain equaliser (TDE) in the receiver before the DFT proc-essor [158-124]. Other approaches extend the FDE to a multi-tap equaliser [163-165] and recent research aims at equalisers for DMT systems with insufficient or even without guard interval [166, 167].

These new advances can serve as a starting point in the search for effective equaliser structures for pulse shaped MCM systems.

T


4 Multicarrier Modulation with Pulse Shaping

In MCM systems with pulse shaping the transmitter and receiver filters have a non-rec-tangular impulse response. Various approaches to determine appropriate pulse shapes can be found in the literature. This chapter presents some approaches and the motiva-tions behind them. In chapter 5, new methods follow.

4.1 Optimisation Criteria

4.1.1 Localisation in Time and Frequency

An often cited objective for pulse shaping is the so called localisation in the time-fre-quency plane, i.e. the basic pulse shape should be well localised in time and its spectrum should be well localised as well. Quantitative measures for time and frequency localisa-tion are the time and frequency dispersion, also called time (frequency) spread, which are defined as a second moment of the time function and the spectrum, respectively. These measures represent the energy localisation around and .

For a time function with signal energy

(4.1)

the time dispersion is defined as

(4.2)

and the frequency dispersion as

(4.3)

Because of the properties of the Fourier transform a small time dispersion leads to a high frequency dispersion and vice versa. For the product, the uncertainty principle states [168]:

(4.4)

where equality holds only for the Gaussian pulse shape which is treated in a following section.

t 0= ω 0=

g t( )

Eg g t( ) 2 td∞–

+∞

∫1

2π------ G ω( ) 2 ωd

∞–

+∞

∫= =

Dt

Dt2 1

Eg------ t

T---

2g t( ) 2⋅ td

∞–

+∞

∫=

Dω

Dω2 1

2πEg------------- ωT( )2 G ω( ) 2⋅ ωd

∞–

+∞

∫=

Dt Dω⋅ 12---≥

73


4.1.2 Out-of-Band Energy

Low out-of-band or stopband energy is desirable in multicarrier systems to reduce the spectral overlap between adjacent subchannels and the spectrum outside the modulated subcarriers. Of course, instead of minimising the out-of-band energy, the in-band energy can be maximised.

The in-band energy of a subchannel according to the continuous-time model of Fig. 3.1 is defined as

(4.5)

The parameter defines the width of the considered frequency band, corre-sponds to a frequency band as wide as the subcarrier spacing .

With the auxiliary function

the in-band energy can be written as

which leads to the expression

(4.6)

For the discrete-time system in Fig. 3.5 with sampling period according to (3.20) the in-band energy is defined via the z-transform of :

with (4.7)

As a function of , the in-band energy is given as

(4.8)

Ein1

2π------ G ω( ) 2 ωd

η∆ω 2⁄–

+η∆ω 2⁄

∫=

η η 1=∆ω

F ω( ) G ω( ) 2 G ω( )G* ω( )= = g t( ) g* t–( ) g τ( )g* τ t–( ) τd∞–

+∞

∫ f t( )= =

Ein1

2π------ f t( )e jωt– td ωd

∞–

+∞

∫η∆ω

2------------–

+η∆ω2

------------

∫1

2π------ g τ( )g* τ t–( ) τ e jωt– ωd td

η∆ω2

------------–

+η∆ω2

------------

∫d∞–

+∞

∫∞–

+∞

∫= =

Einη ∆ω

2π------------ g τ( )g* t( ) si η ∆ω

2------------ t τ–( )

td τd∞–

+∞

∫∞–

+∞

∫=

TAg n[ ]

EinTA2π------ G e

jωTA( )2

ωdη∆ω

2------------–

η∆ω2

------------

∫= G z( ) g n[ ]z n–

n ∞–=

+∞

∑=

g n[ ]

Einη∆ωTA

2π------------------- g n[ ]g* m[ ] si

η∆ωTA2

------------------- n m–( )

m ∞–=

+∞

∑n ∞–=

+∞

∑=

Pulse Shaping Approaches 75

4.2 Pulse Shaping Approaches

4.2.1 Gaussian Pulses

Gaussian impulses were introduced in multicarrier systems in [125] as an optimum trade-off between minimum ISI and small spectral overlapping for a mobile transmis-sion environment. The impulse

(4.9)

is in fact the only one which fulfils the uncertainty principle (4.4) with equality. The inconvenience of this choice is that it does not fulfil the PR condition and thus even for an ideal channel significant ISI and ICI occurs. It is argued in [169] that in a time and fre-quency dispersive channel interference occurs anyway and thus the interference intro-duced by the pulse shaping plays no significant role. Because of the inherent ISI and especially ICI, powerful two-dimensional equaliser structures are required. Details about the design of this technique called MCSIS (multicarrier system with soft impulse shaping) can be found in [125, 126, 169-173].

The energy of is , independent of the parameter which controls the time or frequency width. The time and the frequency dispersion are

,

which gives the minimum dispersion product. The in-band energy can be calculated as

For and the carrier spacing (3.14), this gives the curve shown in Fig. 4.1 where the suggested value of [125, 126] is marked with a cross.

gmcsis t( ) αT

-------eαtT-----

2

–= Gmcsis ω( ) π

α---e

ωT2α-------

2

–=

0.5 1 1.5 2 2.5 30.7

0.75

0.8

0.85

0.9

0.95

1

Fig. 4.1 In-band energy for gaussian pulse shaping as a function of the parameter α.α

EinEg-------

g t( ) Eg 1 T⁄( ) π 2⁄= α

Dt1

2α-------= Dω α=

EinEg------- 1 2Q η∆ωT

2α---------------

–=

η 1=α ∆ωT 2⁄=


4.2.2 Time-Limited Cosine Pulse Shaping

In [174, 175] Li and Stette present a pulse shaping approach which is driven by techno-logical feasibility based on surface acoustic wave chirp lines, instead of mathematical optimisation. The carrier spacing is chosen in accordance with (3.14) and the basic impulse for an MC-OQAM system is given by

The elementary impulses can be calculated easily and are shown in Fig. 4.2. The per-formance degradation due to time and frequency offsets in this system and others has been investigated by Dardari [176].

The time dispersion for this impulse is given by

The frequency dispersion does not converge and thus the product tends to infinity. Therefore this impulse is far from optimum regarding the localisation in fre-quency. The in-band energy is , which is rather low compared to other approaches. However, it has to be recognised that the aim of this approach was neither a good time-frequency localisation nor a low stop-band energy and that improvements compared to rectangular pulse shapes have been achieved.

gLS t( ) πtT-----

cos rect tT---

⋅= GLS ω( ) T2--- si ωT

2------- π

2---–

si ωT2

------- π2---+

+

2πT ωT2

------- cos

π2 ωT( )2–---------------------------------= =

Dt112------ 1

2π2---------–=

Dω DtDω

Ein Eg⁄ 0.6984=

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5r0(t)

Rer10

(t)Imr

10(t)

Rer20

(t)Imr

20(t)

Fig. 4.2 Elementary impulses for impulse with offset modulation.gLS t( )

t T⁄


4.2.3 Hermite Functions

Like the previous technique, the approach of Haas [127, 177, 178] considers a mobile channel and thus derives a pulse that minimises the time-frequency dispersion product. In contrast to MCSIS, the extended Nyquist criterion is taken into account. The deriva-tion of the impulse is based on Hermite functions which have – like the Gaussian func-tion – the special property to be identical to their Fourier transforms. The impulse is derived as

(4.10)

where denotes the n-th normalised Hermite function as given in the appendix 8.2and the series coefficients are given as

The impulse was designed for a carrier spacing , which is the double of (3.14) and thus reduces the spectral efficiency to 50%. However, by applying offset QAM, the impulse can be used with the usual carrier spacing (3.14) without loss in spectral efficiency. The elementary impulses calculated according to (3.50) and drawn in Fig. 4.3 reveal that the orthogonality criterion is satisfied with offset modulation. It is curious to note that this is not pointed out in [178] although MC-OQAM is treated there, too. The time and frequency dispersion are

,

gH t( ) 24 H4kD4k 2t( )k 0=

4

∑=

Dn t( )Hn

H0 1.1850899=

H4 1.9324881– 10 3– H0⋅ ⋅=

H8 7.3110588– 10 6– H0⋅ ⋅=

H12 3.1542096– 10 9– H0⋅ ⋅=

H16 9.6634138 10 13– H0⋅ ⋅=

∆ω 4π T⁄=

gH t( )

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1 r0(t)

Rer10

(t)Imr

10(t)

Rer20

(t)Imr

20(t)

Fig. 4.3 (a) Hermite impulse . Since this function is Fourier-invariant, the spectrum has the same shape. (b) Elementary impulses for MC-OQAM with impulse .

gH t( )gH t( )

(a)t T⁄

g Ht()

(b)t T⁄

Dt 0.2015= Dω 2.5325=


which gives the product , which is only slightly greater than the theoreti-cal minimum. The in-band energy can be calculated numerically as .

This optimisation method based on normalised Hermite functions can be seen as a very elegant and effective method to find an impulse with minimum time-frequency disper-sion while satisfying the extended Nyquist criterion. It probably did not receive its due attention because it has not been published in widely known magazines or conferences.

4.2.4 The Isotropic Orthogonal Transform Algorithm

The isotropic orthogonal transform algorithm (IOTA) is a design approach to determine appropriate pulse shapes for wireless channels. The design involves a mathematical analysis with Hilbertian bases [179]. The impulse shape is described in [179] with orthogonality and Fourier transform operators. A closed form expression was found later [180, 181] and is given as

with

and

The coefficients are given numerically in [180, 181]. The articles [181, 182] describe the IOTA approach in the context of cosine modulated filterbanks and explain how to fit the IOTA function into modulated filterbanks with a given length. A discrete-time analy-sis of MC-OQAM is provided in [183, 144], orthogonality conditions are derived, the link to filterbank theory is established and a comparison of truncated discrete square-root raised cosine filtering and truncated IOTA functions is conducted. A generalisation to biorthogonal frequency division multiplex (BFDM) was carried out in [184]: receiver and transmitter filter are no longer required to be identical. The time-frequency localisation properties of biorthogonal filterbanks are studied in [185]. It seems that not much is to be gained in terms of localisation when generalising to biorthogonal filterbanks, but the overall delay can be reduced at the expense of the localisation properties. Efficient implementations and design examples for the maximisation of the time-frequency local-isation or the in-band energy are given in [186, 187]. A design method for biorthogonal filterbanks which is based on lattice filter structure which structurally ensures the condi-tion for perfect reconstruction is presented in [188, 189].

The time and frequency dispersion of the IOTA-impulse and their product are

DtDω 0.5104=Ein Eg⁄ 0.7806=

giota t( ) 224 T

---------------zα 2 tT---

=

zα t( ) 12--- dk α, gα t k 2+( ) gα t k 2–( )+

k 0=

∞

∑ dl 1 α⁄, 2πlt 2( )cosl 0=

∞

∑⋅=

gα t( ) 2α4 e παt2–=

dk α,


, ,

and the in-band energy is . These values are very similar to those of the impulse based on Hermite functions and indeed Fig. 4.4 shows that both pulse shapes are very similar. They have been obtained by very different mathematical approaches but are based on the same objective function (minimisation of the time-frequency disper-sion) and the same constraints (the extended Nyquist criterion).

4.2.5 Extended Lapped Transforms

Another approach to MCM constitutes the extended lapped transform (ELT) which at first glance seems to have little in common with multicarrier communication systems. The ELT is described as an extension of the modulated lapped transform and has been intended primarily as a substitute for block transforms in transform coding systems for speech and image coding [190, 191]. In Fig. 4.5 the block diagram for a subband coding system with ELTs is given.

The matrix , with , is a matrix, which means that in the analysis filterbank the transform block contains inputs and M outputs. The matrix elements of the transform matrix P are given by

where denotes the prototype filter. Assuming that the prototype is symmetrical, i.e.

, for

the condition for perfect reconstruction is

Dt 0.2018= Dω 2.5360= DtDω 0.5118=

Ein Eg⁄ 0.7842=

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1 r0(t)

Rer10

(t)Imr

10(t)

Rer20

(t)Imr

20(t)

Fig. 4.4 (a) IOTA function . Since this function is Fourier-invariant, the spectrum has the same shape. (b) Elementary impulses for MC-OQAM with impulse .

giota t( )giota t( )

(a)t T⁄

g iota

t()

(b)t T⁄

P pn k,( )= n 0 … LM 1–, ,= k 0 … M 1–, ,= LM M×LM

pn k, g n[ ] 2M----- n M 1+

2--------------+

k 12---+

πM-----

cos⋅=

g n[ ]

g n[ ] g LM 1– n–[ ]= n 0 … LM 1–, ,=


, for (4.11)

where denotes the overlapping factor. The block diagram of the ELT in Fig. 4.5can be substituted by the equivalent filterbank structure depicted in Fig. 4.6. The impulse responses of the filterbank are given by

,

It can be shown [190, 192] that the condition for perfect reconstruction holds as well when the analysis and synthesis filterbank are exchanged. This makes the concepts which were developed for filterbanks and subband coding applicable to multicarrier systems. Thus the condition for perfect reconstruction which is valid for the filterbank in Fig. 4.6 also applies to the transmultiplexer in Fig. 3.6.

g n iM+[ ] g n iM 2sM+ +[ ]⋅i 0=

2K 1– 2s–

∑ δ s[ ]=s 0 … K 1–, ,=n 0 … M 1–, ,=

K L 2⁄=

M

z-1

M

z-1

z-1

M

P'

subband

pro

cessin

g

P

M

M

M

z-1

z-1

z-1

x n[ ]

0

1

LM-1

s k0[ ]

s k1[ ]

s kM-1[ ]

analysis filterbank synthesis filterbank

][ˆ nx

Fig. 4.5 Subband signal processing with extended lapped transform.

fk n[ ] pi k,=

gk n[ ] fk LM 1– n–[ ] pLM 1– n– k,= =k 0 … M 1–, ,=n 0 … LM 1–, ,=

M

M

M

subband

pro

cessin

g

M

M

M

x n[ ] s k0[ ]

s k1[ ]

s kM-1[ ]

g n0[ ]

g n1[ ]

g nM-1[ ]

f n0[ ]

f n1[ ]

f nM-1[ ]

analysis filterbank synthesis filterbank

][ˆ nx

Fig. 4.6 Equivalent filterbank structure to Fig. 4.5.


Until now the concept of the lapped transforms added nothing new – it provides just another viewpoint of the transmultiplexer of section 3.3.1. What the theory of the lapped transforms has to offer is a fast implementation of the structure in Fig. 4.5 which guaran-tees perfect reconstruction, i.e. orthogonality according to (4.11) is ensured by the struc-ture itself. This fast implementation structure is drawn in Fig. 4.7 and contains several simple block transforms and a type-IV DCT.

The matrix of stage is given by

where

and the counter-identity matrix

It can be shown [190, 192] that by this choice of the matrices the condition for perfect reconstruction is satisfied for arbitrary butterfly angles . These angles deter-mine the independent coefficients of and thus instead of a constrained optimi-sation for variables and constraints (see (4.11)), an unconstrained optimisation for only variables can be carried out.

M

M

Mx n[ ]

M

z-1

z-1

z-1

DK-1 DK-2 D0

z-2

z-2

z-2

z-2

z-1

z-1

u 0[ ]

u M[ /2-1]

u M[ /2]

u M[ -1]

DCT-IV

s k0[ ]

s kM-1[ ]

Fig. 4.7 Fast implementation for ELT with DCT (analysis filterbank). This structure inherently fulfils the condition for perfect reconstruction.

……

……

……

……

……

…

Dk k

DkCk– SkJ

JSk JCkJ

=

Ck diag θ0 k,cos θ1 k,cos … θM 2⁄ 1– k,cos, , , =

Sk diag θ0 k,sin θ1 k,sin … θM 2⁄ 1– k,sin, , , =

J

0 … 0 10 … 0 1 0

1 0 … 0

=

… …

Dkθn k, KM 2⁄

KM g n[ ]KM KM

KM 2⁄


This optimisation has been investigated in [192, 193] and the resulting prototype for subcarriers and overlap factor is depicted in Fig. 4.8. The filterbanks

based on the ELT are not directly comparable to the former approaches because they do not fit into the system models of Fig. 3.5 or Fig. 3.13. Nevertheless, the carrier spacing can be determined from Fig. 2 of [193] to and thus the in-band energy can be calculated numerically as for the prototype depicted in Fig. 4.8. The time and frequency dispersion are

, ,

Although only an unconstrained optimisation has to be carried out, the numerical prob-lem of finding the minimum of a high-dimensional function still persists. The objective function is highly nonlinear and thus optimisation with numerical methods is difficult.

4.2.6 Further Approaches and Ideas

A similar approach to the IOTA was presented in [194] which first derives discrete-time orthogonality conditions for MC-OQAM, and then carries out an orthogonalisation pro-cedure similar to that of [179], but which is performed in the frequency domain of the Zak transform instead of the continuous-time domain.

In [195, 196] an OFDM scheme with non-overlapping time waveforms is presented. Because the length of the prototype filter is limited to one symbol interval T, a guard interval can be introduced as usual. However, it is not clear how the prototype filter should be chosen. The filters presented in [196] either reduce the spectral efficiency or introduce ICI which degrades the system performance compared to standard OFDM (see Fig. 5 of [196]).

A very sophisticated approach which adapts the pulse shapes of the transmitter and the receiver to the characteristics of a time-variant channel was presented in [197]. For the

M 128= K 2=

∆ω π T⁄=Ein Eg⁄ 0.813=

Dt 0.4644= Dω 1.2114= DtDω 0.813=

0 0.5 1 1.5 2 2.5 3 3.5 4−0.2

0

0.2

0.4

0.6

0.8

gn[

]

n M⁄Fig. 4.8 Prototype filter for ELT with , .M 128= K 2=


channel a WSSUS (wide sense stationary uncorrelated scattering) model [198] is used and the prototype functions are not required to be orthogonal. The optimisation proce-dure is based on the scattering function of the channel and the noise is taken into account. It is claimed that this impulse design outperforms a system with Hermite impulses when a doubly dispersive channel is assumed.

Wavelet based multicarrier systems are compared with FFT based schemes in [199] and it is found that for time-invariant channels, like the twisted-pair telephone subscriber line, wavelet based MCM performs worse.

The opposite result is obtained in [200] where DWMT systems are compared to standard DMT for transmission over the telephone subscriber line. (The idea of using DWMT instead of DMT for ADSL systems was already introduced in [201] by the same author.) Both systems are compared for various channels and noise environments by simulations and the authors conclude that the DWMT system offers much more robustness ʺthan other multicarrier implementations with regard to ICI and to narrowband channel dis-turbancesʺ. The DWMT system used in the paper is not specified in detail but it is equiv-alent to a cosine modulated filterbank.

Another comparison between DMT systems and cosine modulated filterbanks is carried out in [202] on the basis of analytical calculations and simulation results. The authors conclude that ʺthe cosine modulated filterbank system seems a candidate worthy of con-sideration for the implementation of a multicarrier systemʺ.


5 Optimised Impulses for Multicarrier Offset-QAM

In this chapter a novel pulse shape for MCM is presented. The application we have in mind is the CaTV return path and hence we have to deal with a time-invariant channel with a frequency-selective transfer function and coloured noise, especially with narrow-band interference. Thus, the appropriate optimisation criterion is not a good time-fre-quency localisation of the basic impulse but a low stopband energy in order to keep spectral overlap between adjacent subchannels as small as possible. If a good spectral containment of the subchannels can be achieved, narrowband interferers will affect less subchannels.

Our work is inspired by [203, 204] who derived impulses for time-continuous OFDM and [205] who treated the same task for single carrier systems. MCM systems are imple-mented with digital signal processing and therefore a time-discrete representation of the pulse shapes is needed. The filter coefficients can in principle be derived from the contin-uous-time impulse responses by sampling. However, a direct calculation in the discrete-time domain is not only the straightforward solution but is also more accurate. Moreo-ver, when numerical procedures are involved, the continuous-time functions have to be discretised anyway. Therefore, an approach based on a discrete-time system model leads to a straightforward, precise solution [206, 207].

For the derivation of the pulse shape, a certain set of orthogonal functions plays a central role. The prolate spheroidal wave functions (PSWF) are applied in the optimisation pro-cedure in [204, 205] and their discrete-time counterparts, the discrete prolate spheroidal sequences (DPSS) shall be employed for the derivation of the new pulse shape.

5.1 The Prolate Spheroidal Functions

5.1.1 The Prolate Spheroidal Wave Functions

The prolate spheroidal wave functions [208-211] are a set of functions , with the corresponding eigenvalues . They are defined as solutions of the integral equation

, (5.1)

and depend on the parameter .

The definition can also be written as

ψn t( ) n N0∈λn

λnψn t( )ωgπ

------ ψn τ( ) si ωg t τ–( )( ) τdT 2⁄–

T 2⁄

∫= n N0∈

c ωgT π⁄=

85


(5.2)

This form of the definition reveals that windowing of the function with a rectan-gular window of length and consecutive filtering with an ideal lowpass with cut-off frequency yields a scaled version of the same function , as indicated in the block diagram in Fig. 5.1.

With we can transform the definition into frequency domain:

(5.3)

This equation indicates that the PSWF are bandlimited with cut-off frequency . Thus, filtering of with an ideal lowpass yields the original function:

(5.4)

By comparing (5.1) with (5.4), we recognise that the eigenvalue represents how much of the waveform is concentrated in the interval .

The PSWF are orthogonal in as well as in :

(5.5)

They form a basis for bandlimited energy signals:

with (5.6)

and for time-limited energy signals:

, with (5.7)

These special properties makes the PSWF well suited for a variety of applications [208-213].

λnψn t( ) ψn t( ) rect tT---

ωgπ

------ si ωgt( )=

ψn t( )T

ωg ψn t( )

ψn t( ) ideal lowpass

H ω( ) rect ω2ωg---------

=windowingwith rect t T⁄( )

λnψn t( )

Fig. 5.1 Interpretation of the definition of the PSWF.

Ψn ω( ) F ψn t( ) =

λnΨn ω( ) T2π------ rect ω

2ωg---------

Ψn ζ( ) si T2--- ω ζ–( )

ζd∞–

∞

∫=

ωgψn t( )

ψn t( )ωgπ

------ ψn τ( ) si ωg t τ–( )( ) τd∞–

∞

∫=

λnψn t( ) T 2⁄– T 2⁄,( )

∞– ∞,( ) T 2⁄– T 2⁄,[ ]

ψn t( )ψm t( ) td∞–

∞

∫ δ n m–[ ]=

ψn t( )ψm t( ) tdT 2⁄–

T 2⁄

∫ λnδ n m–[ ]=

f t( ) anψn t( )n 0=

∞

∑= an f t( )ψn t( ) td∞–

∞

∫=

h t( ) bnψn t( )n 0=

∞

∑= t T2---< bn

1λn----- h t( )ψn t( ) td

T 2⁄–

T 2⁄

∫=

The Prolate Spheroidal Functions 87

5.1.2 The Discrete Prolate Spheroidal Sequences

The discrete prolate spheroidal sequences [214-216], which are also known as Slepian sequences are denoted as with , . They are defined as

(5.8)

with the normalisation

, , (5.9)

The are the eigenvalues. The sequences and their eigenvalues depend on the parameters and . In order not to overload the notation we will omit this detail normally in our notation. If it is required we will denote the DPSS as .

The DPSS can be considered as sequences on , or as indexlimited sequences with the domain . In the following, we will stick to the latter.

The Slepian sequences have the following properties:

1. Orthogonality on and

, (5.10)

2. Eigenvalues

, (5.11)

3. The matrix shall be given with , . Then the are the eigenvalues and the are the eigenvec-

tors of R.

4. Symmetry:

, (5.12)

The most important property of the Slepian sequences for our purposes is: The DPSS are the set of orthogonal indexlimited sequences with most concentrated spectrum [214]. This unique characteristic makes them an ideal candidate for the derivation of spectrally concentrated impulses for MCM. Some Slepian sequences are drawn in Fig. 5.2.

vk n[ ] n Z∈ k 0 1 … Np 1–, , , ∈

λkvk n[ ] 2W si 2πW n m–( )( )vk m[ ]m 0=

Np 1–

∑=

vk n[ ] 2

n 0=

Np 1–

∑ 1= vk n[ ]n 0=

Np 1–

∑ 0≥ Np 1– 2n–( )vk n[ ]n 0=

Np 1–

∑ 0≥

λk vk n[ ] λkNp N∈ W R+∈

v Np W,( )k n[ ]

n Z∈n 0 … Np 1–, , ∈

n 0 … Np 1–, , ∈ n Z∈

vi n[ ]vj n[ ]n 0=

Np 1–

∑ λi vi n[ ]vj n[ ]n ∞–=

∞

∑ δ i j–[ ]= = i j, 0 … Np 1–, , ∈

1 λ0 λ1… λNp 1– 0>≥ ≥ ≥ ≥ λk

k 0=

Np 1–

∑ 2WNp=

R rm n,( )= rm n, 2W si 2πW m n–( )( )=m n, 0 … Np 1–, , ∈ λk vk n[ ]

vk n[ ] 1–( )kvk Np 1– n–[ ]=

vNp W,( )

k n[ ] 1–( )kvNp 1– k–Np 1 2⁄ W–,( )

Np 1– n–[ ]=n Z∈ k 0 … Np 1–, , ∈


The eigenvalues of some sets of DPSS with different parameters W are depicted in Fig. 5.3. For small indices the eigenvalues have a value near 1, while for bigger indices they tend to zero. They drop sharply at .

The definition (5.8) allows the following interpretation: ideal lowpass filtering of the indexlimited sequence yields a scaled version of this sequence, and the scaling fac-tor is the eigenvalue . Thus, the eigenvalue stands for the lowpass portion of the sequence .

k 2WNp=

0 100 200 300 400 500 600−0.1

−0.05

0

0.05

0.1

0.15k=1k=3k=5k=7

0 100 200 300 400 500 600−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1k=0k=2k=4k=6

Fig. 5.2 Some Slepian sequences for , , (a) with even indices, (b) with odd indices.

Np 513= WNp 4=v k

n[]

v kn[

]

n n(a) (b)

vk n[ ]λk

vk n[ ]

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W Np =1

W Np =5

W Np =16

W Np =31

Fig. 5.3 Eigenvalues of the DPSS for and various values of .Np 64= W

k

λ k

The Optimisation Procedure 89

5.2 The Optimisation Procedure

For practical implementation with digital signal processing, it is convenient to define the impulse shaping filters as FIR filters with finite length. Also, we assume that the impulse response is symmetrical, thus for the prototype holds:

(5.13a) for , (5.13b)

The filter length is thus .

5.2.1 Orthogonality Condition

We consider the discrete-time system model for MC-OQAM of Fig. 3.13. The delay in the orthogonality condition (3.61) is thus

(5.14)

By inserting (5.13a) into (3.60), it can be shown that , i.e. at the sam-pling instants the elementary impulses take on real values, and we get from (3.60):

(5.15)

It can be shown with elementary calculations that holds for odd , i.e. the orthogonality criterion (3.61) is satisfied. For we obtain

(5.16)

which can be further simplified with (5.14) to

(5.17)

The index range is sufficient since both sides of (5.17) are even in .

With (5.13b) we can limit the index range of the summation variable to

The two time-shifted impulses in (5.17) overlap for , and as both sides are even in k it suffices to consider the range . Thus, we get from (5.17)

g n[ ] g LM n–[ ]=g n[ ] 0= n 0< n LM>

LM 1+

dg L 1+=

Im rν µ, kM[ ] 0=

rν µ, kM[ ] g kM m– mod2 ν( )M2-----– 2π

M------ ν µ–( )m

g m M2-----– mod2 µ 1+( )M

2-----–cos

m ∞–=

∞

∑=

rν µ, kM[ ] 0= ν µ–ν µ– 2χ=

rν µ– kM[ ] g m 2L 1 k–+2

------------------------M+ g m k 1–2

-----------M+ 4πM------χmcos

m ∞–=

∞

∑ δ χ k dg–,[ ]= =!

r χ k,[ ] rν µ– k L– 1–( )M[ ]=

g m L k–2

------------M+ g m L k+2

------------M+ 4πM------χmcos

m ∞–=

∞

∑ δ χ k,[ ]= =!

χ 0 … N 2⁄ 1–, , ∈ χ

m L k–2

------------M– … L k–2

------------M, ,=

L– k L< <k 0 … L 1–, ,=


(5.18)

with

This is an equation system with equations and independent variables, namely . At this point, the motivation for choosing the more complex offset QAM can be explained: the same derivation for the system model in Fig. 3.5 gives an equation system with the double number of equations and thus much less degrees of freedom. Therefore we will only consider the MC-OQAM for the following optimisation. In order to gain some degrees of freedom for the following optimisation, the upsampling factor must be chosen greater than the number of carriers . Since for practical rea-sons it is convenient that the upsampling factor be an integer multiple of the number of carriers, we choose

(5.19)

Notice that the choice of this factor does not affect the spectral efficiency or the carrier spacing. It merely introduces oversampling. In standard OFDM implementations, over-sampling is tacitly introduced by leaving a number of subcarriers unmodulated. In our system model, all subcarriers can be modulated.

5.2.2 Objective Function

With (3.14), (3.20), (5.19) and , the in-band energy follows from (4.8)

(5.20)

The parameter controls the bandwidth of the considered frequency band. Note that in [204] the optimisation was performed only for . For this case, a solution which gives zero out-of-band energy, but requires impulses with infinite length, is the square root raised cosine impulse which was presented in section 3.4.2. Therefore, if the impulse length is chosen sufficiently long, the solution of [204] resembles the square root raised cosine impulse.

With as objective function and the constraints (5.18), the optimisation problem for the filter coefficients is already formulated. However, both the objective function as well as the constraints depend in a highly nonlinear manner on the filter coefficients and thus numerical optimisation with standard algorithms is very difficult, especially when

r χ k,[ ] g m L k–2

------------M+ g m L k+2

------------M+ 4πM------χmcos

m L k–2

------------M–=

L k–2

------------M

∑ δ χ k,[ ]= =!

χ 0 … N2---- 1–, ,=

k 0 … L 1–, ,=

LN 2⁄ LM 2⁄ 1+g 0[ ] … g LM 2⁄[ ], ,

M N

M 2N=

η 1=

Ein1

2N------- g n[ ]g m[ ] si π

2N------- n m–( )

n ∞–=

∞

∑m ∞–=

∞

∑=

ηη 2=

Eing n[ ]


the number of subcarriers is high. A way of simplifying the equations is the series expan-sion of the prototype with the DPSS as base functions: we choose

(5.21)

and because of the symmetry properties (see (5.12), (5.13a)), we consider only Slepian sequences with even indices. Thus, we may write the prototype as

, with (5.22)

By inserting this expansion equation into the expression for the in-band energy we obtain

With (5.23)

and the definition (5.8) follows

which can be simplified with the orthogonality property (5.10) to

(5.24)

The objective function (5.20) can thus be expressed in a simple way with the expansion coefficients .

The constraints (5.18), expressed with the expansion coefficients, are

(5.25)

Now, a numerical optimisation based on the expansion coefficients can be carried out. The numerical minimisation of (5.24) with the constraints (5.25) has been performed with the Matlab function fmincon [217]. The results of this optimisation for 32 and 128 subcarriers and different filter lengths are illustrated in Fig. 5.4. The expansion coeffi-cients decrease rapidly, indicating that only few sequences are needed in the expansion (5.22). This is what was to be expected: the in-band energy of the DPSS is decreasing

Np 2LN 1+=

g n[ ] aiv2i n[ ]i 0=

LN

∑= ai g n[ ]v2i n[ ]n 0=

2LN

∑=

Ein1

2N------- ai aj v2i n[ ] v2j m[ ] si π

M----- n m–( )

m 0=

2LN

∑n 0=

2LN

∑j 0=

LN

∑i 0=

LN

∑=

W 14N-------=

Ein ai ajλ2j v2i n[ ]v2j n[ ]n 0=

2LN

∑j 0=

LN

∑i 0=

LN

∑=

Ein ai2λ2i

i 0=

2LN

∑=

ai

r χ k,[ ] aiaj v2i n kN–[ ]v2j n kN+[ ] 2πN------χn

cosn kN=

2L k–( )N

∑j 0=

LN

∑i 0=

LN

∑ δ χ k,[ ]= =!

ai


with growing index and therefore the sequences with low index are stronger repre-sented. The diagram of the impulses reveals strong variations in the waveform which can only be caused by high-frequency parts that stem from DPSS with high indices. As the coefficients of the high-frequency DPSS are all close to zero, the probable cause of these variations are numerical inaccuracies.

0 200 400 600 800 1000 1200−0.02−0.01

00.010.020.030.040.050.060.070.08

0 2 4 6 8 10 12 14 16 18 20−0.5

0

0.5

1

0 100 200 300 400 500 600−0.04−0.02

00.020.040.060.080.1

0.120.14

0 2 4 6 8 10 12 14 16 18 20−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Fig. 5.4 Impulses and their corresponding eigenvalues after numerical optimisation. (a) optimised impulse for , , (b) expansion coefficients of (a) (c) optimised impulse for , , (d) expansion coefficients of (c)

N 128= L 4=N 32= L 8=

k

a k

n

gn[

]

(a) (b)

kn

gn[

]

(c) (d)

a k

4 5 6 7 8 9 10 11 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2r00

Rer1,0

Imr

1,0

0 1 2 3 4 5 6 7 8−0.2

0

0.2

0.4

0.6

0.8

1

1.2r00

Rer1,0

Imr

1,0

Fig. 5.5 Elementary impulses for optimised pulses of Fig. 5.4. (a) (b) N 128= L, 4= N 32= L, 8=

n M⁄ n M⁄(a) (b)


The elementary impulses in Fig. 5.5 do not show this behaviour and they fulfil the extended Nyquist criterion. Therefore, the impulses determined with the numerical opti-misation are a valid solution, although their varying waveform is quite undesirable and indicates numerical problems.

Another numerical problem is the high complexity which stems from the still quite com-plicated constraints and impedes the calculation of impulses with for a high number of carriers.

5.2.3 Calculation of the Expansion Coefficients

To overcome the numercial problems, a simplified method for the calculation of the expansion coefficients is needed [207]. Based on the observation that only few Slepian sequences are necessary to approximate the optimised impulse, we expand the proto-type filter into only some few sequences:

(5.26)

with . If is sufficiently small, the expansion (5.26) automatically guarantees a high in-band energy as can be seen from the distribution of the eigenvalues in Fig. 5.6, where the eigenvalues are drawn for different filter lengths. The values were calculated for subcarriers, but the diagram is fairly independent of the number of subcarri-ers and also holds for other values of N.

Thus, we can consider the objective function as already satisfied and turn our attention to the constraints.

The constraints (5.25) are replaced by

L 8=

g n[ ] aiv2i n[ ]i 0=

Na

∑=

Na LN< Na

N 32=

0 2 4 6 8 10 120

0.10.20.30.40.50.60.70.80.9

1L =4L =6L =8

Fig. 5.6 Eigenvalues with , for .Np 2LN 1+= W 1 4N( )⁄= N 32=

k

λ k


(5.27)

with , thus ,

As the expansion is now incomplete, the condition can only be approx-imated. We therefore define the quadratic error

(5.28)

which represents the derivation from the extended Nyquist criterion. Separating the terms with leads to

(5.29)

with (5.30)

or in matrix notation with ,

(5.31)

Solution for Expansion into three Sequences

For a filter length the first three sequences with even index already concentrate most spectral energy in the lowpass region as can be seen in Fig. 5.6. Thus, expansion according to (5.26) with already gives a good energy concentration. For this case, an illustrative solution for the minimisation of the squared error exists. Setting the first term in (5.29) to zero gives

(5.32)

This is equivalent to the normalisation of the signal energy of , as can be seen from (5.18). With we can find the expansion coefficient :

(5.33)

which we can substitute in (5.29). Then, the squared error only depends on and and can be visualised like is done in Fig. 5.7. The search for the minimum produced a

r χ k,[ ] aiaj v2i n kN–[ ]v2j n kN+[ ] 2πN------χn

cosn kN=

2L k–( )N

∑j 0=

Na

∑i 0=

Na

∑=

A i j p, ,[ ]

p k N 2⁄ χ+⋅= k divN 2⁄ p( )= χ modN 2⁄ p( )=

r χ k,[ ] δ χ k,[ ]=

J r χ k,[ ] δ χ k,[ ]–( )2

χ 0=

N 2⁄ 1–

∑k 0=

L 1–

∑=

χ k 0= =

J r 0 0,[ ] 1–( )2 cp2

p 1=

LN 2⁄ 1–

∑+=

cp aiajA i j p, ,[ ]j 0=

Na

∑i 0=

Na

∑=

a a0 … aNa, ,( )= Ap A i j p, ,[ ]( )i j, 0 … Na, ,==

cp aApaT=

L 4=

Na 2=J

r 0 0,[ ] ai2

i 0=

Na

∑ 1= =

g n[ ]Na 2= a0

a0 1 a12– a2

2–=

J a1 a2


squared error of (-36.8 dB). The signal energy located inside the subcar-rier frequency band is .

General Solution

For an expansion into more than three Slepian sequences, the solution is obtained by minimisation of the squared error with the constraint (5.32). This can be easily done with standard numerical routines which can be seconded by providing analytical expressions for the gradient of the object function and constraint. In Fig. 5.8 the opti-mised impulse for carriers and filter length is presented. The in-band energy of this impulse is and the squared error is .

The spectrum of the optimised impulse in Fig. 5.9 illustrates the strong energy concen-tration compared to the rectangular impulse used in conventional OFDM. Around the carrier frequency a nearly flat spectrum is achieved while the out-of-band spectral parts fall off much more rapidly. This portends that the system with optimised impulses is much less susceptible to narrowband noise than standard OFDM.

J 2.11 10 4–⋅=Ein Eg⁄ 0.919=

−1−0.5

00.5

1

−1

−0.5

0

0.5

10

10

20

30

40

a1

a2

−10

lg(J

)

Fig. 5.7 Squared error as a function of the expansion coefficients. The function is displayed in negative logarithmic scaling in order to make the minimum more visible. The parameters used for its calculation are , .

J J a1 a2,( )

N 1024= L 4=

J

N 512= L 8=Ein 0.958= J 5.5 10 6–⋅=


5.3 Simulation Results for Narrowband Interference

The ultimate figure of merit for digital communication is not the in-band energy or the time-frequency localisation but the bit error ratio (BER). Therefore, the susceptibility of the optimised system to interferers has been simulated and compared to standard OFDM.

−4 −3 −2 −1 0 1 2 3 4−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Fig. 5.8 Optimised impulse for , , N 512= L 8= Na 6=

n M⁄

gn[

]

−10 −8 −6 −4 −2 0 2 4 6 8 10−120 dB

−100 dB

−80 dB

−60 dB

−40 dB

−20 dB

0 dB

20 dB

40 dB

Fig. 5.9 Magnitude of the spectrum of the optimised impulse in comparison with the spectrum of the rectangular impulse (dotted) used in conventional OFDM.

ω ∆ω⁄

Gejω

T A(

)

Simulation Results for Narrowband Interference 97

For the first simulation, we assume a flat channel with a constant sinusoidal interferer at the frequency and an SNR of -5 dB. The bit error ratios for each subchannel are shown in Fig. 5.10. With the optimised MC-OQAM scheme the interferer affects – as expected – only very few channels in the region around the interferer. For all other subchannels, no bit errors occur. In contrast, the OFDM subchannels from to

are heavily degraded. The total BER over all subchannels is for the MC-OQAM and for the standard OFDM system. For time-invariant channels, like the CaTV return channel or the telephone subscribe line, this advantage can be enhanced by using adaptive bitloading schemes like explained in the next chap-ter.

Simulations with AWGN show identical results for both systems, the BER is equal to the theoretical curve for QAM.

For another simulation, the model for narrowband noise of section 2.3.3 has been used. Narrowband Gaussian noise with a bandwidth of is added to the transmitter out-put signal. The centre frequency of the interferer is , thus the subchannels around index 120 are affected primarily. Fig. 5.11 shows the simulation results. This sto-chastical signal resembles better the nature of a real-world interferer than the determin-istic sinusoidal waveform used above. Nevertheless, the influence on the subchannel BER is quite the same. Again, in standard OFDM a great number of subchannels is

ω 40.2∆ω=

ν 17=ν 63= Pb 4.2 10 6–⋅=

Pb 2.6 10 2–⋅=

15 20 25 30 35 40 45 50 55 60 6510

−4

10−3

10−2

10−1

100

opt.OFDM

Fig. 5.10 Bit error ratios on each subchannels in the presence of a sinusoidal interferer at . Both MCM systems have subcarriers.40.2∆ω N 512=

subcarrier index

bit e

rror

ratio

1.7∆ω120.2∆ω


affected by the narrowband jammer while in the optimised MC-OQAM system only the subchannels around the interferer’s frequency are disturbed. The overall BER for OFDM is and MC-OQAM .

When adaptive bitloading is used, the affected subchannels can be excluded and thus the BER can be kept low while few bandwidth is sacrificed. In another study it has been found that pulse shaping increases the robustness against carrier frequency offsets [218].

Pb 1.6 10 2–⋅= Pb 5.5 10 6–⋅=

60 80 100 120 140 160 18010

−8

10−6

10−4

10−2

100

opt.OFDM

Fig. 5.11 Subchannel BER in the presence of narrowband additive Gaussian noise. The centre frequency of the narrowband noise is and the noise bandwidth is .120.2∆ω 1.7∆ω

6 Subcarrier Allocation and Bitloading

6.1 Channel Capacity of a Single-User Channel

6.1.1 The Waterfilling Theorem

The channel capacity [219] of a flat AWGN channel is given by

, (6.1)

where is the channel bandwidth, is the signal power and is the noise power. The unit of is bit/s and the channel capacity corresponds to the maximum error-free bitrate that can be achieved with the given signal power. It thus marks the upper limit for reliable digital communication.

The basic formula (6.1) can be extended easily to frequency-selective channels with col-oured Gaussian noise [220, 128]. For the channel depicted in Fig. 6.1 which corresponds to the model for the CaTV return channel from Fig. 2.8, the capacity is

, (6.2)

where (6.3)

is the channel gain to noise ratio (CNR) and and are the PSD of the trans-mitter signal and the noise , respectively. As the channel capacity depends on the PSD of the transmitter signal, the question arises of how this PSD should be designed to achieve maximum capacity. The answer to this question is provided by the waterfill-ing theorem [221]. It states that for an available transmit power

(6.4)

CωB2π------- ld 1

σs2

σr2

------+

=

ωB σs2 σr

2

C

Fig. 6.1 AWGN channel with ISI.

s t( )H ω( )

r t( )y t( )

C 12π------ ld 1 Φs ω( )T ω( )+( ) ωd

0

∞

∫=

T ω( ) H ω( ) 2

Φr ω( )-------------------=

Φs ω( ) Φr ω( )s t( ) r t( )

Ps1π--- Φs ω( ) ωd

0

∞

∫=

99


the capacity is maximum if the transmitter PSD is

where (6.5)

The formula (6.5) can be visualised with the diagram in Fig. 6.2. We can interpret as the bottom of a bowl into which we fill an amount of water corresponding to the avail-able transmit power . The water will distribute itself in a way that the depth repre-sents the wanted function . The constant is adjusted to satisfy (6.4).

Thus, to achieve maximum capacity, the PSD of the transmitter output has to approxi-mate as closely as possible the waterfilling solution (6.5). MCM with relatively narrow-band subchannels can approximate the desired PSD closely by applying appropriate constellation sizes and gain factors for the bit-to-symbol mapping on each subchannel. If the length of the cyclic prefix is longer than the length of the channel impulse response, the channel can be decomposed into independent flat channels according to (3.37) and (3.39). Thus, the block diagram in Fig. 3.8 simplifies to the equivalent channel model of Fig. 6.3. The noise sequences are assumed to be independent, white Gaussian, with zero mean and variances . Note, as the variances are not necessarily identical, the noise on the broadband channel is not restricted to be white.

By discretising the frequency axis of the waterfilling diagram, we can adapt the solution (6.5) to the channel model of Fig. 6.3:

, (6.6)

The ʺwater levelʺ must be chosen such that

Φs ω( ) Φ01

T ω( )------------–

+= x[ ]+ x for x 0>

0 else

=

T 1– ω( )

PSΦS ω( ) Φ0

Fig. 6.2 Waterfilling diagram.

πPs

ΦS ω( )

ω0

Φ0

T 1– ω( )

H ω( ) N

rν k[ ]σν

2 Ε rν k[ ] 2 =H ω( )

Eν c0σν

2

Hν2

------------–+

= ν 0 … N 1–, ,=

c0

Channel Capacity of a Single-User Channel 101

(6.7)

stands for the average symbol energy1 of a sample of the sequence , i.e. . The total transmit power is denoted by .

The channel capacity of the discrete-time subchannel is given as [219]

, with , (6.8)

where denotes the signal power at the receiver side for subchannel .

For this channel, the capacity is measured in bits per symbol or, what is the same, in bits per channel use. Note that (6.8) is valid for complex-valued symbols ; with real-valued inputs, the channel capacity would be halved. The waterfilling theorem gives the maximum error-free bitrate that can be achieved theoretically on a given channel, but it gives no hint on how to do so. Therefore, we have to focus on the relationship between available symbol energy, BER and bitrate that holds for coded or uncoded QAM.

According to [116], the necessary symbol energy to transmit bits with a given symbol error probability by QAM modulation is approximated by

(6.9)

where is the ʺSNR gapʺ defined by

1. Note that this is the same as the average transmit power on subchannel .

Fig. 6.3 Channel model for OFDM with cyclic prefix.

X0 k[ ] Y0 k[ ]

r0 k[ ]H0

X1 k[ ] Y1 k[ ]

r1 k[ ]H1

XN 1– k[ ] YN 1– k[ ]

rN 1– k[ ]HN 1–

… …Etot Eν

ν 0=

N 1–

∑=

Eν Xν k[ ]

ν

Eν Ε Xν k[ ] 2 = Etot

ν

Cd ld 1Erx ν( )

σν2

---------------+

= Erx ν( ) Eν Hν2⋅=

Erx ν( ) ν

Xν k[ ]

bPS

EνΓ σν

2⋅

Hν2

-------------- 2b 1–( )=

Γ


, (6.10)

and is the inverse Q-function. Solving (6.9) for gives

(6.11)

Comparing the expression for the channel capacity (6.8) and the formula for the achieva-ble bitrate with QAM (6.11) reveals that the only difference is the factor . If is expressed in dB, it denotes the additional amount of SNR that QAM needs to achieve a bitrate equal to the channel capacity, therefore its name. We can incorporate the SNR gap into the channel gain to noise ratio (CNR):

(6.12)

The SNR gap thus serves as a link between the information theoretic channel capacity (which cannot be attained by any known modulation scheme) and the bitrate which is achievable with QAM modulation. A possible coding gain and a system margin can also be considered in by

(6.13)

Hence, we can base a bitloading algorithm for multicarrier QAM on the waterfilling the-orem.

6.1.2 Single-User Bitloading Algorithm

A simple bitloading algorithm based on the waterfilling theorem is detailed in the struc-ture chart in Fig. 6.4. The channel and the allowed SER (symbol error ratio) are given by

according to (6.12) and (6.13). The waterfilling theorem, adapted to QAM by incorpo-rating the SNR gap, is given by

, (6.14)

and the total transmit energy may not exceed a given limit:

(6.15)

As the number of bits per QAM-symbol is restricted to a set of integers, the solution (6.14) can only be approximated. Before explaining the structure chart in detail, we need to establish some parameters:

Γ 13--- Q 1– PS

4------

2

= Γ 1≥

Q 1– ⋅( ) b

b ld 1Eν Hν

2

Γσν2

------------------+

=

Γ Γ

TνHν

2

Γσν2

------------=

γc γmΓ

Γ 13--- Q 1– PS

4------

2

10γc γm–( ) 10⁄

⋅ ⋅=

Tν

Eν c0 Tν1––[ ]

+= ν 0 … N 1–, ,=

Etot Eνν 0=

N 1–

∑ Emax≤=

Channel Capacity of a Single-User Channel 103

B denotes the set of possible numbers of bits per QAM-symbol. A valid set might be , but it is also possible to exclude less efficient constellations like e.g. 8-

QAM.

denotes the maximum allowed number of bits in the QAM constella-tion which e.g. in ADSL is chosen as .

The variable serves as an index for the set and thus denotes the number of bits per QAM-symbol on subchannel . This additional variable is necessary to allow for discontiguous integer ranges in , like e.g. .

is the desired number of bits per OFDM-symbol.

is the maximum available energy per OFDM-symbol.

The algorithm determines the number of bits per subchannel. It can either minimise the transmit power for a given minimum bitrate or maximise the bitrate for a given transmit power 1. The structure chart in Fig. 6.4 serves for both cases if the parame-ter that is to be optimised is set to infinity. E.g. for and the algo-rithm allocates 100 bits per OFDM symbol in such a way that the transmit power is minimum.

1. A variation of this algorithm can minimise the SER for a given minimum bitrate and a maximum transmit power [222].

B 1 2 … 10, , , =

bmax max B =bmax 15=

cν 1 2 … cmax, , , ∈ B bν B cν( )=ν

B B 2 4 6 8 10, , , , =

Bmin

Emax

bνBmin

Emax

BminEmax

Bmin 100= Emax ∞=Etot

Fig. 6.4 Bitloading algorithm based on waterfilling theorem.

Etot 0= bν, cν 0 ν∀ 0 … N 1–, ,= = =

∆Eν Tν1– 2B 1( ) 1–( ) ν∀ 0 … N 1–, ,==

while bνν 0=N 1–

∑ Bmin and Etot Emax<<

Etot Emax ?≥

i min ∆Eν arg= ,

∆Ei Ti1– 2

B ci 1+( )2B ci( )

– =

ci ci 1+= bi, B ci( )=

Etot Etot ∆Ei+=

yes no

ci cmax ?<yes no

∆Ei ∞=


The basic idea of the algorithm is to allocate a bit to the subchannel which needs less energy for this bit [223]. Starting with and for all subchannels, the algo-rithm calculates for all subchannels the necessary energy to transmit one bit. Then this bit is allocated to the subchannel with minimum energy and the additional energy it would take to transmit another bit on this subchannel is determined. This is repeated until the total transmit power reaches the limit or the desired bitrate

is reached. For subchannels which have already arrived at the maximum number of bits , is set to infinity, so they will not be considered any more in the following steps.

This algorithm gives the optimum approximation to the waterfilling theorem1 for inte-ger constellations, but it may need many iterations to find the solution. Many fast algo-rithms that provide solutions not far from optimum have been found in the last decade [225-234]. Nevertheless, for the simulations described in the following sections, the waterfilling algorithm has been applied as it runs with reasonable performance on the used computer platform.

6.2 The Multiuser Waterfilling Theorem

The extension of the waterfilling theorem to the multiuser case is not obvious. The neces-sary steps for generalising the well-known solution for single-user AWGN channels to multiple users has been developed by Cheng and Verdú [235]. This generalisation involves the idea of an equivalent channel as depicted in Fig. 6.5, which leads to the adoption of an equivalent transmit signal with the equivalent PSD . With these definitions, the waterfilling theorem can be written as

(6.16)

(6.17)

The multiplier must be chosen such that the water level is unity. Now, the waterfilling diagrams for multiple users can be combined to one diagram. For a three-user channel like illustrated in Fig. 6.6, a multiuser waterfilling diagram like in Fig. 6.7 results. The

1. A study on waterfilling algorithms and their properties can be found in [224].

Etot 0= bν 0=∆Eν

∆Ei

Etot EmaxBmin

bmax ∆Ei

H ω( ) H ω( ) λ⁄=λ s t( )

Φs ω( ) λΦs ω( )=

Fig. 6.5 Equivalent channel with equivalent transmitter signal.

λ s t( )H ω( ) λ⁄

r t( )y t( )

Φs ω( ) 1 λT 1– ω( )–[ ]+

=

Ps λPs1π--- ΦS

ˆ ω( ) ωd0

∞

∫= =

λ

The Multiuser Waterfilling Theorem 105

bottom of the bowl is now given by the minimum equivalent CNR and the available spectrum is allocated to the corresponding user. The multiple access scheme is therefore FDMA.

The analytical formulation of the waterfilling theorem for a multiple access AWGN channel with users is given by

(6.18)

, (6.19)

Like in the single-user case, this solution can be easily adapted to MCM with multiple access which is also called OFDMA (orthogonal frequency division multiple access). Based on this theory, the FDMA capacity region of a Gaussian multiple access channel without ISI has been studied by Yu and Cioffi, and a numerical solution for the two-user case has been given [236, 237]. An extension to more than two users can be found in [238, 239]. A related problem, maximising the sum bitrate for multiple access channels with CDMA and multiple antennas, has been studied by Viswanath [240].

min λuTu1– ω( )

Fig. 6.6 Multiple access channel with AWGN and different transfer functions for each user.

λ1 s1 t( ) H1 ω( ) λ1⁄r t( )

y t( )λ2 s2 t( ) H2 ω( ) λ2⁄

λ3 s3 t( ) H3 ω( ) λ3⁄

U

Φsiω( ) 1 λiTi

1– ω( )–[ ]+

for ω with λiTi1– ω( ) λuTu

1– ω( ) u i≠∀≤

0 otherwise

=

λiPsi

1π--- Φsi

ˆ ω( ) ωd0

∞

∫= i 1 … U, ,=

Fig. 6.7 Multiuser waterfilling diagram.

λ1Etot 1( )

λ2Etot 2( )

λ3Etot 3( )

λ2T21– ω( )

1

0

λ3T31– ω( )

λ1T11– ω( )

ω


6.3 OFDM with Multiple Access: OFDMA

The continuous-time channel model, which is illustrated in Fig. 6.6 for three users, can be decomposed for OFDMA into the discrete-time model of Fig. 6.8, provided the length of the cyclic prefix is longer than the longest channel impulse response. This model can be seen as a straightforward extension of the single-user model in Fig. 6.3. In OFDMA systems, each subchannel is allocated to one user. Thus, before the bit and power alloca-tion for the subchannels can be estimated, the subchannels have to be allocated to the users in a suitable way. In a multiuser environment, the transfer function is generally dif-ferent for each user, thus some subchannels might be in deep fade for one user while they are fine for others. As a consequence, the subcarrier allocation should be adapted to the channel and adaptive modulation should be applied on each subchannel. If the chan-nel is time-invariant and known to the transmitter and receiver, it can be shown that OFDMA clearly outperforms other multiple access techniques like TDMA or CDMA [241]. This is intuitively clear, as CDMA and TDMA do not make much use of channel state information (CSI) and are not adapted to a specific channel, and thus are mostly used in systems where no CSI is available.

The process of adapting an OFDMA system to a particular channel hence consists of two steps: first the subcarrier allocation distributes the subchannels among the users, second a bitloading algorithm determines for each user the appropriate bit and power allocation on their subchannels. For the subcarrier allocation there are various optimisation goals which can be considered: maximum overall bitrate, minimum transmit power, minimum number of allocated subcarriers (to reserve remaining subcarriers for future users), etc.

…user

1

X1 0, k[ ]H1 0,

X1 N 1–, k[ ]H1 N 1–,

…

…user

U

XU 0, k[ ]HU 0,

XU N 1–, k[ ]HU N 1–,

…

…

r0 k[ ]

rN 1– k[ ]…

Y0 k[ ]

YN 1– k[ ]

…

Fig. 6.8 Channel model for multiuser OFDM.

OFDM with Multiple Access: OFDMA 107

Furthermore, there might be a noteworthy number of constraints to be considered, as e.g. maximum transmit power, minimum bitrate (both user-individual or for all users together), maximum BER, etc. Of course, the objective and the constraint depend on the application, whereas for nearly every combination of objective function and constraints, a possible application can be imagined.

The subcarrier allocation problem has been studied in the open literature under various premises. Wong et al. [242, 243] presented an algorithm which is based on Lagrange optimisation and minimises the total transmit power under bitrate constraints. This algorithm nearly reaches the optimal solution, but due to its complexity and its slow convergence, it is computationally very expensive. Later, the same authors presented a strongly simplified faster algorithm [244]. Another step towards a fast implementation was made by Yin and Liu [245] who partitioned the task into two steps. Nevertheless, their algorithm still contains a highly complex assignment problem whose solution is shown for only two users. An algorithm based on CSMA (carrier sense multiple access) which maximises the number of simultaneous users under bitrate and power constraints was introduced by García [246], and Rhee [247] developed a suboptimal algorithm which maximises the channel capacity of the user with smallest capacity.

In the following sections we present two algorithms for subcarrier allocation. For both solutions, the channel model of Fig. 6.8 holds, in which the input sequences are QAM-symbols with average energy per symbol and bits per QAM-symbol, with . In analogy to (6.9), the required symbol energy to transmit bits is given as

(6.20)

where the CNR for the multiuser channel is defined as

(6.21)

The SNR gap can be chosen depending on the user, thus allowing different users to employ different coding schemes and possibly different QoS parameters.

6.3.1 An Efficient Waterfilling Algorithm for OFDMA

The Equivalent Channel and the Discrete Waterfilling Theorem

Like in the continuous-time case, we introduce the equivalent channel coefficients as , which leads the adoption of equivalent symbol energies, denoted as

. With these definitions, the multiuser waterfilling theorem can be written as

Xu ν, k[ ]Eu ν, bu ν,

bu ν, B∈ bu ν,

Eu ν, Tu ν,1– 2

bu ν, 1–( )=

Tu ν,Hu ν,

2

Γuσν2

----------------=

Hu ν, Hu ν, λu⁄=Eu ν, λuEu ν,=


, (6.22)

The power constraint of user is given by

, (6.23)

The waterfilling algorithm maximises the sum bitrate under the con-straint of a maximum transmit power per user. Note, that for the case that a total power constraint is given instead of user-individual power constraints

, the subcarrier allocation becomes a trivial task: for each subcarrier the user with the highest CNR is chosen.

From the analytical formulation of the multiuser waterfilling theorem, we observe that inserting (6.22) into (6.23) yields an equation system with equations for the multi-pliers . All other parameters are given. If this system is solved for the , the energy allocation is given by (6.22) and the subchannel allocation can be derived easily. Unfortunately, the equation system is highly nonlinear and standard algorithms which have been applied to this system have not produced satisfying results. Hence, we devel-oped an iterative algorithm which yields a good approximation of the solution [222, 248].

The Iterative Multiuser Waterfilling Algorithm

From (6.22) we recognize: the equivalent transmit power of user , , increases for decreasing . At the same time, the equivalent power budget

decreases. For the values

, , (6.24)

the equivalent transmit power has a saltus, because an additional subcarrier is assigned to user for falling , or a subcarrier is taken away for raising . The follow-ing algorithm varies the multipliers until (6.23) is fulfilled with (almost) equality. This approach can be regarded as a generalisation of the two-user algorithm presented by Diggavi [249].

We define the subcarrier allocation matrix with if user is active on subcarrier and otherwise. This matrix can be derived easily out of the val-ues of and :

Eu ν,1 λuTu ν,

1––[ ]+

for ν with λuTu ν,1– λlTl ν,

1– l u≠∀≤

0 otherwise

=ν 0 … N 1–, ,=u 1 … U, ,=

u

Etot u( ) Eu ν,ν 0=

N 1–

∑ Emax u( )≤= u∀ 1 … U, ,=

bu ν,ν 0=N 1–

∑u 1=U

∑Emax Emax u( )

u 1=U

∑=Emax u( )

Tu ν,

U Uλ1 … λU, , λu

u Etot u( ) Eu ν,ν 0=N 1–

∑=λu

Emax u( ) λu Emax u( )⋅=

λν Tu ν, minl u≠

λlTl ν,1– ⋅= ν 0 … N 1–, ,= u 1 … U, ,=

Etot u( )u λu λu

λu

A au ν,( )= au ν, 1= uν au ν, 0=

λ λ1 … λU, ,( )= T Tu ν,( )=


(6.25)

At the beginning, the algorithm assigns constant (arbitrary) values to the multipliers . Then these values are varied for each user until

(6.26)

For the energy assigned to user is too great and consequently must be increased and a subcarrier has to be taken away from user . For , must be

A g λ T,( )1 for λuTu ν,

1– λlTl ν,1– l u≠∀≤

0 otherwise

= =

λ1 … λU, ,

λu 1 u∀ 1 … U ;, ,= = A g λ T,( )=

until A Aold=

Aold A=

for u U∈

∆E au ν, 1 λu– Tu ν,1–[ ]

+λuEmax u( )–

ν 0=N 1–

∑=

∆E 0 ?≠

λν Tu ν, min l u≠ λlTl ν,1– ⋅=

yes no

λold λu=

ν∀ 0 … N 1–, ,=

∆E 0 ?>

λu min λν λν λold> =

∆Eold ∆E=

yes no

ν1 arg min λν λν λold> =au ν1, 0=

λu max λν λν λold< =ν1 arg max λν λν λold< =

λu ?=yes no

λu 1 ε–( )λold= au ν1, 1=

∆E au ν, 1 λu– Tu ν,1–[ ]

+λuEmax u( )–

ν 0=N 1–

∑=

until ∆Eold( )sgn ∆E( )sgn≠

λu λold∆E λu∆Eold–( ) ∆E ∆Eold–( )⁄=

A g λ T,( )=

U sort meanν Tu ν, ( )=

Fig. 6.9 Multiuser waterfilling algorithm.

∆E u( ) Eu ν,ν 0=

N 1–

∑

λuEmax u( )–= 0≈

∆E 0> u λuu ∆E 0< λu


lowered and eventually an additional subcarrier is assigned to user . In order to accel-erate this procedure, the saltuses are calculated beforehand according to (6.24) and depending on the sign of , the next bigger or the next smaller value out of the set

is assigned to . It might happen, that there is no smaller value for in the set. In this case, the multiplier is decreased by a small amount without assigning a new subcarrier. If , the value of is increased stepwise until a sign change in is encountered. For , the same is done accordingly. In this way, the condition (6.26) is approximated. The optimum value for lies between the two last values and is approximated linearly with

(6.27)

Then, with new values of , the subcarrier allocation matrix A is determined and the algorithm continues with the next user. This procedure is repeated until the allocation matrix is stable.

Channel Capacity for a given Subcarrier Allocation

The described algorithm determines the multipliers and the subcarrier allocation matrix A. The symbol energies can thus be calculated with (6.22) as

, (6.28)

which leads to the signal power at the receiver side:

uλν

∆Eλ0 … λN 1–, , λu λu

λu∆E 0> λu ∆E

∆E 0<λu

λunew( ) λold∆E λuEold–

∆E Eold–---------------------------------------=

λ

λ

Eu ν,au ν,

λu---------- 1 λuTu ν,

1––[ ]+

=

0 10 20 30 40 50 60 70−20 dB

−15 dB

−10 dB

−5 dB

0 dB

5 dB

user 1user 2user 3user 4

Fig. 6.10 Transfer functions for four users.

subcarrier ν

Hu

ν,


(6.29)

In this sum, only one element is nonzero as each subcarrier is allocated to exclusively one user. The channel capacity of the multiple access channel can hence be written according to (6.8) as

(6.30)

Simulation Results

The channel transfer functions for four users were generated assuming a wireless chan-nel as described in [250] and are depicted in Fig. 6.10. White noise was assumed,

, thus the CNRs are just the inverse of the squared magnitude response. For these CNRs and a power budget1 of 31.8 dB per user the described algorithm was exe-cuted. The algorithm determines the subcarrier allocation matrix A and the multipliers

.

1. All powers are normalised to the total noise power .

Erx ν( ) Eu ν, Hu ν,2⋅

u 1=

U

∑=

C ld 1Erx ν( )

σν2

---------------+

ν 0=

N 1–

∑=

σν const.=

N0 σν2

ν 0=N 1–

∑=

λ

user 4user 1

user

1

user

1

user

2

user

2

user

3

user

3

5 dB

0 dB

-5 dB

-10 dB

-15 dB

-20 dB

-25 dB0 2010 30 40 50 60

user 1user 2

user 3

user 4

Fig. 6.11 Waterfilling diagram for the simulated channel.

subcarrier ν

λ uT u

ν,1–⋅


With these multipliers, the multiuser waterfilling diagram can be drawn as in Fig. 6.11, and the subcarrier allocation becomes visible. Each subcarrier is conceded to the user with the smallest equivalent CNR . Upon this value, the equivalent transmit power is added according to (6.22). The sum gives unity, or 0 dB.

The multiuser waterfilling algorithm thus determines which subcarrier is allocated to which user, but not how many bits are to be assigned to the subcarriers. This task is done by the single-user waterfilling algorithm of Fig. 6.4, or by any other bitloading algo-rithm. It has to be performed for each user on the basis of the given subcarrier allocation matrix A. This yields the bit allocation depicted in Fig. 6.12. Note that this already fixes the transmit power through (6.20).

The SNR gap for this simulation was chosen as , , which corresponds to a SER of (uncoded). The channel capacity of this multiple access channel was cal-culated according to (6.30) to and the total achieved bitrate (which corresponds to the sum of the bits per subchannel shown in Fig. 6.12) was 460 bit per OFDM-symbol, which is distributed between the four users as indicated in Table 6.1. The discrepancy between channel capacity and bitrate depends mainly on the SNR gap.

This algorithm has been also tested with higher numbers of users and subcarriers and a variety of different CNRs. The low number of subcarriers and users chosen in the pre-sented simulation is due to illustrative purposes. The computational complexity

Table 6.1 Allocated bitrates to each user by the multiuser waterfilling algorithm.

user 1 2 3 4 sumnumber of bits per OFDM-symbol

87 99 92 182 460

λu Tu ν,1–⋅

Eu ν,

0 10 20 30 40 50 600

1

2

3

4

5

6

7

8


Fig. 6.12 Resulting bit allocation after subcarrier allocation and bitloading.

subcarrier ν

bits

per

QA

M sy

mbo

l

Γu 5= u∀PS 2 10 4–⋅=

C 606 bit OFDM-symbol⁄=


depends approximately linearly on the number of subcarriers, thus making this algo-rithm applicable to OFDMA systems with a high number of carriers. The algorithm pre-sented in [251, 252] is reported to have a complexity with is proportional to .

6.3.2 Subcarrier Allocation with Bitrate and Power Constraints

The following subcarrier allocation algorithm [222, 253] minimises the total transmit power under the constraint of user-individual bitrate and power constraints. In other words, for each user a minimum bitrate and a maximum transmit power are given and the algorithm allocates the subcarriers such that the sum of the users’ transmit powers is minimum. This algorithm is thus suited for upstream transmission in the CaTV environ-ment as well as for the uplink in mobile communications or in a WLAN.

The algorithm is based on an idea of Yin and Liu [245] who divided the subcarrier alloca-tion into two steps, based on the following reasoning:

• the resources for one user, i.e. the number of subcarriers and the transmit power, mainly depend on his desired minimal bitrate and on his mean CNR which is defined as

(6.31)

• which subchannel is assigned to a user depends on the CNR .

Based on these assumptions, the subcarrier allocation can be realised in two steps. First, an estimation about how many subcarriers are conceded to each user is made, consider-ing the users’ average CNRs, the desired minimum bitrates and the users’ maxi-mum transmit powers . In the second step it is determined which subcarriers are given to which user.

Since [245] is aimed at downlink transmission there is just one overall power constraint for the total transmit power whereas the algorithm described in the following considers one power constraint per user. In the second step, the subcarriers are distributed in such a way that the total bitrate is maximised. This is a combinatorial problem with

possibilities, where denotes the number of subcarriers assigned to user . A solution for users is given in [245], but for a greater number of users the com-plexity of Yin and Liu’s algorithm will be enormous. The following two-step algorithm, which is detailed in Fig. 6.13, avoids the complicated combinatorial optimisation.

Step 1: Estimation of the Number of Subcarriers for each User

To each user subcarriers are assigned such that the desired bitrate can be reached with the given maximum symbol energy :

N Nlog

Bmin u( )

Tu1N---- Tu ν,

ν 0=

N 1–

∑=

Tu ν,

Bmin u( )Emax u( )

N! ku!u∑⁄ ku u

U 2=

ku Bmin u( )Emax u( )


(6.32)

For small it might happen that the desired bitrate cannot be attained even if all subcarriers are conceded to user . This is the case for

ku Bmin u( ) bmax⁄= Etot u( ) kuTu1– 2

Bmin u( ) ku⁄1–( )= u∀

while Etot u( ) Emax u( )>

for u 1 … U, ,=

while kuu

∑ N<

ku ku 1+= Etot u( ) kuTu1– 2

Bmin u( ) ku⁄1–( )=

kuu

∑ N ?<yes no

Emax u( ) 1 ε–( )Emax u( )= u∀

for u U∈

while kuu

∑ 0>

while kuu

∑ N>

Enew u( ) ku 1–( )Tu1– 2

Bmin u( ) ku 1–( )⁄1–( )= u∀

u' minu

Enew u( ) Etot u( )– arg=

ku' ku' 1–= Etot u'( ) Enew u'( )=

A 0= p0 u( ) ku N⁄= u∀

U u u max u' 1…U=

ku' arg= =

ν1 min ν M∈

Tu ν,1– arg= M ν au' ν,u' 1=

U∑ 0= =with

ku ku 1;–= au ν1, 1=

p u( ) ku ku'u'

∑⁄=

U u u max p u'( ) p0 u'( )– arg= =

for u U∈

ν1 min ν M∈

Tu ν,1– arg= M ν au' ν,u' 1=

U∑ 0= =with

ku ku 1;–= au ν1, 1=

Fig. 6.13 Two-step subcarrier allocation algorithm.

step

1st

ep 2

Emax u( ) Etot u( )≥ kuTu1– 2

Bmin u( ) ku⁄1–

=

Emax u( )u


(6.33)

In this case the desired bitrate has to be reduced or the allowed transmit power has to be increased.

At the beginning, is calculated as if the maximum number of bits per symbol could be applied to all subcarriers:

(6.34)

Usually, in this first step much less subcarriers are assigned than available (otherwise the desired bitrates would already exceed the system’s transmission capacities). Next, we assign to each user new subcarriers until the required energy does not exceed , in accordance with (6.32). If there are subcarriers left, i.e. , which is normally the case, the maximum energies are lowered by a small step, and the procedure repeats until no subcarriers remain.

As this routine normally assigns some subcarriers more than available, we remove a sub-carrier from the user which has to increase his transmit power by the smallest amount without this subcarrier. This is repeated until exactly subcarriers are granted.

Step 2: Distribution of the Subcarriers

The idea for the subcarrier distribution is that the users choose alternatingly the subcar-rier with the best CNR. This is similar to a procedure that is used in physical education to form to sports teams: beginning with two team captains, the teams choose alternat-ingly one new player until nobody is left. For the subcarrier distribution task, there are more than two users which additionally have unequal numbers of subcarriers. There-fore, the order in which the users choose their subcarriers is important. A procedure based on priorities controls the order: the reference priority is defined as the number of subcarriers of user over the total number of subcarriers:

(6.35)

After user has chosen one subcarrier, is decremented by one; thus here stands for the number of subcarriers that are still to assign. Hence we define the actual priority of user as

, (6.36)

The user with the most subcarriers begins, then after each step the user with the greatest difference between reference and actual priority is picked out for the next turn.

Emin u( ) NTu1– 2

Bmin u( ) N⁄1–( ) Emax u( )>=

ku bmax

ku Bmin u( ) bmax⁄=

Emax u( )kuu∑ N<

N

p0 u( )u

p0 u( )kuN-----=

u ku ku

u

p u( )ku

kuu 1=

U

∑----------------------= u 1 … U, ,=


Simulation Results

The two-step algorithm was performed with the same channel and noise parameters as in the previous section, also the maximum transmit powers per users were chosen iden-tical. The desired bitrates, expressed in bits per OFDM-symbol, are given in Table 6.2. These values may not excel the bitrates in Table 6.1, since these constitute the upper limit for the given channel and power constraints.

The resulting bit allocation is given in Fig. 6.14. Again, we see that for any given subcar-rier the user with the highest CNR is preferred. Of course, this does not apply strictly because the subcarrier distribution is mainly governed by the desired bitrates.

While the multiuser waterfilling algorithm is the practical implementation of the solu-tion of a basic information theoretical problem, the power minimisation with rate con-straints is more relevant in practice. We compared the two-step algorithm with the nearly optimum algorithm of Wong et al. which is based on Lagrange optimisation [242]. The two-step algorithm additionally considers user-individual power constraints. Wong’s algorithm is aimed at the downlink in wireless communication systems and therefore does not consider power constraints. For the simulation, the power budgets were chosen large enough, so that neither algorithm violated these constraints. The allo-cated user energies are given in Table 6.3, indicating that the two-step algorithm only consumes very little more energy than the near optimum solution. A measurement of

Table 6.2 Desired minimum bitrates as input for the two-step algorithm.

user 1 2 3 4

desired bitrate 60 70 65 120u

Bmin u( )

0 10 20 30 40 50 600

1

2

3

4

5

6


Fig. 6.14 Bit allocation achieved with two-step algorithm.

subcarrier ν

bits

per

QA

M sy

mbo

l


the runtimes revealed that the two-step algorithm was several hundred times faster for these input parameters.

This result motivated a more extensive comparison of the execution time of both algo-rithms. The results are shown in Fig. 6.15. A WSSUS channel with exponential delay power spectrum was used and for each user and simulation run, the stochastic channel coefficients were determined. The two-step algorithm was performed with up to

subcarriers and users while the maximum values for Wong’s algo-rithm were , . Especially for many subcarriers and users, the execution times differ by some orders of magnitude while the achieved total transmit powers vary only slightly.

Table 6.3 Allocated energies to each user and total transmit energy per symbol.

user 1 2 4 4 totaltwo-step 24.8 dB 25.2 dB 24.5 dB 25.4 dB 31.0 dBWong 24.6 dB 25.0 dB 24.6 dB 24.8 dB 30.8 dB

Hu ν,N 4096= U 512=

N 128= U 32=

101

102

103

104

105

106

107

10−1

100

101

102

103

104

105

Wongtwo−step

Fig. 6.15 Execution times of the presented algorithm (two-step) in comparison with Wong’s near optimum algorithm for different numbers of users and subcarriers.

U N⋅

exec

utio

n tim

e


7 Conclusion

In this thesis, the return channel of the CaTV network has been examined, a new multi-carrier modulation scheme has been presented and efficient multiple access schemes for multicarrier systems have been developed.

For the CaTV return path, channel models are presented and the noise, which is the major impairment on the return channel, is modelled analytically. The noise is decom-posed into a sum of three noise types: broadband, narrowband and impulse noise. For these noise types, simulation models are developed that allow to generate ingress noise with adjustable parameters. A hardware test bed which has been built up at the Institute of Telecommunications, is described and software models are derived from its network components. Based on the measurements of the network components, a software model for the complete network is deduced and comparisons of the measurements with the simulation results assure the validity of the taken approach. Existing solutions for return channel transmission are reviewed and it is concluded that multicarrier modulation (MCM) offers the potential for remarkable improvements.

The principles of multicarrier modulation are outlined and the basic system models for MCM with and without pulse shaping are explained. For systems with pulse shaping, efficient structures for implementation are derived which replace the high-rate filter-bank by a FFT and a polyphase filterbank that operates on a much lower clock rate. Fast implementations are presented for multicarrier QAM and offset QAM.

Several approaches for MCM with pulse shaping are treated in a common framework and the motivations behind them are outlined. A novel pulse shape for multicarrier off-set QAM which maximises the signal energy in the frequency band of one subchannel while maintaining orthogonality is derived. This pulse shape is calculated as a series expansion, in which discrete prolate spheroidal sequences act as base functions. The sequences are especially well-suited for this task as they are the set of orthogonal sequences with the most concentrated spectra. The susceptibility to narrowband inter-ference of a multicarrier offset QAM system with the new pulse shape is compared to that of a standard OFDM system. It is shown that the new system is much more robust against this type of noise and that the resulting BER is significantly lower.

The return path of a CaTV network is a multiuser environment and thus a multiple access scheme must be employed to avoid collisions and interference between the users. MCM systems can be extended straightforwardly to multiple access systems by assign-ing the subchannels to different users. A result from information theory, the multiuser waterfilling theorem, states that for a frequency-selective channel with Gaussian noise this access scheme is optimum with respect to channel capacity. A fast algorithm which calculates the subcarrier allocations according to this theorem is presented. This approach maximises the sum bitrate of all users. Depending on the desired bitrate allo-cation strategy, a minimum bitrate for each user may be required. For this case, a second

119

120 Conclusion

algorithm has been developed. This algorithm considers the required bitrate and the maximum transmit power of each user and allocates the subcarriers in a way that the total transmit power is minimised. Both algorithms, although first intended for the use in CaTV networks, are well-suited for wireless applications like WLAN or mobile com-munication systems.

8 Appendix

8.1 The Equivalent Lowpass Channel

Before defining the equivalent low pass channel and the complex envelope, some words about the Hilbert transform and the analytical signal are in order.

The Hilbert transform of the signal is defined1 in the time domain as

(8.1)

and in the frequency domain as

(8.2)

This operation can be considered as filtering of the signal with a Hilbert filter like illustrated in Fig. 8.1, where

(8.3)

Thus, the term Hilbert transform is rather misleading, since there is no change of domain involved as e.g. in the Fourier, z, or Laplace transform, but it is a mere filtering operation. Properties of the Hilbert transform can be found in [41, 168].

The analytical signal or pre-envelope of an arbitrary real-valued signal is defined as:

(8.4)

We assume that the lowpass signal is bandlimited with and . Then the corresponding bandpass signal and its Fourier transform are given by:

1. Note that the convolution integral in (8.1) is an improper integral because the integrand has a singularity at . Therefore, the integral has to be taken as Cauchy’s principal value.

x t( )

τ t=

H x t( ) x t( ) x t( ) 1πt----- 1

π--- x τ( ) 1

t τ–---------- τd

∞–

+∞

∫= = =

H X ω( ) jX ω( ) ω( )sgn– X ω( )ejπ2--- ω( )sgn–

= =

x t( )

hH t( )1πt----- for t 0≠

0 for t 0=

= HH ω( ) j ω( )sgn–=

Fig. 8.1 Hilbert filter

hH t( )x t( ) x t( )

f + t( ) f t( )

f + t( ) := f t( ) j f t( )+ F+ ω( ) F ω( ) 1 ω( )sgn+( )⋅2F ω( ) for ω 0>F 0( ) for ω 0=

0 for ω 0<

= =

x t( ) ωg ωg ωc<xBP t( ) XBP ω( )

121

122 Appendix

The output lowpass signal is given by

As and are zero for , it follows

and thus the equivalent lowpass transfer function is defined as

(8.5a)

Solving for the bandpass functions yields

(8.5b)

This can be shown by solving (8.5a) for , making use of the fact that this signal is real-valued. Now, the relationship between a (real-valued) passband signal and its equivalent lowpass signal , also called the complex envelope can be established as

(8.6a)

(8.6b)

The signal can be identified with the signals or from Fig. 8.2 and their cor-responding passband equivalents. The perceptive reader might have noted that the choice of the factor 2 in the receiver lowpass filter is somewhat arbitrary. Another valid definition of the equivalent lowpass distributes this factor by assigning factors to the real part operator and the lowpass filter. The definition used here has the convenience that the relation (8.6a,b) can be applied to both the signals and . Another inter-esting observation is that the signal in Fig. 8.2 is the analytical signal of , i.e.

.

xBP t( ) Re x t( )ejωct = XBP ω( ) 1

2--- X ω ωc–( ) X∗ ω– ωc–( )+( )=

x(t)HBP(ω)

H(ω)

xBP(t) yBP(t) y(t)uc(t)Re 2 rect(ω/2ωg)

Fig. 8.2 Definition of the complex envelope and the equivalent lowpass channel

Y ω( ) YBP ω ωc+( ) 2 rect ω2ωg---------

⋅ XBP ω ωc+( )HBP ω ωc+( )2rect ω2ωg---------

= =

X ω( ) rect ω 2ωg⁄( ) ω ωg>

Y ω( ) rect ω2ωg---------

HBP ω ωc+( ) X ω( )⋅=

H ω( ) rect ω2ωg---------

HBP ω ωc+( ) 12---HBP

+ ω ωc+( )= = h t( ) 12---hBP

+ t( )ejωct–

=

HBP ω( ) H ω ωc–( ) H∗ ω– ωc–( )+= hBP t( ) 2Re h t( )ejωct =

hBP t( )fBP t( )

f t( )

f t( ) f BP+ t( )e

jωct–= F ω( ) FBP

+ ω ωc+( )=

fBP t( ) Re f t( )ejωct = FBP ω( ) 1

2--- F ω ωc–( ) F ω– ωc–( )+( )=

f t( ) x t( ) y t( )

2

x t( ) y t( )uc t( ) xBP t( )

uc t( ) xBP+ t( )=

Hermite Polynomials and Functions 123

8.2 Hermite Polynomials and Functions

The Hermite polynomial of degree n is defined in [178] by the Rodrigues’ formula

, (8.7)

Note, that this definition does not correspond exactly to the one given in the mathemati-cal literature [254, 255] where the Hermite polynomials are defined as

. Both definitions fulfil the differential equation:

(8.8)

The recurrence relation for the calculation of the (n+1)th polynomial is given by

(8.9)

The Hermite polynomials are orthogonal:

(8.10)

The Hermite function is defined as

(8.11)

It solves the linear integral equation

, with (8.12)

and fulfils the orthogonality relation

(8.13)

The normalised Hermite function is defined as

(8.14)

The normalised Hermite functions satisfy the integral equation

(8.15)

Pn x( )

Pn x( ) 1–( )n

2n------------- ex2 dn

dxn--------e x2–⋅ ⋅= n N0∈

Hn x( ) 2n Pn x( )⋅=

d2

dx2--------Pn x( ) 2x d

dx------Pn x( )– 2nPn x( )+ 0=

Pn 1+ x( ) xPn x( ) n2---Pn 1– x( )–=

e x2– Pn x( )Pm x( ) xd∞–

+∞

∫πn!2n

------------δ n m–[ ]=

hn x( )

hn x( ) 1–( )n2ne x2 2⁄– Pn x( ) ex2

2---- dn

dxn--------e x2–= =

λnhn t( ) h ω( )ejωt ωd∞–

+∞

∫= λn jn 2π=

hn t( )hm t( ) td∞–

∞

∫ π2nn! δ n m–[ ]⋅=

Dn t( ) hn 2π t( ) 1–( )n2nPn 2π t( )e πt2–= =

jnDn t( ) Dn f( )ej2πft fd∞–

+∞

∫=

124 Appendix

This signifies that for , the normalised Hermite functions are invar-iant with respect to the Fourier transformation:

(8.16)

This holds as well for any linear combination . In Fig. 8.3 the first four Fourier-invariant Hermite functions are drawn.

8.3 Calculation of the Peak-to-Average Power Ratio

Based on the system model in Fig. 3.1, the PAPR is defined as

(8.17)

We assume that all subchannels use the same QAM constellation and that the sequences are uncorrelated, thus the mean signal energy per symbol is the same for all

subchannels. For -QAM with equiprobable signal points and square constellation ( ), the mean signal power1 of the sequence is given by [220]

(8.18)

where is the distance between two adjacent signal points. For general signals , it holds and thus

(8.19)

1. The mean signal energy per symbol corresponds to the mean signal power.

n 4k= k 0 1 …, , ∈

D4k f( ) F D4k t( ) =

g t( ) akD4k t( )k 0=

∞∑=

−3 −2 −1 0 1 2 3−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

D0(t)

D4(t)/16

−3 −2 −1 0 1 2 3−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

D8(t)

D12

(t)/128

Fig. 8.3 Normalised Hermite functions . These functions are identical with their Fourier transforms.

D4k t( )

tt

γ max s t( ) 2

Ε s t( ) 2 -------------------------------=

Xν k[ ]Mq

Mq 4 16 64 …, , ,= Xν k[ ]

EX 2Mq 1–

3----------------a2=

2a x t( )Ε x t( ) 2 Ε x t( ) jωνt( )exp 2 =

Ε s t( ) 2 NΕ sν t( ) 2 NΕ s0 t( ) 2 = =

Calculation of the Peak-to-Average Power Ratio 125

where . Assuming that the samples of the sequence are uncor-related, the autocorrelation function of is

(8.20)

which leads to the PSD of :

and the PSD of :

This gives the mean signal power of :

from which follows with (8.19)

(8.21)

The output signal is given by

(8.22)

The magnitude of this expression takes on its maximum in the case that all terms have the same phase. With , the maximum output signal

power is

(8.23)

This leads with (8.21) to the PAPR

(8.24)

For square constellations, it holds , thus

s0 t( ) x0 t( ) g t( )= X0 k[ ]X0 k[ ]

ϕX k[ ] EXδ k[ ]=

x0 t( )

ΦX ω( ) T ϕX k[ ]e jωkT–

k ∞–=

∞

∑ TEX= =

s0 t( )

Φs0ω( ) ΦX ω( ) G ω( ) 2⋅ TEX G ω( ) 2= =

s0 t( )

Ε s0 t( ) 2 12π------ Φs0

ω( ) ωd∞–

+∞

∫ TEX g t( ) 2 td∞–

+∞

∫= =

Ε s t( ) 2 NTEX g t( ) 2 td∞–

+∞

∫=

s t( )

s t( ) T ejωνt

Xν k[ ]g t kT–( )ν 0=

N 1–

∑k ∞–=

∞

∑=

ejωνt

Xν k[ ] A max Xν k[ ] =

max s t( ) 2 A N T g kT( )k ∞–=

∞

∑ 2

=

γ N A2

EX------

T g kT( )k ∞–=

∞

∑ 2

g t( ) 2 td∞–

+∞

∫-------------------------------------------------⋅ ⋅=

A 2 Mq 1–( )a=

126 Appendix

(8.25)

For a normalised transmitter filter, i.e. a transmitter filter with unit signal energy, and square constellations, the PAPR is

(8.26)

A2

EX------

3 Mq 1–( )2

Mq 1–-------------------------------=

γ N3 Mq 1–( )

2

Mq 1–------------------------------- T g kT( )

k ∞–=

∞

∑ 2

⋅ ⋅=

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