Compressed Sensing Crash Intro

Compressed Sensing Crash Intro

Thomas [email protected]

Technology Platforms Section

Dept. of Electronic Systems, Aalborg University

UC Louvain, Louvain-la-Neuve, November 30th, 2012

mailto:[email protected]

8

Traditional Nyquist Sampling

Classic Nyquist sampling:

z(t)

{zn}t

z

z =

z0...zN

8

Traditional Nyquist SamplingCompression


� We sample z.

� Then compress; e.g. MP3, JPEG etc.

� Conclusion: we sample a lot of data, but throw most of itaway.

8



� We sample z.



8



� We sample z.



8



� We sample z.



8

Compressed SensingBasics

What is compressed sensing?

� New signal acquisition/compression theory from around 2004.

� Combines sampling and compression of signals.

� Interesting implication: Signals can be sampled significantlybelow Nyquist rate!

8






8






8






8

Compressed SensingRequirements

The signal must be sparse in a known dictionary:

sparsedictionary

=·

original

vector zvector xmatrix Ψ signal

8

Compressed SensingAcquisition

compression compressed

=·

originalmatrix Φ

vector zvector ysignal

� The signal vector is mixed with a measurement matrix beforesampling.

� Sample the (fewer) mixed “measurements”.

8



=·

originalmatrix Φ




8



=·

originalmatrix Φ




8

Compressed SensingThe Reconstruction Principle

sparsedictionary reconstructedcompressed

==· ·

original

vector zvector y

vector zvector xmatrix Ψ

optimization-based reconstruction

signal signalcompressionmatrix

8

Compressed SensingConclusion

Is compressed sensing (CS) always a good idea?

� No!

When to use CS

� When samples are otherwise too “expensive” to take...For example:

� Sensor is expensive to build with sufficiently highresolution/rate.

� Collecting enough samples takes too long.� Collecting fewer samples saves significant energy.

8



� No!

When to use CS




8



� No!

When to use CS




8



� No!

When to use CS




8



� No!

When to use CS




8



� No!

When to use CS



� Collecting enough samples takes too long.

� Collecting fewer samples saves significant energy.

8



� No!

When to use CS




Compressed Sensing in RF Communication andAnalog-to-Digital Conversion

Thomas [email protected]

Technology Platforms Section

Dept. of Electronic Systems, Aalborg University

UC Louvain, Louvain-la-Neuve, November 30th, 2012

mailto:[email protected]

34

CompressedSensing in RFCommunication

andAnalog-to-Digital

Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

D/A Conversion

TPS, Dept. ofElectronic Systems,Aalborg University

Agenda

Projects and People

RF Communication

Reconstruction

D/A Conversion

34



Conversion

Thomas Arildsen

3 Projects andPeople

RFCommunication

Reconstruction

D/A Conversion


Agenda

Projects and People

RF Communication

Reconstruction

D/A Conversion

34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion


Projects and PeopleSparse Signal Processing in Wireless Communications (SparSig)

� New ways to sample, quantize, process andre-synthesize analog signals.

� Focus on wireless communication and signals for such asystem.

� Main contributions: to develop a framework for handlingrealistic signals (which are noisy, distorted etc.) and toprovide a convincing validation (including experiments).

� Objective: to reduce the power consumption ofconverters and digital signal processing in devicestypically used in wireless communication systems.

34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion



Participants

� Prof. Torben Larsen

� Prof. Søren Holdt Jensen (MISP section)

� Assist. Prof. Thomas Arildsen

� Post doc. Tobias Lindstrøm Jensen (partly MISP)

� PhD stud. Karsten Fyhn (partly MISP)

� PhD stud. Peng Li

� PhD stud. Pawe�l Pankiewicz

34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion


Projects and People4th Generation Mobile Communication and Test Platform (4GMCT)

The compressed sensing part of this project

� Wireless communication in 4G communication systems.

� Concepts and electronic circuits to performenergy-efficient sampling, quantization and processingof signals.

� Purpose: investigate theoretical basis, analyze andpropose solutions for implementing compressedsampling techniques in communication receivers.

� Main issue: to propose solutions for sub-Nyquistsampling and quantization, as well as a reconstructionalgorithm design.

34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion








34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion








34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion



Participants (compr. sens. part)


� PhD stud. Hao Shen

� PhD stud. Jacek Pierzchlewski

34



Conversion

Thomas Arildsen


RFCommunication

Reconstruction

D/A Conversion


Projects and PeopleCompressive Sensing in Signal Analyzer

� Main purpose: to advance the compressed sensingtheory and calculation process to enable utilizingcompressed sensing in real-life applications.

� In particular signals analyzer equipment.

Participants


� PhD stud. Ruben Grigoryan

34



Conversion

Thomas Arildsen

Projects andPeople

9 RFCommunication

Reconstruction

D/A Conversion


Agenda

Projects and People

RF Communication

Reconstruction

D/A Conversion

34



Conversion

Thomas Arildsen

Projects andPeople

10 RFCommunication

Reconstruction

D/A Conversion


RF CommunicationDownsampling of DFT Precoded Signals for the AWGN Channel

� We connect so-called k-simple vectors to decoding ofdigital signals.

� Data vector not sparse: x = [0 1 1 0 1 0 1 1 . . .]T.

� We extend an existing linear programming recoverymethod with a semidefinite recovery method.

Figure: Encoder-decoder principle

34



Conversion

Thomas Arildsen

Projects andPeople

11 RFCommunication

Reconstruction

D/A Conversion


RF CommunicationDownsampling of DFT Precoded Signals for the AWGN Channel

Figure: Uncoded BER for QPSK signaling – downsampling attransmitter.

34



Conversion

Thomas Arildsen

Projects andPeople

12 RFCommunication

Reconstruction

D/A Conversion


RF CommunicationCompressive Sensing for Spread Spectrum Signals

� Compressive sensing in a CDMA or DSSS receiver.

� Possibility to decrease the sampling rate.

� Made possible by the spreading codes used to bettercombat interference or low SNR at the receiver.

� These spreading codes decrease the information rate perchip or bit sent, which enables a sparse decompositionapproach.

34



Conversion

Thomas Arildsen

Projects andPeople

12 RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

12 RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

13 RFCommunication

Reconstruction

D/A Conversion



� Example with IEEE 802.15.4 physical layer (for exampleZigBee).

� Notice no random linear mixing necessary in receiver.� No compressed sensing reconstruction – simpler

compressive classification in stead.

Figure: Encoder-decoder principle

34



Conversion

Thomas Arildsen

Projects andPeople

14 RFCommunication

Reconstruction

D/A Conversion



−9 −6 −3 0 3 6 9 12 1510

−5

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BE

R

Non−coherent theoretical BER

Coherent theoretical BER

LS estimator

CS − κ=0.5

Figure: Bit-error-rate – downsampling at the receiver.

34



Conversion

Thomas Arildsen

Projects andPeople

15 RFCommunication

Reconstruction

D/A Conversion


RF CommunicationCompressed Sensing-Based Direct Conversion Receiver

� Computational power of modern data receivers enablesmoving more processing from the analog to the digitaldomain.

� Use compressed sensing to relax the analog filteringrequirements in a direct conversion receiver.

� The filtered, down-converted radio signal is randomlysampled with a sub-Nyquist average sampling frequency.

� Exploits frequency-domain sparsity of thedown-converted radio signals.

34



Conversion

Thomas Arildsen

Projects andPeople

15 RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

15 RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

15 RFCommunication

Reconstruction

D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

16 RFCommunication

Reconstruction

D/A Conversion



Figure: Spectral content around desired signal.

34



Conversion

Thomas Arildsen

Projects andPeople

17 RFCommunication

Reconstruction

D/A Conversion



Figure: Ordinary direct conversion receiver.

34



Conversion

Thomas Arildsen

Projects andPeople

18 RFCommunication

Reconstruction

D/A Conversion



Figure: Proposed CS-based direct conversion receiver.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

19 Reconstruction

D/A Conversion


Agenda

Projects and People

RF Communication

Reconstruction

D/A Conversion

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

20 Reconstruction

D/A Conversion


ReconstructionCorrelation Between Measurements and Noise

� What happens when compressed measurements areaffected by noise correlated with the measurements?

y = Ax

y = y + n,

� Where does this happen?For example when quantizing measurements at lowresolution.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

20 Reconstruction

D/A Conversion



� What happens when compressed measurements areaffected by noise correlated with the measurements?

y = Ax

y = y + n,

� Where does this happen?For example when quantizing measurements at lowresolution.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

21 Reconstruction

D/A Conversion



� Linear correlated noise model:

y = αAx +w,

� Leads to noise correlated with the measurements

n = y − y

= (α− 1)y +w

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

21 Reconstruction

D/A Conversion



� Linear correlated noise model:

y = αAx +w,

� Leads to noise correlated with the measurements

n = y − y

= (α− 1)y +w

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

22 Reconstruction

D/A Conversion



� What to do about it?

� We can simply rescale the estimate of x after basispursuit de-noising (BPDN) reconstruction:

x =1

αargmin

v: �y−Av�2≤��v�1. (1)

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

22 Reconstruction

D/A Conversion



� What to do about it?

� We can simply rescale the estimate of x after BPDNreconstruction:

x =1

αargmin

v: �y−Av�2≤��v�1. (1)

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

23 Reconstruction

D/A Conversion



0 0.1 0.2 0.3 0.4 0.5 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Measurement density ρ (K/M) [–]

NM

SE[–

]

BPDNBPDN-scale

Figure: Example of improvement for 1 bit/sample quantization.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

24 D/A Conversion


Agenda

Projects and People

RF Communication

Reconstruction

D/A Conversion

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

25 D/A Conversion


D/A ConversionCompressive Parameter Estimation

� Instead of reconstruction, it is possible to usecompressive sensing for parameter estimation.

� Especially interesting: manifold models.

� Allow parameters drawn from a continuous space,rather than from a discrete dictionary as usual.

� We are currently investigating such models for use intime delay estimation and sinosoidal parameterestimation.

� We show that it is possible to use compressive sensingand still attain good mean squared error on theparameter estimate.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

25 D/A Conversion








34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

25 D/A Conversion








34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

25 D/A Conversion








34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

25 D/A Conversion








34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

26 D/A Conversion


D/A ConversionQuantization in CS with a Fixed Bit Budget

� Investigation of reconstruction performance incompressive sensing with quantized measurements.

� Trade-off between quantizer resolution and number ofcompressed measurements for fixed total numbers ofbits.

� We compare two methods by Laska et al. tailored tosaturated measurements for the Basis PursuitDe-Noising method.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

26 D/A Conversion






34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

26 D/A Conversion






34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

27 D/A Conversion



Figure: Normalized mean-squared error. Saturation rate 5%.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

28 D/A Conversion



� Existing approaches tailored for saturation effects donot consider information spent on identifying saturatedmeasurements.

� We propose reserving quantization indices to marksaturated measurements.

� Applicable to current quantizer models.

Figure: Reserving quantizer indices for indicating saturation.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

28 D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

28 D/A Conversion






Quantization q bits/meas.

Signal reconstruction

Unsaturated meas.

Saturated meas.

2q

indices2q-n

indices

Quantization q bits/meas.

Signal reconstruction

Unsaturated meas.

Saturated meas.

(a) (b)

n reserved indices


34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

29 D/A Conversion



Figure: Normalized mean-squared error vs. quantizer resolution.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

30 D/A Conversion


D/A ConversionFilter Imperfections in Random Demodulator

� Applying compressed sensing (CS) to continuoussignals: analog sampling front-end providing a signalrepresentation compatible with CS.

� The random demodulator provides pseudo-randomlinear projections of the analog input signal.

� The analog front-end has to be modeled accurately inthe reconstruction algorithm.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

30 D/A Conversion






34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

30 D/A Conversion






34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

31 D/A Conversion



yyy

y...

M

2

3

1chipping sequencefilter

ADC

1 2 3

ideal impulse response of the filterdeviated response value fluctuation #1deviated response value fluctuation #2

time domain response

pn sequence

...

.

.

.

CS reconstruction

DSP

x(n)

3

2

1

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

32 D/A Conversion


D/A ConversionMeasurement Matrix Deviations due to Filter Imperfections in RandomDemodulator

� Simulations of imperfect filter: component variation 5%and 10% for capacitors and inductors respectively.

� Simulations of 16 worst-case scenarios using 4th orderButterworth filter indicated up to 40 dB loss in SNR

2 4 6 8 10 12 14 160

10

20

30

40

50

Basis Pursuit (CVX)

Orthogonal Matching Pursuit

Basis Pursuit De-Noising

[dB]

ideal1c c c3c 5c cc cccc7c 9c 11c 13c 15c

SNR

Figure: Reconstruction error corner cases for imperfect filter.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

33 D/A Conversion


D/A ConversionCalibration of Filter Imperfections in the Random Demodulator

� Mismatch considered an additive error in the discretizedimpulse response: y = (A+ E)x

� Estimate error by sampling a known signal, enablingleast-squares estimation of the impulse response error.

� Error and known problem structure are used to correctthe measurement matrix.

� Simulation results demonstrate the effectiveness of thecalibration technique even for highly deviating low-passfilter responses.

34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

33 D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

33 D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

33 D/A Conversion







34



Conversion

Thomas Arildsen

Projects andPeople

RFCommunication

Reconstruction

34 D/A Conversion



1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

instances (Table 1)

SNR [d

B]

CS reconstrucion using (BP - SPGL1)

precise model

calibrated model

uncertain model

Figure: Example: improvements from calibration of all filtercomponents deviating up to 2%.

Documents

Compressed Sensing Crash Intro