Compressed Sensing Crash Intro
Thomas [email protected]
Technology Platforms Section
Dept. of Electronic Systems, Aalborg University
UC Louvain, Louvain-la-Neuve, November 30th, 2012
8
Traditional Nyquist Sampling
Classic Nyquist sampling:
z(t)
{zn}t
z
z =
z0...zN
8
Traditional Nyquist SamplingCompression
Classic Nyquist sampling:
� We sample z.
� Then compress; e.g. MP3, JPEG etc.
� Conclusion: we sample a lot of data, but throw most of itaway.
8
Traditional Nyquist SamplingCompression
Classic Nyquist sampling:
� We sample z.
� Then compress; e.g. MP3, JPEG etc.
� Conclusion: we sample a lot of data, but throw most of itaway.
8
Traditional Nyquist SamplingCompression
Classic Nyquist sampling:
� We sample z.
� Then compress; e.g. MP3, JPEG etc.
� Conclusion: we sample a lot of data, but throw most of itaway.
8
Traditional Nyquist SamplingCompression
Classic Nyquist sampling:
� We sample z.
� Then compress; e.g. MP3, JPEG etc.
� Conclusion: we sample a lot of data, but throw most of itaway.
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
3 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Agenda
Projects and People
RF Communication
Reconstruction
D/A Conversion
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
4 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
4 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
4 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
5 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Projects and PeopleSparse Signal Processing in Wireless Communications (SparSig)
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
6 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
6 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
6 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
7 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Projects and People4th Generation Mobile Communication and Test Platform (4GMCT)
Participants (compr. sens. part)
� Prof. Torben Larsen
� PhD stud. Hao Shen
� PhD stud. Jacek Pierzchlewski
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
8 Projects andPeople
RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
� Prof. Torben Larsen
� PhD stud. Ruben Grigoryan
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
9 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Agenda
Projects and People
RF Communication
Reconstruction
D/A Conversion
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
10 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
11 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationDownsampling of DFT Precoded Signals for the AWGN Channel
Figure: Uncoded BER for QPSK signaling – downsampling attransmitter.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
12 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
12 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
12 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
13 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationCompressive Sensing for Spread Spectrum Signals
� 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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
14 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationCompressive Sensing for Spread Spectrum Signals
−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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
15 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
15 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
15 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
15 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
16 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationCompressed Sensing-Based Direct Conversion Receiver
Figure: Spectral content around desired signal.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
17 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationCompressed Sensing-Based Direct Conversion Receiver
Figure: Ordinary direct conversion receiver.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
18 RFCommunication
Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
RF CommunicationCompressed Sensing-Based Direct Conversion Receiver
Figure: Proposed CS-based direct conversion receiver.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
19 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Agenda
Projects and People
RF Communication
Reconstruction
D/A Conversion
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
20 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
20 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
21 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
ReconstructionCorrelation Between Measurements and Noise
� Linear correlated noise model:
y = αAx +w,
� Leads to noise correlated with the measurements
n = y − y
= (α− 1)y +w
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
21 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
ReconstructionCorrelation Between Measurements and Noise
� Linear correlated noise model:
y = αAx +w,
� Leads to noise correlated with the measurements
n = y − y
= (α− 1)y +w
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
22 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
ReconstructionCorrelation Between Measurements and Noise
� 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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
22 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
ReconstructionCorrelation Between Measurements and Noise
� 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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
23 Reconstruction
D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
ReconstructionCorrelation Between Measurements and Noise
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
24 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
Agenda
Projects and People
RF Communication
Reconstruction
D/A Conversion
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
25 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
25 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
25 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
25 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
25 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
26 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
26 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
26 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
27 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionQuantization in CS with a Fixed Bit Budget
Figure: Normalized mean-squared error. Saturation rate 5%.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
28 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionQuantization in CS with a Fixed Bit Budget
� 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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
28 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionQuantization in CS with a Fixed Bit Budget
� 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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
28 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionQuantization in CS with a Fixed Bit Budget
� 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.
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
Figure: Reserving quantizer indices for indicating saturation.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
29 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionQuantization in CS with a Fixed Bit Budget
Figure: Normalized mean-squared error vs. quantizer resolution.
34
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
30 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
30 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
30 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
31 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionFilter Imperfections in Random Demodulator
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
32 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
33 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
33 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
33 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
33 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
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
CompressedSensing in RFCommunication
andAnalog-to-Digital
Conversion
Thomas Arildsen
Projects andPeople
RFCommunication
Reconstruction
34 D/A Conversion
TPS, Dept. ofElectronic Systems,Aalborg University
D/A ConversionCalibration of Filter Imperfections in the Random Demodulator
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%.