Phase Retrieval and Analog to Digital Compression - · PDF file1/20 Workshop on the Interface...
Transcript of Phase Retrieval and Analog to Digital Compression - · PDF file1/20 Workshop on the Interface...
1/20
Workshop on the Interface of Statistics and Optimization
Feb. 2017
Phase Retrieval and Analog to Digital Compression
Yonina Eldar
Department of Electrical EngineeringTechnion – Israel Institute of Technology
http://www.ee.technion.ac.il/people/[email protected]
2
Signal Recovery from Compressed Measurements
There has been an explosion of work on recovery of sparse signals from linear measurements Basis of compressed sensing
Many applications with nonlinear measurements y=g(x)Phase retrievalQuantization
Many applications where we don’t need to estimate x but only its statistics
3
Signal Recovery from Nonlinear Measurements
Can x be recovered from y=g(x)?How many measurements are needed for recovery?What type of measurements should be used?How do the nonlinearities in the sampling and processing affect sampling rates?
Combine tools of optimization, statistics, and sampling theory
Part I: Phase retrieval Part II: Analog to digital compression
4
Goal: Recover signals from their Fourier magnitudeKnown to be impossible for 1D problems
No known stable methods for 2D problems
Recent methods rely on random measurements rather than Fourier
Proper design of deterministic Fourier measurements together with optimization methods allows for recovery even in 1D problems!
Measurement Design for Phase Retrieval
Can we design measurement schemes to enable phase retrieval from Fourier measurements?
Fourier + Absolute value
5
Sampling in the presence of quantization with Andrea Goldsmith and Alon Kipnis
Second-order statistics estimation with Geert Leus and Deborah Cohen
Applications to cognitive radio and ultrasound imaging
Analog to Digital Compression
Can we reduce sampling rates without assuming structure in the presence of nonlinearities?
6
Phase Retrieval
7
Phase Retrieval: Recover a signal from its Fourier magnitude
Fourier + Absolute value
Arises in many fields: crystallography (Patterson 35), astronomy (Fienup 82), optical imaging (Millane 90), and moreGiven an optical image illuminated by coherent light, in the far field we obtain the image’s Fourier transformOptical devices measure the photon flux,which is proportional to the magnitudePhase retrieval can allow direct recoveryof the image
Crystallography
Astronomy
8
Phase Is Important!
Fourier Transform
Magnitude
Fourier Transform
Magnitude
Inverse Fourier Transform
Phase
Phase
Inverse Fourier Transform
8
9
Theory of Phase Retrieval
Difficult to analyze theoretically when recovery is possible
No uniqueness in 1D problems (Hofstetter 64)
Uniqueness in 2D if oversampled by a factor of 2 (Hayes 82)
No guarantee on stability
No known algorithms to achieve unique solution
Recovery from Fourier Magnitude Measurements is Difficult!
10
Progress on Phase RetrievalAssume random measurements to develop theory (Candes et. al, Rauhut et. al, Gross et. al, Li et. al, Eldar et. al, Netrapalli et. al, Fannjiang et. al …)
𝑦𝑦𝑖𝑖 = 𝑎𝑎𝑖𝑖 , 𝑥𝑥 2 + 𝑤𝑤𝑖𝑖 noise
Introduce prior to stabilize solutionSupport restriction (Fienup 82)
Sparsity (Moravec et. al 07, Eldar et. al 11, Vetterli et. al 11, Shechtman et. al 11)
Add redundancy to Fourier measurementsImpulse addition and least-squares recovery (Huang et. al 15)
Short-time Fourier transform (Nawab et. al 83, Eldar et. al 15, Jaganathan et. al 15)
Masks (Candes et. al 13, Bandeira et. al 13, Jaganathan et. al 15)
random vector
11
Analysis of Random Measurements:𝑦𝑦𝑖𝑖 = 𝑎𝑎𝑖𝑖 , 𝑥𝑥 2 + 𝑤𝑤𝑖𝑖 noise 𝑥𝑥 ∈ 𝑅𝑅𝑁𝑁
4𝑁𝑁 − 2 measurements needed for uniqueness (Balan, Casazza, Edidin o6, Bandira et. al 13)
Analysis of Phase Retrieval
random vector
Stable Phase Retrieval (Eldar and Mendelson 14):
𝑂𝑂(𝑁𝑁) measurements needed for stability𝑂𝑂(𝑘𝑘log(𝑁𝑁/𝑘𝑘)) measurements needed for stability with sparse inputSolving provides stable solution
How to solve objective function?
12
Phase Retrieval Outline
Algorithms for phase retrieval
Semidefinite relaxation
Gradient methods
Sparse recovery from nonlinear measurements
Adding redundancy: STFT measurements
Applications to optics
Recent overview:Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev,“Phase retrieval with application to optical imaging”,SP magazine 2015
13
Part 1:Algorithms for PR
14
Alternating Projections-FienupAlgorithms
Basic scheme:
[Sussman 62], [Gerchberg 74], [Fienup 82][H. H. Bauschke, P. L. Combettes, and D. R. Luke 02]
Fourier
Impose Fourier Constraints
Inverse Fourier
Impose temporal Constraints:
Support, positivity ,…
Initial guess
measurements
15
Drawbacks
Algorithm may converge slowly
Even when the algorithm converges, it does not necessarily converge to the correct solution
Support has to be pre-defined
Does not perform well when only part of the Fourier measurements are given (super-resolution)
16
𝑎𝑎𝑘𝑘 , 𝑥𝑥 2 = Tr 𝐴𝐴𝑘𝑘𝑋𝑋 with 𝐴𝐴𝑘𝑘 = 𝑎𝑎𝑘𝑘𝑎𝑎𝑘𝑘𝑇𝑇, 𝑋𝑋 = 𝑥𝑥𝑥𝑥𝑇𝑇
Phase retrieval can be written asminimize rank (X)subject to A(X) = b, X ≥ 0
SDP relaxation: replace rank 𝑋𝑋 by Tr 𝑋𝑋 or by logdet(𝑋𝑋 + 𝜀𝜀𝜀𝜀) and apply reweightingPhaseCut: semidefinite relaxation based on MAXCUT (Waldspurger et. al 12)
Advantages / DisadvantagesYields the true vector whp for Gaussian meas. (Candes et al. 12)
Recovers sparse vectors whp for Gaussian meas. (Candes et al. 12)
Computationally demandingFourier measurements?
Candes, Eldar, Strohmer ,Voroninski 12
Modern Optimization Meets PR
17
We have seen that SDP relaxation can recover the true signal for sufficiently many random Gaussian measurementsWe can show that in fact SDP relaxation for Fourier phase retrievalis tight!
Create the correlation sequenceAny choice of x such that is optimal
Least-Squares Phase RetrievalHuang, Eldar and Sidiropoulos 15
Theorem (Huang, Eldar and Sidiropoulos 15)
18
Spectral Factorization
Minimum phase solution can always be found in polynomial time
Theorem
Minimum phase solution:
19
By solving two SDPs we can always solve the LS phase retrieval problem from Fourier measurementsThe solution can be found by implementing:
The minimum phase solution is optimal namely minimizes the LS errorHowever, solution may not be equal to the true x since there are no uniqueness guarantees in 1D phase retrieval
Summary: LS Phase RetrievalHuang, Eldar and Sidiropoulos 15
Convert any signal into a minimum phase signal and then measure it!
20
Any signal can be made minimum phase by adding an impulse at zero
Add an impulse at zeroTake Fourier magnitude measurementsRecover the minimum phase signalSubtract the impulse
Impulse AdditionHuang, Eldar and Sidiropoulos 15
Theorem (Huang, Eldar and Sidiropoulos 15)
Robust recovery of any 1D complex signal from Fourier magnitude measurements using SDP!
21
Simulation: Exact Recovery
Compare with Fourier measurements with recovery using PhaseLift(semidefinite relaxation) initialized by FienupBoth produced zero fitting error but only our approach led to recovery of the true signal
22
Simulation: Gaussian NoiseSignal length: 𝑁𝑁 = 128Left: 𝑀𝑀 = 4𝑁𝑁, SNR increases from 30dB to 60dBRight: SNR=50dB, 𝑀𝑀 increases from 2𝑁𝑁 to 16𝑁𝑁
We achieve the Cramer-Rao bound in all cases
23
Gradient Methods: Wirtinger Flow
Candes et. al 14
24
Gradient Methods: Truncated Amplitude Flow
Wang, Giannakis, Eldar 16
25
TAF RecoveryWang, Giannakis, Eldar 16
In the presence of bounded noise we have
Theorem (Wang, Giannakis, Eldar 16)
26
TAF IntuitionWang, Giannakis, Eldar 16
Truncated spectral initialization
Our TAF initialization
TAF 100 iterations
Milky way galaxy image recovery (measured with 6 masks):
27
Part 2:Sparse Phase Retrieval
28
Phase Retrieval: Sparsity
Sparsity can be added to improve performance in the presence of noise, missing data and more(Moravec et. al 07, Shechtman et. al 11, Bahmani et. al 11, Ohlsson et. al 12, Janganathan 12)
Both Fienup methods and SDP-based techniques can be modified to account for sparsity (Shechtman et. al 11, Mukherjee et. al 12)
Sparsity in Fienup Iterations:In the time (space) domain we choose the k-largest valuesNo proof of optimality
28
29
Quadratic Compressed SensingShechtman, Eldar, Szameit and Segev, 11
Recover a sparse vector from quadric measurements 𝑦𝑦𝑘𝑘 = 𝑥𝑥𝑇𝑇𝐴𝐴𝑘𝑘𝑥𝑥𝑥𝑥𝑇𝑇𝐴𝐴𝑘𝑘𝑥𝑥 = Tr 𝐴𝐴𝑘𝑘𝑋𝑋 with 𝑋𝑋 = 𝑥𝑥𝑥𝑥𝑇𝑇If x is sparse then X is row sparsePhase retrieval can be written as
minimize rank 𝑋𝑋subject to 𝐴𝐴 𝑋𝑋 = 𝑏𝑏
𝑋𝑋 ≥ 0𝑋𝑋 1,2 ≤ 𝑘𝑘 𝑜𝑜𝑜𝑜 𝑋𝑋 1 ≤ 𝑘𝑘
SDP relaxation: replace rank(𝑋𝑋) by Tr(𝑋𝑋) or by logdet 𝑋𝑋 + 𝜖𝜖𝜀𝜀 and apply reweightingSDP methods require 𝑘𝑘2log(𝑁𝑁) measurements (Candes et. al 12, Li 12)
30
Greedy Methods: General Theory
min f(x) s.t. ||x||0 ≤ 𝑘𝑘
General theory and algorithms for nonlinear sparse recovery
Derive conditions for optimal solution
Use them to generate algorithms
Since our problem is non-convex no simple necessary and sufficient conditions
We develop two necessary conditions
L-stationarityCW-minima
Beck and Eldar, 13
Iterative Hard ThresholdingGreedy Sparse Simplex (OMP)
31
L-StationarityFor constrained problems min 𝑓𝑓 𝑥𝑥 : 𝑥𝑥 ∈ 𝐶𝐶 where C is convex, a necessary condition is stationarity
< 𝛻𝛻𝑓𝑓 𝑥𝑥∗ , 𝑥𝑥 − 𝑥𝑥∗ > ≥ 0 for all 𝑥𝑥 ∈ 𝐶𝐶
For any 𝐿𝐿 > 0 a vector 𝑥𝑥∗ is stationary if and only if
𝑥𝑥∗ = 𝑃𝑃𝑐𝑐(𝑥𝑥∗ − 1𝐿𝐿𝛻𝛻𝑓𝑓 𝑥𝑥∗ )
Does not depend on L!
For our nonconvex setting we define an L-stationary point:𝑥𝑥∗ ∈ 𝑃𝑃𝑐𝑐(𝑥𝑥∗ − 1
𝐿𝐿𝛻𝛻𝑓𝑓 𝑥𝑥∗ )
The projection is no longer unique
Let 𝛻𝛻𝑓𝑓 𝑥𝑥 be Lipschitz continuous. Then L-stationarity with L>L(f) is necessary for optimality
Theorem (Beck and Eldar, 13)
32
L-Stationarity: DiscussionAdvantages:
L–stationarity is necessary for optimalitySimple algorithm that finds L–stationary points
Algorithm popular for compressed sensing when 𝑦𝑦 = 𝐴𝐴𝑥𝑥(Blumensath and Davies, 08)
Disadvantages:Requires knowledge of Lipschitz constant
Often converges to wrong solution – not a strong condition
Iterative Hard Thresholding:𝑥𝑥𝑘𝑘+1 ∈ 𝐻𝐻(𝑥𝑥𝑘𝑘 − 1
𝐿𝐿𝛻𝛻𝑓𝑓 𝑥𝑥𝑘𝑘 )
where 𝐻𝐻(𝑥𝑥)=hard thresholding of 𝑥𝑥
33
Coordinate–Wise Minima
For an unconstrained problem 𝑥𝑥∗ is a coordinate-wise minima (CW) if 𝑥𝑥𝑖𝑖∗ is a minimum with respect to the ith component
𝑥𝑥𝑖𝑖∗ ∈ argmin 𝑓𝑓(𝑥𝑥1∗, . . , 𝑥𝑥𝑖𝑖−1∗ , 𝑥𝑥𝑖𝑖, 𝑥𝑥𝑖𝑖+1∗ , … , 𝑥𝑥𝑛𝑛∗ )
For our constrained problem we define CW as:
1. 𝑥𝑥∗ 0 < 𝑘𝑘 and 𝑓𝑓 𝑥𝑥∗ = min𝑡𝑡∈𝑅𝑅
𝑓𝑓(𝑥𝑥∗ + 𝑡𝑡𝑒𝑒𝑖𝑖)
2. 𝑥𝑥∗ 0 = 𝑘𝑘 and for every 𝑖𝑖 ∈ 𝑆𝑆𝑓𝑓 𝑥𝑥∗ ≤ min
𝑡𝑡∈𝑅𝑅𝑓𝑓(𝑥𝑥∗ − 𝑥𝑥𝑖𝑖∗𝑒𝑒𝑖𝑖 + 𝑡𝑡𝑒𝑒𝑗𝑗)
Any optimal solution is a CW minima
Theorem (Beck and Eldar, 13)
34
CW–Minima: Discussion
Does not require Lipschitz continuityAny optimal solution is a CW minimaIf gradient is Lipschitz continuous:
In fact, CW minima implies L’ stationarity with 𝐿𝐿𝐿 < 𝐿𝐿
Stronger than L- stationarity
Optimal solution CW-minima L’-stationarity L-stationarity
Optimal solution CW-minima L-stationarity
35
Greedy Sparse Simplex Method
Generalizes orthogonal matching pursuit to nonlinear objectives
Additional correction step which improves OMP
Converges to CW minima (stronger than IHT guarantee)
Does not require Lipschitz continuity
General step 𝑥𝑥∗ 0 < 𝑘𝑘 : find coordinate that minimizes 𝑓𝑓(𝑥𝑥)
𝑡𝑡𝑖𝑖 ∈ arg min𝑡𝑡𝑓𝑓(𝑥𝑥𝑘𝑘 + 𝑡𝑡𝑒𝑒𝑖𝑖) 𝑓𝑓𝑖𝑖 = min
𝑡𝑡𝑓𝑓(𝑥𝑥𝑘𝑘 + 𝑡𝑡𝑒𝑒𝑖𝑖)
𝑥𝑥𝑘𝑘+1 = 𝑥𝑥𝑘𝑘 + 𝑡𝑡𝑖𝑖∗𝑒𝑒𝑖𝑖∗
Swap 𝑥𝑥∗ 0 = 𝑘𝑘 : Swap index i with best j if lower objective
36
GESPAR: GrEedy Sparse PhAse RetrievalSpecializing our general algorithm to phase retrieval
Local search method with update of support
For given support solution found via Damped Gauss Newton
Efficient and more accurate than current techniques
Shechtman, Beck and Eldar, 13
1. For a given support: minimizing objective over support by linearizing the function around current support and solve for 𝑦𝑦𝑘𝑘
𝑧𝑧𝑘𝑘 = 𝑧𝑧𝑘𝑘−1 + 𝑡𝑡𝑘𝑘(𝑦𝑦𝑘𝑘 − 𝑧𝑧𝑘𝑘−1)
2. Find support by finding best swap: swap index with small value 𝑥𝑥𝑖𝑖with index with large value 𝛻𝛻𝑓𝑓(𝑥𝑥𝑗𝑗)
determined by backtracking
37
Performance Comparison
Each point = 100 signals
n=64m=128
38
Nonlinear Sparse Recovery:
We can also consider a regularized approach for
where x is k-sparse and has length n
If f is an invertible function, and A satisfies RIP-like properties, then x
can be recovered from measurements
Any stationary point of
with will recover the true x with vanishing error
Such a point can be found by using a Gradient-descent like algorithm
combined with soft thresholding
Yang, Wang, Liu, Eldar and Zhang 15
39
Part 3:Short-Time Fourier
Magnitude
Delay [fs]
Freq
uenc
y [P
Hz]
Measured FROG trace
-400 -200 0 200 400
-0.5
0
0.5
40
Recovery From the STFT Magnitude
Easy to implement in optical settingsFROG – measurements of short pulses (Trebino and Kane 91)
Ptychography – measurement of optical images (Hoppe 69)
Also encountered in speech processing (Griffin and Lim 84, Nawab et. al 83)
Almost all signals can be recovered as long as there is overlap between the segmentsAlmost all signals can be recovered using semidefinite relaxationGradient methods with proper initialization
L – step sizeN – signal lengthW – window length
41
Method for measuring ultrashort laser pulsesThe pulse gates itself in a nonlinear medium and is then spectrally resolved
In XFROG a reference pulse is used for gating leading to STFT-magnitude measurements:
Frequency-Resolved Optical Gating (FROG)
Trebino and Kane 91
41
L – step sizeN – signal lengthW – window length
Delay [fs]
Freq
uenc
y [P
Hz]
Measured FROG trace
-400 -200 0 200 400
-0.5
0
0.5
42
PtychographyHoppe 69
Plane waveScanning
For all positions (𝑋𝑋𝑖𝑖 ,𝑌𝑌𝑗𝑗) record
diffraction pattern 𝜀𝜀𝑘𝑘 𝜀𝜀3𝜀𝜀2𝜀𝜀1
𝜀𝜀𝐾𝐾
Method for optical imaging with X-raysRecords multiple diffraction patterns as a function of sample positionsMathematically this is equivalent to recording the STFT
43
Theoretical Guarantees
43
Theorem (Eldar, Sidorenko, Mixon et. al 15)The STFT magnitude with L=1 uniquely determines any x[n] that is everywhere nonzero (up to a global phase factor) if:1. The length-N DTFT of is nonzero2.3. N and W-1 are coprime
Theorem (Jaganathan, Eldar and Hassibi 15)The STFT magnitude uniquely determines almost any x[n] that is everywhere nonzero (up to a global phase factor) if:1. The window g[n] is nonzero2.
Uniqueness condition for L=1 and all signals:
Uniqueness condition for general overlap and almost all signals:
Strong uniqueness for 1D signals and Fourier measurements
44
Recovery From STFT via SDPTheorem (Jaganathan, Eldar and Hassibi 15)
SDP relaxation uniquely recovers any x[n] that is everywhere nonzero from the STFT magnitude with L=1 (up to a global phase factor) if
In practice SDP relaxation seems to works as long as (at least 50% overlap) No proof yet …Strong phase transition at L=W/2
Probability of Success for N=32 for differentL and W
Can prove the result assuming the first W/2 values of x[n] are known
L=W/2
L
W
N = 32
45
Nonconvex Recovery From STFT Magnitude
We consider the data in a transformed domain (1D DFT with respect to the frequency variable)
where is the (non-Hermitian) measurement matrix We suggest using gradient descent to minimize the non-convex loss
Initialization by the principle eigenvector of a matrix, constructed as the solution of a least-squares problem Under appropriate conditions, initialization is close to the true solution
Bendory and Eldar 16
46
Nonconvex Recovery From STFT Magnitude
Simple example (N=23, W=7, L=1, SNR=20db)
Initialization Recovery
47
Subwavelength Imaging + Phase Retrieval
Diffraction limit: The resolution of any optical imaging system is limited by half the wavelengthThis results in image smearing Furthermore, optical devices only measure magnitude, not phase
100 nm
Collaboration with the groups of Moti Segev and Oren Cohen
Sketch of an optical microscope: the physics of EM waves acts
as an ideal low-pass filter
Nano-holes as seen in
electronic microscope
Blurred image seen in
optical microscope
λ=514nm
48
Sparsity Based Subwavelength CDI
Circles are 100 nm diameter
Wavelength 532 nm
SEM image Sparse recoveryBlurred image
Diffraction-limited (low frequency)
intensity measurements
Model Fourier transform
Frequency [1/λ]
Freq
uenc
y [1
/ λ]
-5 0 5
-6
-4
-2
0
2
4
6
Szameit et al., Nature Materials, 12
49
Sparsity Based Ankylography
Concept:A short x-ray pulse is scattered from a 3D molecule combined of known elements. The 3D scattered diffraction pattern is then sampled in a single shot
Recover a 3D molecule using 2D sampleShort pulse X-ray
K.S. Raines et al. Nature 463, 214 ,(2010).Mutzafi et. al., (2013).
50
Analog to Digital Compression
Phase retrieval: using optimization techniques and statistical analysis to recover from nonlinear measurements
In the presence of nonlinear processing (quantization, statistics estimation), can we reduce the sampling rate?
Sampling rate reduction in the presence of quantizationwith Andrea Goldsmith and Alon Kipnis
Power spectrum estimation from sub-Nyquist samples with Geert Leus and Deborah Cohen
51
Shannon-Nyquist theorem: Bandlimited signal with bandwidth 2𝐵𝐵Minimal sampling rate: 𝑓𝑓𝑁𝑁𝑁𝑁𝑁𝑁 = 2𝐵𝐵
Landau rate:Multiband signal with known support of measure ΛMinimal sampling rate: Λ
Extension to arbitrary subspaces:Signal in a subspace with dimension 𝐷𝐷 requires sampling at rate 𝐷𝐷Shift-invariant subspaces ∑𝑛𝑛∈ℤ 𝑎𝑎 𝑛𝑛 ℎ(𝑡𝑡 − 𝑛𝑛𝑛𝑛) require sampling at rate 1/𝑛𝑛
Traditional SamplingSampling rate required in order to recover 𝑥𝑥(𝑡𝑡) from its samples
52
Exploit analog structure to reduce sampling rateMultiband signal with unknown supportof measure ΛMinimal sampling rate: 2Λ (Mishali and Eldar ‘09)
Stream of k pulses (finite rate of innovation) Minimal sampling rate: 2k (Vetterli et. al ‘02)
Union of subspaces (Lu and Do ‘08, Mishali and Eldar ‘09)
Sparse vectors(Candes, Romberg, Tau ‘06, Donoho ‘06)
Sampling of Structured Signals
Sampling rate below Nyquist for recovery of 𝑥𝑥 𝑡𝑡 by exploiting structure
53
Many examples in which we can reducesampling rate by exploiting structureXampling: practical sub-Nyquist methodswhich allow low-rate sampling and low-rate processing in diverse applications
Sub-Nyquist Sampling
Cognitive radioRadar
UltrasoundPulsesDOA Estimation
54
Xampling Hardware
sums of exponentials
The filter H(f) shapes the tones and reduces bandwidth
The channels can be collapsed to a single channel
x(t) AcquisitionCompressed sensing and processing
recovery
Analog preprocessing Low rate (bandwidth)
55
Achieves the Cramer-Rao bound for analog recovery given a sub-Nyquist sampling rate (Ben-Haim, Michaeli, and Eldar 12)Minimizes the worst-case capacity loss for a wide class of signal models (Chen, Eldar and Goldsmith 13)Capacity provides further justification for the use of random tones
Optimality of Xampling Hardware
)(th ][ny
( )n t)(tx
EncoderMessage
signal structurecaptured by channel
capacity-achieving sub-Nyquist sampler
binary entropy function
α: undersampling factor
β: sparsity ratio
56
Until now we ignored quantizationQuantization introduces inevitable distortion to the signalSince the recovered signal will be distorted due to quantization do we still need to sample at the Nyquist rate?
Reducing Rate with Quantization
01001001001010010…
quantizer
Source Coding [Shannon]Sampling Theory
ˆ[ ]y n[ ]y n2log (#levels)
bit/secsR f=
Goal: Unify sampling and rate distortion theory
( )x t
Kipnis, Goldsmith and Eldar 15
57
Standard source coding: For a given discrete-time process y[n] and a given bit rate R what is the minimal achievable distortion
Our question:For a given continuous-time process x(t) and a given bit rate R what is the minimal distortion
What sampling rate is needed to achieve the optimal distortion?
Unification of Rate-Distortion and Sampling Theory
)(th( )x t[ ]y n
( )n t
ENC DEC
R
f s
ˆ( )x t
2ˆ( ) inf [ ] [ ]D R y n y n= −
2ˆinf ( , ) inf ( ) ( )sf sD f R x t x t= −
[ ]y n ENC DEC
R ˆ[ ]y n
58
Quantizing the Samples:Source Coding Perspective
Preserve signal components above “noise floor” q , dictated by RDistortion corresponds to mmse error + signal components below noise floor
Theorem (Kipnis, Goldsmith, Eldar, Weissman 2014)
2
2
1( , ) log ( ) /2
fs
fss X YR f S f dfθ θ+
− = ∫
2
2
( , ) ( ) min{ ( ), }fs
fss sX Y X YD f mmse f S f dfθ θ−
= + ∫
59
Can we achieve D(R) by sampling below fNyq?
Yes! For any non-flat PSD of the input
Optimal Sampling Rate
( , ) ( ) for ( )!
s
s DR
D R f D Rf f R
=≥
Shannon [1948]:“we are not interested in exact transmission when we have a continuous source, but only in transmission to within a given tolerance”
No optimality loss when sampling at sub-Nyquist (without input structure)!
60
Sometimes reconstructing the covariance rather than the signal itself is enough:
Support detectionStatistical analysisParameter estimation (e.g. DOA)
Assumption: Wide-sense stationary ergodic signalIf all we want to estimate is the covariance then we can substantially reduce the sampling rate even without structure!
Power Spectrum Reconstruction
What is the minimal sampling rate to estimate the signal covariance?
Cognitive Radios Financial timeSeries analysis
61
Let 𝑥𝑥 𝑡𝑡 be a wide-sense stationary ergodic signalWe sample 𝑥𝑥 𝑡𝑡 with a stable sampling set at times �𝑅𝑅 = {𝑡𝑡𝑖𝑖}𝑖𝑖∈ℤWe want to estimate 𝑜𝑜𝑥𝑥 𝜏𝜏 = 𝔼𝔼[𝑥𝑥(𝑡𝑡)x(𝑡𝑡 − 𝜏𝜏)]
Covariance Estimation
What is the minimal sampling rate to recover 𝑜𝑜𝑥𝑥 𝜏𝜏 ?Sub-Nyquist sampling is possible!
Intuition: The covariance 𝑜𝑜𝑥𝑥 𝜏𝜏 is a function of the time lags 𝜏𝜏 = 𝑡𝑡𝑖𝑖 − 𝑡𝑡𝑗𝑗To recover 𝑜𝑜𝑥𝑥 𝜏𝜏 , we are interested inthe difference set R:
Sampling set�𝑅𝑅 = {𝑡𝑡𝑖𝑖}𝑖𝑖∈ℤ
Difference set𝑅𝑅 = {𝑡𝑡𝑖𝑖 − 𝑡𝑡𝑗𝑗}𝑖𝑖,𝑗𝑗∈ℤ
𝑡𝑡𝑖𝑖 > 𝑡𝑡𝑗𝑗
𝑡𝑡1𝑡𝑡2
𝑡𝑡3𝑡𝑡4
𝑡𝑡5
𝑡𝑡2 − 𝑡𝑡1
𝑡𝑡4 − 𝑡𝑡1𝑡𝑡4 − 𝑡𝑡2𝑡𝑡3 − 𝑡𝑡2
𝑡𝑡5 − 𝑡𝑡4
𝑡𝑡4 − 𝑡𝑡3𝑡𝑡5 − 𝑡𝑡2
𝑡𝑡3 − 𝑡𝑡1
𝑡𝑡5 − 𝑡𝑡3
Cohen, Eldar and Leus 15
62
It is possible to create sampling sets with Beurling density 0 for which the difference set has Beurling density ∞!
There should be enough distinct differences so that the size of the difference set goes like the square of the size of the sampling setThe density of the set should go to 0 slower than the square root
the density of the square (difference set) goes to ∞
Difference Set Density
Theorem
63
Under the previous conditions on the sampling set, we can reconstruct 𝑜𝑜𝑥𝑥(𝜏𝜏) from {𝑥𝑥 𝑡𝑡𝑖𝑖 }𝑖𝑖∈ℤ
Universal Minimal Sampling Rate
We can reconstruct the covariance from signal samples with density 0!
Theorem
64
Cantor ternary set: repeatedly delete the openmiddle third of a set of line segments, startingwith the interval [0,1]
Sampling set: 𝐷𝐷− �𝑅𝑅𝐶𝐶 ⟶ 0Difference set: 𝐷𝐷− 𝑅𝑅𝐶𝐶 ⟶ ∞ (both conditions hold)
Uniform sampling: let �𝑅𝑅𝑈𝑈 = {𝑘𝑘𝑛𝑛}𝑘𝑘∈ℤ be a uniform sampling set spaced by T. It holds that 𝑅𝑅𝑈𝑈 = �𝑅𝑅𝑈𝑈. If 𝑛𝑛 ⟶ ∞, then
Sampling set: 𝐷𝐷− �𝑅𝑅𝑈𝑈 ⟶ 0
Difference set: 𝐷𝐷− 𝑅𝑅𝑈𝑈 ⟶ 0 (not enough distinct differences)
Sampling Sets Examples
Can we analyze practical sampling sets with positive Beurling density?
65
Practical sampling set with finite rateDivide the Nyquist grid into blocks of 𝑛𝑛 consecutive samples (cosets)Keep 𝑚𝑚 samples from each blockSampling set: 𝐷𝐷− �𝑅𝑅 = 𝑚𝑚
𝑛𝑛𝑇𝑇
Multicoset Sampling
What is the minimal sampling rate for perfect covariance recovery from multicoset samples with 𝑛𝑛 cosets?
𝑛𝑛𝑡𝑡 = 𝑘𝑘𝑛𝑛𝑛𝑛
𝑥𝑥(𝑡𝑡)
𝑐𝑐1𝑛𝑛
𝑐𝑐𝑚𝑚𝑛𝑛
𝑡𝑡 = 𝑘𝑘𝑛𝑛𝑛𝑛
𝑥𝑥𝑐𝑐1[𝑘𝑘]
𝑥𝑥𝑐𝑐𝑚𝑚[𝑘𝑘]
Time shifts
66
Achieved when the differences between two distinct cosets are unique, namely 𝑐𝑐𝑖𝑖 − 𝑐𝑐𝑗𝑗 ≠ 𝑐𝑐𝑘𝑘 − 𝑐𝑐𝑙𝑙 ,∀𝑖𝑖 ≠ 𝑘𝑘, 𝑗𝑗 ≠ 𝑙𝑙Known as the Golomb ruler
Multicoset – Bandlimited Signal
Signal recovery: 𝑚𝑚 ≥ 𝑛𝑛Covariance recovery: 𝑚𝑚 ≿ 𝑛𝑛
Theorem
67
Let 𝑥𝑥(𝑡𝑡) be sparse with unknown support with occupancy 𝜀𝜀 < ⁄1 2
Minimal sampling rate for signal recovery: ⁄2𝜀𝜀𝑇𝑇 (Mishali and Eldar ‘09)
Multicoset – Sparse Signal
Signal recovery: 𝑚𝑚 ≥ 2𝜀𝜀𝑛𝑛Covariance recovery: 𝑚𝑚 ≿ 2𝜀𝜀𝑛𝑛
Theorem
68
The Modulated Wideband Converter
~ ~~~
Time Frequency
Mishali and Eldar, 11
B
B
69
Single Channel Realization
~ ~
Mishali and Eldar, 11
2𝑛𝑛𝑁𝑁𝑛𝑛𝑝𝑝𝑁𝑁𝑛𝑛𝑝𝑝
𝑥𝑥(𝑡𝑡)
𝑝𝑝(𝑡𝑡)
12𝑛𝑛𝑝𝑝
22𝑛𝑛𝑝𝑝
BandwidthNB
~~
𝐻𝐻(𝑓𝑓)
𝑦𝑦 𝑛𝑛
𝐻𝐻(𝑓𝑓)
The MWC does not require multiple channelsDoes not need accurate delaysDoes not suffer from analog bandwidth issues
70
Application:Cognitive Radio
“In theory, theory and practice are the same.In practice, they are not.”
Albert Einstein
71
Cognitive RadioCognitive radio mobiles utilize unused spectrum ``holes’’Need to identify the signal support at low rates
Federal Communications Commission (FCC)frequency allocation
Licensed spectrum highly underused: E.g. TV white space, guard bands and more
Shared Spectrum Company (SSC) – 16-18 Nov 2005
72
Power Spectrum RecoveryCohen and Eldar, 2013
Power spectrum detection outperforms signal detection
Can be used to reduce sampling rate by ½ and to improve robustness
3 signals 80 MHz eachNyquist rate 10 GhzSampling rate 1.04 Ghz
73
Cyclic Spectrum Recovery
Cyclostationary processes: periodic statisticsCyclic spectrum: cyclic frequencies of modulated signals depend on theircarrier frequencies and bandwidthsCyclic spectrum recovery: correlation of frequency-shifted versions of the MWC sampleswhere is mapped to the cyclic spectrumIncreased robustness to stationary noise
AM BPSK
Cyclostationary detection outperforms energy detection at low SNRs
3 AM 100 MHz eachNyquist rate 10 GHzSampling rate 1.09 GHz
74
Nyquist rate: 6 GHzXampling rate: 360 MHz (6% of Nyquist rate)
Wideband receiver mode: 49 dB dynamic range, SNDR > 30 dB ADC mode: 1.2v peak-to-peak full-scale, 42 dB SNDR = 6.7 ENOB
Parameters:
Performance:
Nyquist: 6 GHzSampling Rate: 360MHz
MWC analog front-end
Mishali, Eldar, Dounaevsky, and Shoshan, 2010Cohen et. al. 20146% of Nyquist rate!
75
76
UltrasoundRelatively simple, radiation free imaging
Tx pulse
Ultrasonic probe
Rx signal Unknowns
Echoes result from scattering in the tissueThe image is formed by identifying the scatterers
Cardiac sonography Obstetric sonography
77
To increase SNR and resolution an antenna array is usedSNR and resolution are improved through beamforming by introducing appropriate time shifts to the received signals
Requires high sampling rates and large data processing ratesOne image trace requires 128 samplers @ 20M, beamforming to 150 points, a total of 6.3x106 sums/frame
Processing Rates
Scan Plane
Xdcr
Focusing the receivedbeam by applying nonlinear delays
( )2 2
1
1 1( ; ) 4( ) sin 4( )2
M
m m mm
t t t t c t cM
θ ϕ δ θ δ=
Φ = − − − +
∑
128-256 elements
78
79
Same beamforming idea can be used in radar in order to obtain high resolution radar from low rate samplesOur radar prototype is robust to noise and clutterDoppler Focusing (beamforming in frequency):
Pulse-Doppler Radar
Optimal SNR scalingCS size does not increase with number of pulsesNo restrictions on the transmitterClutter rejection and the ability to handle large dynamic range
Bar-Ilan and Eldar, 13
80
Conclusions
Phase retrieval is possible with a combination of new measurements and optimization tools
Leads to an extension of sparse recovery to nonlinear measurements
Sub-Nyquist sampling of arbitrary signals by exploiting processing task and taking nonlinearities into account
Robust cognitive radio at sub-Nyquist rates
Ultrasound and radar at very low rates
Can substantially reduce rates and recover from nonlinearities by careful measurement and algorithm design!
82
AcknowledgementsGraduate students (Past and Present)
DeborahCohen
TanyaChernyakova
ShaharTsiper
TamirBen-Dori
Engineers: Eli Shoshan, Yair Keller, Robert Ifraimov, Alon Eilam,
Idan Shmuel
Collaborators
Funding: ERC, ISF, BSF, SRC, Intel
Industry: GE, NI, Agilent
AlonKipnis
PavelSidorenko
KejunHuang
KishoreJaganathan
83
If you found this interesting …Looking for a post-doc!
Thank you!