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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/YoninaEldaryonina@ee.technion.ac.il

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

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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!

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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!

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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

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Gradient Methods: Wirtinger Flow

Candes et. al 14

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Gradient Methods: Truncated Amplitude Flow

Wang, Giannakis, Eldar 16

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TAF RecoveryWang, Giannakis, Eldar 16

In the presence of bounded noise we have

Theorem (Wang, Giannakis, Eldar 16)

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TAF IntuitionWang, Giannakis, Eldar 16

Truncated spectral initialization

Our TAF initialization

TAF 100 iterations

Milky way galaxy image recovery (measured with 6 masks):

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Part 2:Sparse Phase Retrieval

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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

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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)

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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)

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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)

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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

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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)

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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

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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

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Performance Comparison

Each point = 100 signals

n=64m=128

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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

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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

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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

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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

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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

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Theoretical Guarantees

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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

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Nonconvex Recovery From STFT Magnitude

Simple example (N=23, W=7, L=1, SNR=20db)

Initialization Recovery

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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).

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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

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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

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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

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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?

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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

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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

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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

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The Modulated Wideband Converter

~ ~~~

Time Frequency

Mishali and Eldar, 11

B

B

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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

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Application:Cognitive Radio

“In theory, theory and practice are the same.In practice, they are not.”

Albert Einstein

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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

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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

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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

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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!

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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

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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

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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

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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!

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SAMPL Lab Website

www.sampl.technion.ac.il

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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

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Thank you!