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SAMPL 2015 Workshop
“Xampling The Future”
Monday, June 22nd, 2015

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Workshop Goal
Expose SAMPL team activity in the area of sub-Nyquist sampling
and super-resolution with applications to
MRI
Ultrasound
Body sensor networks
Cognitive radio
Radar
Microscopy and optical imaging
Quantum systems
Defect detection in microchips
Initiate industrial and academic collaborations
Recruit talented researchers to join our activities

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SAMPL Group
Research Group:
M.Sc.
David
Cohen
Gal
Mazor
Kfir
Aberman
Tanya
Chernyakova
Amir
Kiperwas
Ph.D. Post Doc
Deborah
Cohen
Lior
Weizman
Shahar
Tsiper
Regev
Cohen
Oren
Solomon

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SAMPL Staff
Yair Keller
Lab Engineer
Eli Shoshan
Systems Advisor
Yoram Or-Chen
Advisor
Idan Shmuel
RF engineer
Robert Ifraimov
Hardware Engineer
Alon Eilam
System Engineer
Aviad Arobas
Computer
Administrator
Shahar Tsiper
RF activity support
David Cohen
Radar activity
support
Oren Solomon
Bioimaging activity
support
Anat Zaslavsky
Personal Assistant

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Undergraduate Projects
20 projects each semester
All projects are part of research
activities
Students participate in conferences
and demos worldwide
Many of the projects are performed
in collaboration with industry
Optimized Micro-
Beamforming
for Medical US
Silent MRI with
Steady Gradients
Increased Sampling
Capacity in Optical
Communication
Joint Spectrum
Blind Reconstruction
Synthetic Aperture
Radar Simulator

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Optics Team
Prof. Moti Segev Prof. Oren Cohen
Ph.D. Maor
Mutzafi
Ph.D. Dikla
Oren
Ph.D. Yoav
Shechtman
Ph.D. Pavel
Sidorenko

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Close Collaborators (Partial)
Technion Labs: SIPL, HSDSL, VLSI
Intersection of information theory and sampling theory: Prof. Andrea Goldsmith
(Stanford), Alon Kipnis, Yuxin Chen
ADC design aspects and sub-Nyquist sampling: Prof. Boris Murmann (Stanford),
Doug Adams, Niki Hammler, John Spaulding
Ultrasound: Prof. Dan Adam (Technion), Avinoam Bar-Zion, Zvi Friedman and
Arcady Kempinski (GE), Jeremy Bercoff (Supersonic US), Dr. Shai Tejman-Yarden
(Sheba Medical Center)
MRI: Dr. Dafna Ben-Bashat (Ichilov), Dr. Assaf Tal (Weizmann)
Phase Retrieval and Optics: Prof. John Miao (UCLA), Prof. Emmanuel Candes
(Stanford), Prof. Babak Hassibi (Caltech), Kishore Jaganathan, Prof. Shahar
Mandelson (Technion), Prof. Amir Beck (Technion)
Radar: Prof. Alex Haimovich (NJIT), Marco Rossi, Prof. Arye Nehorai (WUSTL),
Zhao Tan

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Industry and Medical Partners
Industry Partners:
National Instruments
General Electric
Keysight Technologies
Texas Instruments
Rafael
Medical Partners:
Rambam Hospital
Sheba Medical Center
Tel Aviv Sourasky
Medical Center
Funding:
Intel University Industry
Research Corporation
Magnet Metro 450
ICORE center
ERC Consolidator Grant
ISF - Israel Science
Foundation
BSF Program for
Transformative Science

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SAMPL Lab
Lab inauguration: Spring 2013
Main areas: graduate students + discussion
room, communication, medical imaging,
computer space

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SAMPL Lab
About 20 student projects each semester
One of the biggest labs in the department
Many awards:
Best demo award at ICASSP2014 for radar
Best demo award at ICASSP2014 for
cognitive radio
Herschel Rich innovation award for ultrasound
Kasher Prize
Magnet award, and many more …

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SAMPL Lab
Undergraduate students participate in research papers:
Many students participate in writing conference papers and many
attend conferences worldwide to present their work
Several projects have led to full journal papers
Many undergraduate students in our lab continued to higher
degrees and received the Meyer award
Tens of live demos of our technology are presented in major
conferences worldwide

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SAMPL Vision
Tight connection between deep theory and engineering
Impact basic science as well as technology and society
Teaching and mentoring of students
All undergraduate students in the lab are involved in state-of-
the-art research and demo development
Train students to do research and write scientific papers early
on in their academic studies
Only lab worldwide that develops wideband sub-Nyquist
receivers
Continue to develop new theory combined with
technology design through industry involvement while
creating human capital and training the future
generation of researchers

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Agenda
General overview of sub-Nyquist sampling
Part I: Medical Imaging (MRI, Ultrasound, ECG)
Part II: Communications and Radar
Lunch Break
Lab Awards
Part III: Optics and superresolution
Throughout the day there will be posters outside which
expand on the ideas presented
Sessions will include live demos of our prototypes

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Thanks
Research students and lab team
Eli Shoshan
Suzie Eid
Ina Rivkin
Yoram Or-Chen
Anat Zaslavsky

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Xampling:
Sub Nyquist Sampling
Cognitive radioRadar
Ultrasound Pulses DOA Estimation
15

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Very high sampling rates:
hardware excessive solutions
High DSP rates
Digital worldAnalog world
Signal processing
Image denoising
Analysis…
Processing
Music
Radar
Speech
Image…
ADCs, the front end of every digital
application, remain a major bottleneck
Sampling: “Analog Girl in a Digital World…”
Judy Gorman 99
Sampling rate must be at least
twice the highest frequency
Sampling
Analog-to-Digital
(ADC)
H. Nyquist C. Shannon

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Today’s Paradigm
The Separation Theorem:
Circuit design experts design samplers at
Nyquist rate or higher
DSP/machine learning experts process the data
Typical first step: Throw away (or combine in a “smart” way e.g.
dimensionality reduction) much of the data …
Logic: Exploit structure prevalent in most applications to reduce DSP
processing rates
However, the analog step is one of the costly steps
Can we use the structure to reduce sampling rate + first
DSP rate (data transfer, bus …) as well?
ADC
first DSP steps, bus,
data transfer
DSP
high rate, generic low rate
exploits structure
x(t) c[n]

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Proposed Paradigm
The Separation Theorem:
Circuit design experts design samplers at
Nyquist rate or higher
DSP/machine learning experts process the data
Typical first step: Throw away (or combine in a “smart” way e.g.
dimensionality reduction) much of the data …
Can we use the structure to reduce sampling rate + first
DSP rate (data transfer, bus …) as well?
ADC first DSP steps DSP
high bandwidth
exploits structure
low rate
exploits structure
x(t) c[n]Pre-Processing

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Reduce storage/reduce sampling rates
Reduce processing rates
Reduce power consumption and energy
Increase resolution
Increase speed such as frame rate in imaging
Increase the number of signals that can be concurrently processed
Enable technologies that are currently infeasible (such as fast 3D
imaging, wideband sensing, rural and portable imaging and more)
We exploit structure in the analog domain in order to reduce
sampling and processing rates and enable new technologies
Xampling: Low-Rate Sampling
Sample only the info that is needed
Xampling = Compression+Sampling

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Group Objectives
We examine the basic pillars of sampling and information theory:
Theory: Developing the fundamental limits of rate reduction based on signal
structure and the information theory of sub-sampled and structured channels
Hardware: Developing the theory and hardware of mixed analog-digital
hardware prototypes that exploit signal structure at sub-Nyquist rates
Applications: Demonstrating the broad benefit of low rate sampling in
applications ranging from wireless communication to medical imaging
Goal of the presentation:
Survey the main principles in exploiting analog structure
Provide a variety of different applications and benefits
Technical details can be found in the references
and in lectures throughout the day

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Motivation
Xampling: Compression + sampling of analog signals
Sub-Nyquist sampling without structure
Applications of sub-Nyquist sampling:
Sub-Nyquist spectrum sensing
Sub-Nyquist radar
Sub-Nyquist ultrasound
Nonlinear compressed sensing and optics
Talk Outline

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Part 1:
Motivation

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Structured Analog Models
Can be viewed as bandlimited (subspace)
But sampling at rate is a waste of resources
For wideband applications Nyquist sampling may be infeasible
Multiband communication:
Question:
How do we treat structured analog models efficiently?
Unknown carriers – non-subspace

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Cognitive Radio
Cognitive radio mobiles utilize unused spectrum ``holes’’
Spectral map is unknown a-priori, leading to a multiband model
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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Sometimes reconstructing the covariance rather than the
signal itself is enough:
Support detection
Statistical analysis
Parameter estimation (e.g. DOA)
Assumption: Wide-sense stationary ergodic signal
If 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 time
Series analysis
Deborah
Cohen

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Structured Analog Models
Digital matched filter or super-resolution ideas (MUSIC etc.) (Quazi,Brukstein,
Shan,Kailath,Pallas,Jouradin,Schmidt,Saarnisaari,Roy,Kumaresan,Tufts …)
But requires sampling at the Nyquist rate of
The pulse shape is known – No need to waste sampling resources!
Medium identification:
Unknown delays – non-subspace
Channel
Question (same):
How do we treat structured analog models efficiently?
Similar problem arises in radar, UWB
communications, timing recovery problems …

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Ultrasound
Relatively simple, radiation free imaging
Tx pulse
Ultrasonic probe
Rx signal Unknowns
Echoes result from scattering in the tissue
The image is formed by identifying the
scatterers
Cardiac sonography Obstetric sonography

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To increase SNR and resolution an antenna array is used
SNR and resolution are improved through beamforming by introducing
appropriate time shifts to the received signals
Requires high sampling rates and large data processing rates
One image trace requires 128 samplers @ 20M, beamforming to 150
points, a total of 6.3x106 sums/frame
Processing Rates
Scan Plane
Xdcr
Focusing the received
beam by applying nonlinear
delays
 2 2
1
1 1
( ; ) 4( ) sin 4( )
2
M
m m m
m
t t t t c t c
M
    

 
      
 


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Can we reduce analog sampling rates?
Can we perform nonlinear beamforming on the sub-Nyquist samples
without interpolating back to the high Nyquist-rate grid digitally?
Challenges
Compressed Beamforming
Goal: reduce ultrasound machine size at same resolution
Enable 3D imaging
Increase frame rate
Enable remote wireless ultrasound
Re
Tanya
Chernyakova

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Subwavelength Imaging + Phase Retrieval
Diffraction limit: The resolution of any optical imaging system is
limited by half the wavelength
This results in image smearing
Furthermore, optical devices only measure magnitude, not phase
100 nm
474 476 478 480 482 484 486
462
464
466
468
470
472
474
476
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

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Imaging via “Sparse” Modeling
Model FT intensity
Frequency [1/]Frequency[1/]
-5 0 5
-6
-4
-2
0
2
4
6
Diffraction-limited
(low frequency)
intensity measurements
Model
Fourier transform
Circles are
100 nm
diameter
Wavelength
532 nm
SEM image Sparse recovery
474 476 478 480 482 484 486
462
464
466
468
470
472
474
476
Blurred image
Szameit et al., Nature Materials, ‘12
Recovery of
sub-wavelength images
from highly truncated
Fourier power spectrum
Sparse phase retrieval
methods

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Part 2:
Xampling Theory
Cognitive radioRadar
Ultrasound Pulses DOA Estimation

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Theory: Union of Subspaces
Model:
Mixed estimation detection problem
Lu and Do 08, Mishali and Eldar 09

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Union of Subspaces
Model:
Standard approach: Look at sum of all subspaces
Signal bandlimited to
High rate
Lu and Do 08, Mishali and Eldar 09

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Union of Subspaces
Model:
Allows to keep low dimension in the problem model
Low dimension translates to low sampling rate
Theorem
Lu and Do 08, Mishali and Eldar 09
Multiband Sampling: 2NB
Pulse streams: 2L

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Xampling
Xampling: Compression + Sampling
Prior to analog sampling reduce bandwidth by projecting data onto
low dimensional analog space
Creates aliasing of the data
Sample the data at low rate using standard ADCs in such a way that in
the digital domain we get a compressed sensing problem
Typically set up problem in frequency: low rate processing, robustness
Results in low rate, low bandwidth, simple hardware and low
computational cost
x(t) Acquisition
Compressed
sensing and
processing
recovery
Analog preprocessing Low rate (bandwidth)
Mishali and Eldar, 10

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~~
~~
Rate should be 2L if we have L pulses
Naïve attempt: direct sampling at low rate
Most samples do not contain information!!
Multiband problem: Rate should be 2NB
Most bands do not have energy – which band should be sampled?
Low Rate Acquisition: Difficulty
Low rate
ADC
Analog preprocessing

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Alias all energy to baseband before sampling (analog projection)
Can sample at low rate
Resolve ambiguity in the digital domain
~~
~~
Smear pulse before sampling
(analog projection – bandwidth reduction)
Each sample contains energy
Resolve ambiguity in the digital domain
Intuitive Solution: Pre-Processing
Low rate
ADC

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Subspace techniques developed in the context of array
processing (such as MUSIC, ESPRIT etc.)
Compressed sensing
Connections between CS and subspace methods:
Malioutov, Cetin, and Willsky, Davies and Eldar, Lee and Bresler, Kim, Lee and Ye, Fannjiang, Austin, Moses, Ash
and Ertin
For nonlinear sampling:
Quadratic compressed sensing (Shechtman et. al 11, Eldar and Mendelson 12,
Ohlsson et. al 12, Janganathan 12)
More generally, nonlinear compressed sensing
(Beck and Eldar 12, Bahman et. al 11)
Digital Recovery

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Compressed Sensing
Candes, Romberg, Tau 06, Donoho 06

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Compressed Sensing

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Optimal Xampling Hardware
Sampling Reconstruction
AWGN
We derive two lower bounds on the performance of UoS estimation:
Fundamental limit – regardless of sampling technique or rate
Lower bound for a given sampling rate
Allows to determine optimal sampling method
Can compare practical algorithms to bound
(det. by )
Sampling with sinusoids is optimal under
a wide set of inputs!
Ben-Haim, Michaeli, and Eldar 11Aliasing
The minimal MSE is obtained with where are
the eigenfunctions of
Theorem (Generalized KLT)

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Xampling Hardware
sums of exponentials
The filter H(f) allows for additional freedom in shaping the
tones and reduces the bandwidth
The channels can be collapsed to a single channel

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Shannon Meets Nyquist:
Capacity Perspective
We can formulate our problem in a communication setting and design
sub-Nyquist sampling methods that optimize the channel capacity
What is the capacity-achieving sub-Nyquist sampler and optimal input?
It turns out that our Xampling methods are optimal in terms of capacity
as well under a wide range of settings!
Chen, Eldar and Goldsmith 13
)(th ][ny
( )n t
)(tx
EncoderMessage
signal structure

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Channel-Blind (Universal) Sampling
For each channel state, there is a sampled capacity loss with respect to the
known channel case
Robustness measure: Minimax Sampled Capacity Loss over all realizations
Chen, Goldsmith and Eldar 13
Model: Multiband channel
The sampler is designed independent of instantaneous channel realization!
maximize capacity
Encoder
x(t)

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Minimax Capacity Loss
Xampling system (with random modulator) achieves Minimax Capacity Loss!
 Complementary viewpoint on optimality of Xampling hardware
 Proves optimality of random sampling
)(th
)(t
LPF 1[ ]y n
[ ]iy n
[ ]my n

q1(t)

qi (t)

qm (t)r(t)
y1(t)
yi (t)
ym (t)
)(tx
LPF
LPF
random modulation coefficients
α: undersampling factor
β: band sparsity ratio
Optimal channel-blind sampling strategy
from information-theoretic perspective
achieved with sub-Gaussian distribution
binary entropy function

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Part 3:
Xampling Without
Structure

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Until now we exploited analog structure to reduce sampling rates
Two examples of reducing rate without structure:
Covariance estimation
Joint work With Prof. Geert Leus and Deborah Cohen
Accounting for quantization effects
Joint work with Prof. Andrea Goldsmith and Alon Kipnis
Reducing Rate Without Structure
In both cases optimal performance can be achieved at rates lower than
Nyquist since we are not interested in recovering the full analog signal!

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Until now we ignored quantization
Quantization introduces inevitable distortion to the signal
Since the recovered signal will be distorted due to quantization
do we still need to sample at the Nyquist rate?
Reducing Rate with Quantization
01001001001
010010…
quantizer
Source Coding [Shannon]Sampling Theory
ˆ[ ]y n[ ]y n
2log (#levels)
bit/sec
sR f
Goal: Unify sampling and rate distortion theory
( )x t
Kipnis, Goldsmith and Eldar 15

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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
2
ˆ( ) inf [ ] [ ]D R y n y n 
2
ˆinf ( , ) inf ( ) ( )sf sD f R x t x t 
[ ]y n ENC DEC
R
ˆ[ ]y n

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Quantizing the Samples:
Source Coding Perspective
Preserve signal components above “noise floor” q , dictated by R
Distortion corresponds to mmse error + signal components below noise floor
Theorem (Kipnis, Goldsmith, Weissman, Eldar 2013)
2
2
1
( , ) log ( ) /
2
fs
fss X Y
R f S f df 

 
 
2
2
( , ) ( ) min{ ( ), }
fs
fss sX Y X Y
D f mmse f S f df 

  

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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 R
f 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)!

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Part 4:
Applications
“In theory, theory and practice are the same.
In practice, they are not.”
Albert Einstein

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The Modulated Wideband Converter
~~
~~
Time Frequency
Mishali and Eldar, 11
B
B

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Single Channel Realization
~~
Time Frequency
Mishali and Eldar, 11
2𝑛𝑁𝑇𝑝
𝑁
𝑇𝑝
𝑥(𝑡)
𝑝(𝑡)
1
2𝑇𝑝
2
2𝑇𝑝
Bandwidth
NB
~~
𝐻(𝑓)
𝑦 𝑛
𝑇𝑝 – periodic 𝑝(𝑡) gives the desired aliasing effect
𝐻(𝑓)

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Nyquist rate: 6 GHz
Xampling 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:
Cognitive Radio
MWC analog front-end
Mishali, Eldar, Dounaevsky, and Shoshan, 2010
Cohen et. al. 2014
6% of Nyquist rate!

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Cognitive Radio Demo at ICASSP 2014
Lustig et al., 2008

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Further Details Later Today
Live Demo
Robustness to noise
DoA estimation
Distributed collaborative detection
Shahar Tsiper Deborah Cohen

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Streams of Pulses
Xampling requires using a small set of Fourier coefficients
Pulses can be entirely recovered from only 2L Fourier coefficients
Efficient hardware:
Gedalyahu, Tur, Eldar 10, Tur, Freidman, Eldar 10
Theorem (Tur, Eldar and Friedman 11)
Sum-of-Sincs filter with compact support
𝑥 𝑡 𝑠∗
−𝑡 𝐹𝐹𝑇 𝑐 𝑘

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Selecting The Active Frequencies
For good resolution and CS properties we need wide frequency aperture
To avoid ambiguities we need at least two close frequencies
Can randomly place frequencies over wide aperture
Our choice: Use a small set of bandpass filters spread randomly over a
wide frequency range
BPF4
BPF3
BPF2
Analog
signal
Band-pass
Filter 1
Low rate
ADC
Baseband down-convertor
Multichannel filter:
Wide aperture
Close frequencies

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Xampling in Radar
Distance to target
Target velocity
Demand for high range resolution radar requires high bandwidth
signals on the order of 100s Mhz to several Ghz
Classic matched filtering requires sampling and processing
at the Nyquist rate
Long time-on-target needed for
good Doppler resolution
When using multiple antennas need
to space them at the spatial Nyquist
rate – many antennas!
Targets
k
Tx/Rx sensors
Plane
wave
Bar-Ilan and Eldar 14, Itzhak et. al. 14,
Rossi, Haimovich and Eldar 14

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Later Today
Omer Bar-Ilan: sub-Nyquist Pulse-Doppler
radar – detect targets at high resolution from
sub-Nyquist samples
Deborah Cohen: Reduced time-on-target –
target detection with a small number of pulses
Alex Dikopoltsev: Cognitive radar –
transmitting only where the bandwidth is free
David Cohen: sub-Nyquist MIMO radar –
detecting targets using a small number of
antennas, each sampled at a sub-Nyquist rate
In addition:
Deborah Cohen: Resolve range ambiguity in
Doppler radar
Kfir Aberman: Applications to SAR
Deborah CohenOmer Bar-Ilan

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Xampling of Radar Pulses
1/30 of the Nyquist Rate
analog filter banks ADCs
splitters
low pass filter
Itzhak et. al. 2012 in collaboration with NI
Robert Ifraimov Idan Shmuel

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Radar Demo System at ICASSP 2014

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-80 -60 -40 -20 0 20 40 60 80
0
20
40
60
80
100
120
140
160
Standard Imaging
We obtain a 32-fold rate reduction
Enable handheld wireless devices for rural
medicine,
emergency imaging in the field/ambulance
Enable 3D imaging
High frame rate for cardiac imaging
3328 real-valued samples, per line 360 complex-valued samples, per line
-80 -60 -40 -20 0 20 40 60 80
0
20
40
60
80
100
120
140
160
-80 -60 -40 -20 0 20 40 60 80
0
20
40
60
80
100
120
140
160
100 complex-valued samples, per line
~1/10 of Nyquist per element ~1/32 of Nyquist per element
Sub-Nyquist Ultrasound Imaging
Chernyakova and Eldar 14
Tanya
Chernyakova
Alon Eilam

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Wireless Ultrasound Imaging
A wireless probe performs Xampling
and transmits the low rate data to a
server for processing
Frequency Domain Beamforming and
image reconstruction is performed
by the server
The image is sent for display on a monitor
Xampler
Alon Eilam

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Phase Retrieval:
Nonlinear Compressed Sensing
Arises in many fields: microscopy, crystallography,
astronomy, optical imaging, and more
Given an optical image illuminated by coherent light, in the far
field we obtain the Fourier transform of the image
Optical measurement devices measure the photon flux, which is
proportional to the magnitude squared of the field
Fourier +
Absolute value
2
[ ] [ ]y k X k[ ]x n
Can we extend compressed sensing results to
the nonlinear case?
Crystallography

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Phase Is Important!
Fourier
Transform
Magnitude
Fourier
Transform
Magnitude
Inverse
Fourier
Transform
Phase
Phase
Inverse
Fourier
Transform

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Phase Retrieval Applications
Ultra-short optical pulse measurement
Coherent Diffractive Imaging
Crystallography
[1] R. Trebino et al., JOSA A 10, 5 1101-1111 (1993)
[2] MM Seibert et al. Nature 470, 78-81 (2011)
[3] D Shechtman et al. PRL 53, 20, 1951-1952 (1984)
[1]
[2]
[3]

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Phase Retrieval
Difficult to analyze theoretically when recovery is possible
No uniqueness in 1D problems (Hofstetter 64)
Uniqueness in 2D if oversampled by factor 2 (Hayes 82)
No guarantee on stability
No known algorithms to achieve unique solution
Analysis of Random Measurements:
𝑦𝑖 = 𝑎𝑖, 𝑥 2 + 𝑤𝑖 noise 𝑥 ∈ 𝑅 𝑁
4𝑁 − 2 measurements needed for uniqueness (Balan, Casazza, Edidin o6, Bandira et. al 13)
random vector
How to solve objective function?
Stable Phase Retrieval (Eldar and Mendelson 14):
𝑁log(𝑁) measurements needed for stability
𝑘log(𝑁/𝑘) measurements needed for stability with sparse input
Solving 𝑖=1
𝑀
𝑦𝑖 − 𝑎𝑖, 𝑥 2 𝑝
1 < 𝑝 ≤ 2 provides stable solution

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Nonlinear Sparse Recovery
General theory and algorithms for nonlinear sparse recovery
Derive conditions for optimal solution
Use them to generate algorithms
Necessary Conditions:
L-stationarity Iterative Hard Thresholding
CW-minima Greedy Sparse Simplex (OMP)
Beck and Eldar, 13
min 𝑓 𝑥 s.t. 𝑥 0 ≤ 𝑘
Generalization of compressed sensing algorithms to the nonlinear
setting!

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GESPAR:
GrEedy Sparse PhAse Retrieval
Generalization of matching pursuit 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
1. For a given support: minimizing objective over support by linearizing
the function around current support and solve for 𝑦 𝑘
𝑧 𝑘 = 𝑧 𝑘−1 + 𝑡 𝑘(𝑦 𝑘 − 𝑧 𝑘−1)
2. Find support by finding best swap: swap index with small value 𝑥𝑖
with index with large value 𝛻𝑓(𝑥𝑗)
Shechtman, Beck and Eldar, 13
determined by backtracking

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Introducing Redundancy
Oblique illumination Candes, Eldar, Strohmer , Voroninski 12
Multiple masks (phase diversity) Candes, Li and Soltanolkotabi 13
Short-time Fourier transform (STFT) Jaganathan, Eldar and Hassibi 15
FROG/XFROG
Ptychogrpahy – scanning CDI: CDI with several overlapping
illumination patterns
All of these techniques lead to redundant
magnitude measurements which enhance recovery
STFT Recovery Results (Jaganathan, Eldar and Hassibi 15):
does not vanish anywhere then almost all
signals can be recovered uniquely
 If / 2 andL W N x n 

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Sparsity Based Subwavelength CDI
Circles are
100 nm
diameter
Wavelength
532 nm
SEM image Sparse recovery
474 476 478 480 482 484 486
462
464
466
468
470
472
474
476
Blurred image
Diffraction-limited
(low frequency)
intensity measurements
Model
Fourier transformModel FT intensity
Frequency [1/]
Frequency[1/]
-5 0 5
-6
-4
-2
0
2
4
6
Szameit et al., Nature Materials, 12

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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 sample
Short pulse X-ray
K.S. Raines et al. Nature 463, 214 ,(2010).
Mutzafi et. al., (2013).

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Compressed sampling and processing of many analog signals
Wideband sub-Nyquist samplers in hardware
Significant rate reduction in both analog and digital while retaining
sufficient quality
Merging information theory and sampling theory
Extensions to nonlinear measurements
Many applications and many research opportunities: extensions to
other analog and digital problems, robustness, hardware, many open
theoretical questions in nonlinear domain …
Exploiting structure can lead to a new sampling
paradigm which combines analog + digital and to
superresolution imaging
Conclusions

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Xampling Website
webee.technion.ac.il/people/YoninaEldar/xampling_top.html
Y. C. Eldar and G. Kutyniok, "Compressed Sensing: Theory and Applications",
Cambridge University Press, 2012
Y. C. Eldar, “Sampling Theory: Beyond Bandlimited Systems", Cambridge
University Press, 2015

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SAMPL Lab Website
www.sampl.technion.ac.il

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Thank you