Sub-Nyquist Sampling and
Identification of LTV Systems
Yonina Eldar
Department of Electrical Engineering
Technion – Israel Institute of Technology
Electrical Engineering and Statistics at Stanford
http://www.ee.technion.ac.il/people/YoninaEldar
yonina@ee.technion.ac.il
Joint Work with Waheed Bajwa (Duke University) and Kfir Gedalyahu (Technion)
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Summary
Identify LTV systems from a single output using minimal resources
exploiting the connection with sub-Nyquist sampling
x(t)
LTV system
y(t)=H(x(t))
Analog compressed sensing: Xampling
LTV system identification
Application to super-resolution radar
Real-world issues
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Compressed Sensing
Explosion of interest in the idea of CS: Recover a vector x from a small
number of measurements y=Ax
Many beautiful papers covering theory, algorithms, and applications
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Analog Compressed Sensing
Can we use these ideas to build new sub-Nyquist A/D converters?
Prior work: Yu et. al., Ragheb et. al., Tropp et. al.
Input
Sparsity
Measurement
Recovery
Standard CS
vector x
few nonzero values
random/det. matrix
convex optimization
greedy methods
Analog CS
analog signal x(t)
?
RF hardware
need to recover analog input
or specific data (demodulation)
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Sampling/Compressed Sensing
One approach to treating continuous-time signals within the CS
framework is via discretization
Thomas and Ali discussed at length this morning in thier beautiful
talks!
Alternative: Use more standard sampling techniques to convert the
signal from analog to digital and then rely on CS methods in the
digital domain (Xampling = CS + Sampling)
Possible benefits: Simple hardware, compatibility with existing
methods, smaller size digital problems
Possible drawbacks: SNR sensitivities
Can we tie the two worlds together?
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Signal Models
Unknown delays
Unknown pulse shape
degrees of freedom
per time unit
Applications: multipath communication channels,
ultrawideband, radar, bio-imaging (ultrasound) ….
(Mishali and Eldar, 08-10)
Unknown carriers
More general abstract frameworks – Union of subspaces
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Streams of Pulses
degrees of freedom
per time unit
Special case of Finite Rate of Innovation (FRI) signals (Vetterli , Marziliano& Blu)
Minimal sampling rate – the rate of innovation:
Previous work:
The rate of innovation is not achieved (Dragotti, Vetterli & Blu)
Pulse shape often limited to diracs (Kusuma & Goyal, Seelamantula & Unser)
Unstable for high model orders
Alternative approach based on discretization and CS (Herman and Strohmer)
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(Gedalyahu and Eldar 09-10)
Serial-to-Parallel
Sampling Stage: Stream of Pulses
The analog sampling filter “smoothes” the input signal :
Allows sampling of short-length pulses at low rate
CS interpretation: each sample is a
linear combination of the signal’s values.
The digital correction filter-bank:
Removes the pulse and sampling kernel effects
Samples at its output satisfy:
The delays can be recovered using ESPRIT as long as
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What Happens When There is Noise?
If the transmitted pulse is “flat” then effectively after sampling we obtain
Can use known methods for ESPRIT with noise
In our case an advantage is that we can accumulate several values of n so
as to increase robustness
Systematic study of noise effects: work in progress (Ben-Haim, Michaeli,
Eldar 10)
Develop Cramer-Rao bounds under no assumptions on the delays
and amplitudes for a given rate independent of sampling method
Examine various methods in light of the bound
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What Happens When There is Noise?
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Robustness in the Presence of Noise
Proposed scheme:
Mix & integrate
Take linear combinations
from which Fourier coeff.
can be obtained
Samples
Fourier coeff.
vector
Gedalyahu, Tur & Eldar (2010)
Supports general pulse shapes (time limited)
Operates at the rate of innovation
Stable in the presence of noise – achieves the Cramer-Rao bound
Practical implementation based on the MWC
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Xampling: Sub-Nyquist Sampling
(Mishali and Eldar, 08-10)
Sub-Nyquist sampler in hardware
Combines analog preprocessing with digital post processing
Supporting theory proves the concept and robustness for a variety of
applications including multiband signals
Allows time delay recovery from low-rate samples (Gedalyahu and
Eldar 09-10)
Applications to ultrasound (Tur and Eldar 09)
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Online Demonstrations
GUI package of the MWC
Video recording of sub-Nyquist sampling + carrier recovery in lab
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Degrees of Freedom
Low rate sampling means the signal can be represented using fewer
degrees of freedom
The Xampling framework implies that many analog signals have
fewer DOF than previously assumed by Nyquist-rate sampling
Can these ideas be exploited to characterize fundamental limits in
other areas?
Today: Applications to linear time-varying (LTV) system identification
Sub-Nyquist sampling of pulse streams can be used to
identify LTV systems using low time-bandwidth product
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Outline
Identify LTV systems from a single output using minimal resources
x(t)
LTV system
y(t)=H(x(t))
Previous results
Channel model: Structured LTV channel
Algorithm for system identification
Application to super-resolution radar
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LTV Systems
Many physical systems can be described as linear and time-varying
Identifying LTV systems can be of great importance in applications:
Improving BER in communications
Integral part of system operation (radar or sonar)
Examples:
Multipath identification
Probing signal
LTV system
LTV
channel
K targets
propagation paths
pulses per period
Received signal
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LTV Systems
Any LTV system can be written as (Kailath 62, Bello 63)
delay-Doppler spreading function
Assumption:
Underspread systems
(more generally the footprint in the delay-Doppler space has area less than 1)
Typical in communications (Hashemi 93):
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System Identification
Probe the system with a known input x(t)
LTV system
y(t)=H(x(t))
Identify the system from H(x(t)) i.e. recover the spreading function
Theorem (Kailath 62, Bello 63, Kozek and Pfander 05):
Difficulties:
Proposed algorithms require inputs with infinite bandwidth W and
infinite time support T
W – System resources, T – Time to identify targets
Can we identify a class of LTV systems with finite WT?
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Structured LTV Systems
Problem: minimize WT and provide concrete recovery method
Solution: Add structure to the problem
Finite number of delays and Dopplers:
Examples:
Multipath fading: finite number of paths between Tx and Rx
Target identification in radar and sonar: Finite number of targets
Goal: Minimize WT for identifying
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Main Identification Result
Probing pulse:
Theorem (Bajwa, Gedalyahu and Eldar 10):
WT is proportional only to the number of
unknowns!
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Implications
Without noise: Infinite resolution with finite resources
Performance degrades gracefully in the presence of noise up to a threshold
Low bandwidth allows for recovery from low-rate samples
Low time allows quick identification
Efficient hardware implementation and simple recovery
Application:
Super-resolution radar from low-rate samples and fast varying targets
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Super-resolution Radar
Main tasks in radar target detection:
Disambiguate between multiple targets, even if they have similar
velocities and range (super-resolution)
Identify targets using small bandwidth waveforms: helps
in interference avoidance and sampling (bandwidth)
Identify targets in small amount of time (time)
Matched-filtering based detection: Resolution is limited by time and
bandwidth of the radar waveform (Woodward’s ambiguity function)
CS radar (Herman and Strohmer 09): assumes discretized grid
Proposed Method: No grid assumptions and characterization of the
relationship between WT and number of targets
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Super-resolution Radar
Setup
Nine targets
Max. delay = 10 micro secs
Max. Doppler = 10 kHz
W = 1.2 MHz
T = 0.48 milli secs
N = 48 pulses in x(t)
Sequence = random binary
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Behavior With Noise
Setup: Estimation of nine delay-Doppler pairs (targets)
Time-Bandwidth Product = 5 times oversampling of the noiseless limit
Observations
Performance degrades gracefully in the presence of noise
Doppler estimation has higher MSE due to two-step recovery
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Main Tools
The results and recovery method rely on sub-Nyquist methods for
streams of pulses (Gedalyahu and Eldar 09):
Such signals can be recovered from the output of a LPF with
Allows to reduce the bandwidth W of the probing signal
The doppler shifts can be recovered from
using DOA methods by
exploiting the structure of the probing signal: reduces time
Combining sub-Nyquist sampling with DOA methods
leads to identifiability results and recovery techniques
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Conclusion
Parametric LTV systems can be identified using finite WT
Concrete polynomial time recovery method
Uses ideas from sub-Nyquist sampling: Efficient hardware
Super-resolution radar: no restrictions on delays and Dopplers
when the noise is not too high
Sub-Nyquist methods have the potential to lead to
interesting results in related areas
More details in: W. U. Bajwa, K. Gedalyahu and Y. C. Eldar, "Identification of Parametric
Underspread Linear Systems and Super-Resolution Radar,“ to appear in IEEE Trans. Sig. Proc.
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Next Step
Can we combine analog sampling methods with CS on the digital
side to improve robustness?
~
~
Can be done e.g. in the multiband problem
Extend to radar problem
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References
M. Mishali and Y. C. Eldar, “Blind multiband signal reconstruction: Compressed
sensing for analog signals,” IEEE Trans. Signal Processing, vol. 57, pp. 993–1009,
Mar. 2009.
M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of
sparse wideband analog signals,” IEEE Journal of Selected Topics on Signal
Processing, vol. 4, pp. 375-391, April 2010.
M. Mishali, Y. C. Eldar, O. Dounaevsky and E. Shoshan, " Xampling: Analog to
Digital at Sub-Nyquist Rates," to appear in IET.
K. Gedalyahu and Y. C. Eldar, "Time-Delay Estimation From Low-Rate Samples:
A Union of Subspaces Approach," IEEE Trans. Signal Processing, vol. 58, no. 6, pp.
3017–3031, June 2010.
R. Tur, Y. C. Eldar and Z. Friedman, "Low Rate Sampling of Pulse Streams with
Application to Ultrasound Imaging," to appear in IEEE Trans. on Signal Processing.
K. Gedalyahu, R. Tur and Y. C. Eldar, "Multichannel Sampling of Pulse Streams at
the Rate of Innovation," to appear in IEEE Trans. on Signal Processing.
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Thank you
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