1/20

Sub-Nyquist Sampling and Sub-Nyquist Sampling and Identification of LTV SystemsIdentification of LTV Systems

Yonina Eldar

Department of Electrical EngineeringTechnion – Israel Institute of Technology

Electrical Engineering and Statistics at Stanford

http://www.ee.technion.ac.il/people/YoninaEldaryonina@ee.technion.ac.il

Joint Work with Waheed Bajwa (Duke University) and Kfir Gedalyahu (Technion)

2

Summary

Analog compressed sensing: XamplingLTV system identification Application to super-resolution radarReal-world issues

Identify LTV systems from a single output using minimal resources exploiting the connection with sub-Nyquist

sampling

x(t) x(t) LTV system LTV system y(t)=H(x(t)) y(t)=H(x(t))

3

Compressed Sensing

Explosion of interest in the idea of CS: Recover a vector x from a small number of measurements y=AxMany beautiful papers covering theory, algorithms, and applications

4

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.

InputInput Sparsity Sparsity MeasuremenMeasurementt Recovery Recovery

Standard CS Standard CS vector xfew nonzero values random/det. matrix convex optimizationgreedy methods

Analog CSAnalog CSanalog signal x(t) ? RF hardware need to recover analog input or specific data (demodulation)

5

One approach to treating continuous-time signals within the CS framework is via discretizationThomas 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 problemsPossible drawbacks: SNR sensitivitiesCan we tie the two worlds together?

Sampling/Compressed Sensing

6

Signal Models

Applications: multipath communication channels, ultrawideband, radar, bio-imaging (ultrasound) ….

More general abstract frameworks – Union of subspaces

Unknown carriers

(Mishali and Eldar, 08-10)

degrees of freedom per time unit

Unknown delaysUnknown pulse shape

7

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 Pulse shape often limited to diracs Unstable for high model orders

Streams of Pulses

degrees of freedom per time unit

(Kusuma & Goyal, Seelamantula & Unser)

(Dragotti, Vetterli & Blu)

Alternative approach based on discretization and CS (Herman and Strohmer)

8

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

(Gedalyahu and Eldar 09-10)

9

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

10

What Happens When There is Noise?

11

Robustness in the Presence of Noise

Gedalyahu, Tur & Eldar (2010)

Proposed scheme:Mix & integrateTake linear combinations

from which Fourier coeff. can be obtained

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

Fourier coeff. vector

Samples

12

Sub-Nyquist sampler in hardwareCombines analog preprocessing with digital post processingSupporting theory proves the concept and robustness for a variety of applications including multiband signalsAllows time delay recovery from low-rate samples (Gedalyahu and Eldar 09-10) Applications to ultrasound (Tur and Eldar 09)

Xampling: Sub-Nyquist Sampling

(Mishali and Eldar, 08-10)

13

Online Demonstrations

GUI package of the MWC

Video recording of sub-Nyquist sampling + carrier recovery in lab

14

Can these ideas be exploited to characterize fundamental limits in other areas?

Degrees of Freedom

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

Low rate sampling means the signal can be represented using fewer degrees of freedomThe Xampling framework implies that many analog signals have fewer DOF than previously assumed by Nyquist-rate sampling

15

Outline

Previous resultsChannel model: Structured LTV channelAlgorithm for system identificationApplication to super-resolution radar

Identify LTV systems from a single output using minimal resources

x(t) x(t) LTV system LTV system y(t)=H(x(t)) y(t)=H(x(t))

16

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)

LTV channel propagation

paths

LTV channel propagation

paths

pulses per period

Multipath identificationExamples:

LTV system K targets

LTV system K targets

Probing signal

Received signal

17

LTV Systems

Assumption:

Underspread systems

(more generally the footprint in the delay-Doppler space has area less than 1)

Typical in communications (Hashemi 93):

delay-Doppler spreading function

Any LTV system can be written as (Kailath 62, Bello 63)

18

System Identification

Difficulties:

Proposed algorithms require inputs with infinite bandwidth W and

infinite time support T

W – System resources, T – Time to identify targets

Theorem (Kailath 62, Bello 63, Kozek and Pfander 05):

Can we identify a class of LTV systems with finite WT?

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

19

Structured LTV Systems

Problem: minimize WT and provide concrete recovery

method

Solution: Add structure to the problemFinite number of delays and Dopplers:

Goal: Minimize WT for identifying

Examples:

Multipath fading: finite number of paths between Tx and

Rx

Target identification in radar and sonar: Finite number of

targets

20

Main Identification Result

Probing pulse:

Theorem (Bajwa, Gedalyahu and Eldar 10):

WT is proportional only to the number of unknowns!

21

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

22

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

23

Super-resolution Radar

23

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

24

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 noiseDoppler estimation has higher MSE due to two-step recovery

25

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

26

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 highSub-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.

27

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

~ ~~~

28

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.

References

29

Thank you