1/20 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 Joint Work with Waheed Bajwa (Duke University) and Kfir Gedalyahu (Technion)

2 Summary Analog compressed sensing: Xampling LTV system identification Application to super-resolution radar Real-world issues 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)) x(t) LTV system 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=Ax Many 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. InputSparsityMeasurementRecovery 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)

5 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? 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 delays Unknown 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 Serial-to-Parallel 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 & integrate Take 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 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) 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 freedom The Xampling framework implies that many analog signals have fewer DOF than previously assumed by Nyquist-rate sampling

15 Outline Previous results Channel model: Structured LTV channel Algorithm for system identification Application to super-resolution radar Identify LTV systems from a single output using minimal resources x(t) LTV system y(t)=H(x(t)) x(t) LTV system 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 identification Examples: 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 problem Finite 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 noise Doppler 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 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.

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 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 , April 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 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