1. Richard Baraniuk Rice University dsp.rice.edu/cs Lecture 2: Compressive Sampling for Analog Time Signals

2. Analog-to-Digital Conversion

3. Sensing by Sampling Foundation of Analog-to-Digital conversion: Shannon/Nyquist sampling theorem –periodically sample at 2x signal bandwidth Increasingly, signal processing systems rely on A/D converter at front-end –radio frequency (RF) applications have hit a performance brick wall

5. Sensing by Sampling Foundation of Analog-to-Digital conversion: Shannon/Nyquist sampling theorem –periodically sample at 2x signal bandwidth Increasingly, signal processing systems rely on A/D converter at front-end –RF applications have hit a performance brick wall –“Moore’s Law” for A/D’s: doubling in performance only every 6 years” Major issues: –limited bandwidth (# Hz) –limited dynamic range (# bits) –deluge of bits to process downstream

6. “Analog-to-Information” Conversion [Dennis Healy, DARPA]

7. Signal Sparsity Shannon was a pessimist –sample rate N times/sec is worst-case bound Sparsity: “information rate” K per second, K << N Applications: Communications, radar, sonar, … wideband signal samples large Gabor (TF) coefficients time frequency time

8. Local Fourier Sparsity (Spectrogram) time frequency

9. Signal Sparsity wideband signal samples large Gabor (TF) coefficients Fourier matrix

10. Compressive Sampling Compressive sampling “random measurements” measurements sparse signal information rate

11. Compressive Sampling Universality

12. Streaming Measurements measurements Nyquist rate information rate streaming requires special Streaming applications: cannot fit entire signal into a processing buffer at one time

13. A Simple Model for Analog Compressive Sampling

14. Analog CS analog signal digital measurements information statistics A2IDSP Analog-to-information (A2I) converter takes analog input signal and creates discrete (digital) measurements Much of CS literature involves exclusively discrete signals First, define an appropriate signal acquisition model

15. A Simple Analog CS Model K-sparse vector analog signal digital measurements information statistics A2IDSP Operator takes discrete vector and generates analog signal from a (wideband) subspace

16. A Simple Analog CS Model K-sparse vector analog signal digital measurements information statistics A2IDSP Operator takes analog signal and generates discrete vector

17. Analog CS K-sparse vector analog signal digital measurements information statistics A2IDSP is a CS matrix

18. Architectures for A2I: 1. Random Sampling

19. A2I via Random Sampling [Gilbert, Strauss, et al] Can apply “random” sampling concepts from Anna Gilbert’s lectures directly to A2I Average sampling rate < Nyquist rate Appropriate for narrowband signals (sinusoids), wideband signals (wavelets), histograms, … Highly efficient, one-pass decoding algorithms

20. Sparsogram Spectrogram computed using random samples

21. Example: Frequency Hopper Random sampling A2I at 13x sub-Nyquist average sampling rate spectrogram sparsogram

22. Architectures for A2I: 2. Random Filtering

23. A2I via Random Filtering Analog LTI filter with “random impulse response” Quasi-Toeplitz measurement system y(t)y(t)

24. Comparison to Full Gaussian Fourier-sparse signals N = 128, K = 10 y(t)y(t) B = length of filter h in terms of Nyquist rate samples = horizontal width of band of A2I conv

25. Architectures for A2I: 3. Random Demodulation

26. A2I via Random Demodulation

27. Theorem [Tropp et al 2007] If the sampling rate satisfies then locally Fourier K -sparse signals can be recovered exactly with probability

28. Empirical Results

29. Example: Frequency Hopper Random demodulator AIC at 8x sub-Nyquist spectrogram sparsogram

30. Summary Analog-to-information conversion: Analog CS Key concepts of discrete-time CS carry over Streaming signals require specially structured measurement systems Tension between what can be built in hardware versus what systems create a good CS matrix Three examples: –random sampling, random filtering, random demodulation

31. Open Issues New hardware designs New transforms that sparsity natural and man-made signals Analysis and optimization under real-world non-idealities such as jitter, measurement noise, interference, etc. Reconstruction/processing algorithms for dealing with large N dsp.rice.edu/cs