1. Instructor: Yonina Eldar Teaching Assistant: Tomer Michaeli Spring 2009 Modern Sampling Methods 049033

2. 2 Sampling: “Analog Girl in a Digital World…” Judy Gorman 99 Digital worldAnalog world Signal processing Denoising Image analysis … Reconstruction D2A Sampling A2D (Interpolation)

3. 3 Applications Sampling Rate Conversion Common audio standards: 8 KHz (VOIP, wireless microphone, …) 11.025 KHz (MPEG audio, …) 16 KHz (VOIP, …) 22.05 KHz (MPEG audio, …) 32 KHz (miniDV, DVCAM, DAT, NICAM, …) 44.1 KHz (CD, MP3, …) 48 KHz (DVD, DAT, …) …

4. 4 Lens distortion correction Image scaling Applications Image Transformations

5. 5 Applications CT Scans

6. 6 Applications Spatial Superresolution

7. 7 Applications Temporal Superresolution

8. 8

9. 9 Signal generator carrier @ 500 MHz (11 dBm) Power splitter (m=2 channels) MixerLowpass filter Sign wavefor generator @ 54 MHz M=32 5 th order Chebyshev Type-I Filter Spectrum Analyzer Scope MATLAB ™ (reconstruction) Applications Low-Rate Wide Band Conversion

10. 10 Our Point-Of-View The field of sampling was traditionally associated with methods implemented either in the frequency domain, or in the time domain Sampling can be viewed in a broader sense of projection onto any subspace or union of subspaces Can choose the subspaces to yield interesting new possibilities (below Nyquist sampling of sparse signals, pointwise samples of non bandlimited signals, perfect compensation of nonlinear effects …)

11. 11 Cauchy (1841): Whittaker (1915) - Shannon (1948): A. J. Jerri, “The Shannon sampling theorem - its various extensions and applications: A tutorial review”, Proc. IEEE, pp. 1565-1595, Nov. 1977. Bandlimited Sampling Theorems

12. 12 Limitations of Shannon’s Theorem Input bandlimited Impractical reconstruction (sinc) Ideal sampling Towards more robust DSPs: General inputs Nonideal sampling: general pre-filters, nonlinear distortions Simple interpolation kernels

13. 13 Generalized anti- aliasing filter Sampling Process Linear Distortion Sampling functions Electrical circuit Local averaging

14. 14 Replace Fourier analysis by functional analysis, Hilbert space algebra, and convex optimization Original + Initial guess Reconstructed signal Sampling Process Nonlinear Distortion Nonlinear distortion Linear distortion

15. 15 Employ estimation techniques (different course …) Sampling Process Noise The statistical connection:

16. 16 Signal Priors x(t) bandlimited x(t) piece-wise linear Different priors lead to different reconstructions

17. 17 Shift invariant subspace: General subspace in a Hilbert space Signal Priors Subspace Priors Common in communication: pulse amplitude modulation (PAM) Bandlimited Spline spaces

18. 18 Two key ideas in bandlimited sampling: Avoid aliasing Fourier domain analysis Beyond Bandlimited Misleading concepts! Suppose that with Signal is clearly not bandlimited Aliasing in frequency and time Perfect reconstruction possible from samples Aliasing is not the issue …

19. 19 Example: Bandlimited Sampling Can be recovered even though it is not bandlimited? YES ! 1. Compute convolutional inverse of 2. Convolve the samples with 3. Reconstruct with

20. 20 Signal Priors Smoothness Priors Minimize the worst-case difference: Infinite dimensional non-convex optimization problem … Complicated problem but … simple solution optimal interpolation kernel

21. 21 Signal Priors Stochastic Priors Original ImageBicubic InterpolationMatern Interpolation

22. 22 Signal Priors Sparsity Priors Wavelet transform of images is commonly sparse STFT transform of speech signals is commonly sparse Fourier transform of radio signals is commonly sparse

23. 23 Sparse Modelling - Motivation Original 2500 KB 100% Compressed 950 KB 38% Compressed 392 KB 15% Compressed 148 KB 6% “Can we not just directly measure the part that will not end up being thrown away ? ” Donoho OR: It works in digital… Can it work in analog ?

24. 24 Compressed Sensing “Can we not just directly measure the part that will not end up being thrown away ?” Donoho “sensing … as a way of extracting information about an object from a small number of randomly selected observations” Candès et. al. Nyquist rate Sampling Analog Audio Signal Compression (e.g. MP3) High-rateLow-rate Compressed Sensing

25. 25 The Modulated Wideband Converter

26. 26 Does The Dream Come True ? Signal generator carrier @ 500 MHz (11 dBm) Power splitter (m=2 channels) MixerLowpass filter Sign wavefor generator @ 54 MHz M=32 5 th order Chebyshev Type-I Filter Spectrum Analyzer Scope MATLAB ™ (reconstruction)

27. 27 Reconstruction Constraints Unconstrained Schemes SamplingReconstruction

28. 28 Reconstruction Constraints Predefined Kernel SamplingReconstructionPredefined Minimax methods Consistency requirement

29. 29 Reconstruction Constraints Dense Grid InterpolationPredefined (e.g. linear interpolation) To improve performance: Increase reconstruction rate

30. 30 Reconstruction Constraints Dense Grid Interpolation Bicubic InterpolationSecond Order Approximation to Matern Interpolation with K=2 Optimal Dense Grid Matern Interpolation with K=2

31. 31 Course Outline (Subject to change without further notice) Motivating introduction after which you will all want to take this course (1 lesson) Crash course on linear algebra (basically no prior knowledge is assumed but strong interest in algebra is highly recommended) (~3 lessons) Subspace sampling (sampling of nonbandlimited signals, interpolation methods) (~2 lessons) Nonlinear sampling (~1 lesson) Minimax recovery techniques (~1 lesson) Constrained reconstruction: minimax and consistent methods (~2 lessons) Sampling sparse signals (1 lesson) Sampling random signals (1 lesson)