EE 123 Discussion Section 6. Frank Ong March 14th, 2016

Transcription

1 EE 123 Discussion Section 6 Frank Ong March 14th, 2016

2 Plan Sparse FFT Magnitude Filter Design with convex optimization

3 Sparse FFT Given a length-n signal, FFT takes O(N log N) time to compute its DFT What if our DFT is k-sparse, ie, has only k non-zero entries, How much faster can we compute the DFT?

4 Sparse FFT Given a length-n signal, FFT takes O(N log N) time to compute its DFT What if our DFT is k-sparse, ie, has only k non-zero entries, How much faster can we compute the DFT? O(k log k) if non-zero locations are randomized The FFAST algorithm:

5 Three Ideas behind FFAST Sparse spectrum does not alias much 1-sparse DFT is easy Different sampling factor gives different aliasing pattern Uses basic DSP sampling theory

6 Aliasing with Dense Spectrum 2D Signal 2D Spectrum

7 Aliasing with Dense Spectrum Everything gets aliased on top of eachsubsampled other Alias DFT Signals 2D Signal Subsampled 2D Signal 2D-DFT 2D Spectrum 2D Signal Su bs am pl e Subsampling Aliased 2D Spectrum Subsample Aliasing 2D-DFT sa b Su

8 Aliasing with Sparse Spectrum 2D Signal 2D Spectrum

9 Aliasing with Sparse Spectrum Most entries do not have aliasing! 2D Signal Subsampled 2D Signal Subsampling 2D Spectrum Aliased 2D Spectrum Aliasing

10 1-Sparse DFT

11 1-Sparse DFT Magnitude DFT Phase

12 1-Sparse DFT Magnitude DFT Phase Slope = location

13 1-Sparse DFT For noise-less, needs only 2 samples, Magnitude Constant computation time Phase

14 Different subsampling produces different aliasing Sparse spectrum

15 Different subsampling produces different aliasing Sparse spectrum Subsampling by 3 in signal domain

16 Different subsampling produces different aliasing Sparse spectrum Subsampling by 3 in signal domain Subsampling by 2 in signal domain

17 Different subsampling produces different aliasing Sparse spectrum Subsampling by 3 in signal domain Subsampling by 2 in signal domain Red and green are not aliased for different sampling factors

18 Combining three ideas Sparse spectrum

19 Combining three ideas Sparse spectrum Subsampling by 3 in signal domain Subsampling by 2 in signal domain

20 Combining three ideas Sparse spectrum Subsampling by 3 in signal domain Subsampling by 2 in signal domain Shift sampling pattern by 1 Shift sampling pattern by 1

21 Combining three ideas Sparse spectrum Can recover red and green locations via phase differences Subsampling by 3 in signal domain Subsampling by 2 in signal domain Shift sampling pattern by 1 Shift sampling pattern by 1

22 Combining three ideas Sparse spectrum Can peel off red and green Subsampling by 3 in signal domain Subsampling by 2 in signal domain Shift sampling pattern by 1 Shift sampling pattern by 1

23 Combining three ideas Sparse spectrum Can recover blue location via phase differences Subsampling by 3 in signal domain Subsampling by 2 in signal domain Shift sampling pattern by 1 Shift sampling pattern by 1

24 Combining three ideas Sparse spectrum Recovered all sparse entries Subsampling by 3 in signal domain Subsampling by 2 in signal domain Shift sampling pattern by 1 Shift sampling pattern by 1

25 FFAST Architecture Sampling factor: f i ~ k Number of stages: ~3 Computational complexity: O(k log k)

26 Sampling Factors How to pick sampling factors that gives different aliasing patterns? Does undersample by 2 and 4 work?

27 Sampling Factors How to pick sampling factors that gives different aliasing patterns? Does undersample by 2 and 4 work? No, undersampling by 4 does not give additional info. Chinese Remainder Theorem: Factors should be relatively co-prime For example, subsample by 5, 6, and 7

28 Theoretical Guarantee Proof techniques from coding theory The FFAST algorithm is fast: Uses <4 k samples O(k log k) computation complexity Trades off robustness to noise compared to compressed sensing For more detail:

29 The Art of Filter Design Frank Ong

30 Filter Design Given an ideal filter response profile How can you design to match?

31 Reduce to a linear problem Key observation: Filter response is linear in h[n]

32 Reduce to a linear problem Key observation: Filter response is linear in h[n]

33 Filter design as a linear program Discretize z into z i Find such that for all z i you care about

34 Filter design as a linear program Discretize z into z i Find such that for all z i you care about This is a linear program

35 Magnitude filter design Oftentimes, we don t care about the phase of our filter

36 Magnitude filter design Oftentimes, we don t care about the phase of our filter Given an ideal magnitude filter response profile How can you design to match?

37 Magnitude filter design Oftentimes, we don t care about the phase of our filter Given an ideal magnitude filter response profile How can you design to match? Is h[n] still linear in magnitude response?

38 Reduce to a quadratic problem Key observation: Magnitude filter response is quadratic in h[n]

39 Reduce to a quadratic problem Key observation: Magnitude filter response is quadratic in h[n]

40 Reduce to a quadratic problem Key observation: Magnitude filter response is quadratic in h[n]

41 Reduce to a quadratic problem Key observation: Magnitude filter response is quadratic in h[n]

42 Filter design as a rank-1 quadratic program Discretize z into z i Find such that for all z i you care about

43 Filter design as a rank-1 quadratic program Discretize z into z i Find such that for all z i you care about

44 Filter design as a rank-1 quadratic program Discretize z into z i Find such that for all z i you care about The rank-1 constraint is non-convex!

45 Filter design as a Semi-definite Program Discretize z into z i Find such that for all z i you care about Relax to a convex semi-definite constraint (lifting)! Hot topic in convex optimization

46 Filter design as a Semi-definite Program H is in general not unique Outputs the minimum phase

47 Example

48 Zero-phase filter using linear program

49 Minimum phase filter using SDP

50 Minimum phase filter using SDP

51 Github package Filter design package using SDP: Uses CVX