Signal Reconstruction from its Spectrogram Radu Balan IMAHA 2010, Northern Illinois University, April 24, 2010

2/23 Overview 1.Problem formulation 2.Reconstruction from absolute value of frame coefficients 3.Our approach –Embedding into the Hilbert-Schmidt space –Discrete Gabor multipliers –Quadratic reconstruction 4.Numerical example

3/23 1. Problem formulation Typical signal processing “pipeline”: Analysis Processing Synthesis In Out Features:  Relative low complexity O(Nlog(N))  On-line version if possible

4/23 x AnalysisSynthesis cy The Analysis/Synthesis Components: Example: Short-Time Fourier Transform

5/23 * = fft = * Data frame index (k) f c k,0 c k,F-1 c k+1,F-1 c k+1,0 g(t) x(t+kb:t+kb+M-1) x(t+kb+M:t+kb+2M-1) x(t+kb)g(t)x(t+(k+1)b)g(t)

6/23 * = ifft = * ĝ(t) c k,0 c k,F-1 c k+1,F-1 c k+1,0 +

7/23 Problem: Given the Short-Time Fourier Amplitudes (STFA): we want an efficient reconstruction algorithm:  Reduced computational complexity  On-line (“on-the-fly”) processing |.| Reconstruction c k,f d k,f x

8/23 Where is this problem important: –Speech enhancement –Speech separation –Old recording processing

9/23 Setup: –H=E n, where E=R or E=C –F={f 1,f 2,...,f m } a spanning set of m>n vectors Consider the map: Problem 1: When is N injective? Problem 2: Assume N is injective, Given c=N(x) construct a vector y equivalent to x (that is, invert N up to a constant phase factor) 2. Reconstruction from absolute value of frame coefficients

10/23 Theorem [R.B.,Casazza, Edidin, ACHA(2006)] For E = R : if m  2n-1, and a generic frame set F, then N is injective; if m  2n-2 then for any set F, N cannot be injective; N is injective iff for any subset J  F either J or F\J spans R n. if any n-element subset of F is linearly independent, then N is injective; for m=2n-1 this is a necessary and sufficient condition.

11/23 Theorem [R.B.,Casazza, Edidin, ACHA(2006)] For E = C : if m  4n-2, and a generic frame set F, then N is injective. if m  2n and a generic frame set F, then the set of points in C n where N fails to be injective is thin (its complement has dense interior).

12/23 3. Our approach First observation: Hilbert-Schmidt Signal space: l 2 (Z) x KxKx K nonlinear embedding K g k,f E=span{K g k,f } Hilbert-Schmidt: HS(l 2 (Z)) Recall:

13/23 Assume {K g k,f } form a frame for its span, E. Then the projection P E can be written as: where {Q k,f } is the canonical dual of {K g k,f }. Frame operator

14/23 Second observation: since: it follows:

15/23 However: Explicitely:

16/23 Short digression: Gabor Multipliers Goes back to Weyl, Klauder, Daubechies More recently: Feichtinger (2000), Benedetto- Pfander (2006), Dörfler-Toressani (2008) Theorem [F’00] Assume {g,  Lattice} is a frame for L 2 (R). Then the following are equivalent: 1.{ g,  Lattice} is a frame for its span, in HS(L 2 (R)); 2.{ g,  Lattice} is a Riesz basis for its span, in HS(L2(R)); 3.The function H does not vanish,

17/23 Return to our setting. Let Theorem Assume {g k,f } (k,f)  ZxZ F is a frame for l 2 (Z). Then 1.is a frame for its span in HS(l 2 (Z)) iff for each m  Z F, H( ,m) either vanishes identically in , or it is never zero; 2.is a Riesz basis for its span in HS(l 2 (Z)) iff for each m  Z F and , H( ,m) is never zero.

18/23 Third observation. Under the following settings: –For translation step b=1; –For window support supp(g)={0,1,2,...,L-1} –For F  2L The span of is the set of 2L-1 diagonal band matrices. 

19/23 The reproducing condition (i.e. of the projection onto E) implies that Q must satisfy: By working out this condition we obtain :

20/23 The fourth observation: We are able now to reconstruct up to L-1 diagonals of K x. This means we can estimate Assuming we already estimated x s for s<t, we estimate x t by a minimization problem: for some J  L-1 and weights w 1,...,w J. Remark: This algorithm is similar to Nawab, Quatieri, Lim [’83] IEEE paper.

21/23 Reconstruction Scheme Putting all blocks together we get: IFFTIFFT |c k,0 | 2 |c k,F-1 | 2  W0W0 W L-1   Least Square Solver Stage 1 Stage 2

22/23 3. Numerical Example

23/23 Conclusions All is well but... For nice analysis windows (Hamming, Hanning, gaussian) the set {K g k,f } DOES NOT form a frame for its span! The lower frame bound is 0. This is the (main) reason for the observed numerical instability! Solution: Regularization.