On Time-Frequency Sparsity and Uncertainty

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Pearl Norton
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1 CNRS & E cole Normale Supe rieure de Lyon, France * partially based on joint works with Franc ois Auger, Pierre Borgnat and E ric Chassande-Mottin

2 Heisenberg (classical) from or to and Heisenberg refined Moments Second-order (variance-type) measures for the individual and spreadings of a signal x(t) L 2 (R) with spectrum X (ω): 2 t (x) := 1 t 2 x(t) 2 dt ; 2 E ω(x ) := 1 ω 2 X (ω) 2 dω x E x 2π Theorem (Weyl, 27; Gabor, 46,... ) t (x) ω (X ) 1 2, with lower bound attained for Gabor logons, i.e., Gaussian waveforms x (t) = C e α t2, α < 0

3 Heisenberg (-) from or to and Heisenberg refined Joint moment Second-order (variance-type) measure for the - spreading of a signal x(t) with energy distribution C x (t, ω; ϕ): tω (C x ) := 1 E x ( t 2 ) T 2 + T 2 ω 2 C x (t, ω; ϕ) dt dω 2π Theorem (Janssen, 91) Wigner tω (W x ) 1 Spectrogram tω (S h x ) 2, with lower bounds attained for logons and matched Gaussian windows, i.e., h(t) = x(t) = x (t)

4 - covariance from or to and Heisenberg refined Definition (Cohen, 95) c(x) := t x(t) 2 d arg x(t) dt dt Interpretation Covariance quantifies the coupling between and instantaneous c(x) = t ω x (t) Intuition Covariance is zero in case of no coupling c(x) = t ω x (t) = t ω x (t) = t ω = 0

5 Schrödinger from or to and Heisenberg refined Theorem (Schrödinger, 30) t (x) ω (X ) c 2 2 (x), with lower bound attained for Gaussian waveforms of the form x (t) = e α t2 +β t+γ, with Re{α} < 0. Terminology Waveforms x attaining the lower bound correspond to squeezed states in quantum mechanics and linear chirps in signal theory Interpretation Possibility of localization in the plane beyond pointwise energy concentration

6 from logons to chirps

7 effective localization Wigner and beyond lim W x (t, ω) = δ (ω (β + 2 Im{α}t)) Re{α} 0 1 Generalization: localization on more arbitrary curves of the plane by modifying the symmetry rules underlying the Wigner distribution (Gonçalvès and F., 96) 2 Caveat: global localization holds for monocomponent signals only Ways out constrained EMD (Pustelnik et al., 12 + poster)

8 rationale Fourier Duality between distribution and correlation: C x (t, ω; ϕ) = F ξ t F τ ω {ϕ(ξ, τ) A x (ξ, τ)}, with A x (ξ, τ) = F t ξ F ω τ {W x (t, ω)} a TF correlation function ( Ambiguity Function ) Consequences 1 signal terms located around the origin of the AF plane 2 cross-terms located outside 3 low-pass filtering (e.g., A h (ξ, τ) for the spectrogram) trade-off between localization and cross-terms reduction

9 example Wigner!Ville Distribution Ambiguity Function Doppler Data!adaptive TFR delay RGK mask Doppler delay

10 perspective Discrete Signal of dimension N TFD of dimension N 2 when computed over N bins Few AM-FM components K N at most KN N 2 non-zero values in the TF plane Sparsity Minimizing l 0 -norm not feasible, but near-optimal solution by minimizing l 1 -norm (in the spirit of basis pursuit or compressed sensing )

11 sparse solution From l 2 to l 1 (F. & Borgnat, 08) 1 Select a domain Ω neighbouring the origin of the AF plane 2 Find the sparsest - distribution ρ(t, ω) by solving the program min ρ 1 s.t. F{ρ} A x = 0 ρ (ξ,τ) Ω 3 The exact equality over Ω can be relaxed according to min ρ 1 s.t. F{ρ} A x ρ 2 ɛ (ξ,τ) Ω

12 toy example 2-component AM-FM signal, 128 data points

13 Wigner-Ville

14 ambiguity function

15 selection

16 sparse solution

17 domain selection '($)*!+,-. /0/1211"34 '($)*!+,-. /0/ '($)*!+,-. /0/121"14 '($)*!+,-. /0/12"78 '($)*!+,-. /0/12697! "!#$$!5$('!#! "!#$$!%& '($)*!+,-. /0/1211"34 '($)*!+,-. /0/ '($)*!+,-. /0/121"14 '($)*!+,-. /0/12"78 '($)*!+,-. /0/12697

18 domain selection Interpretation Result Ω too small not enough auto-terms TFD not enough sharp : penalty with entropy (H) Ω too large inclusion of cross-terms TFD sharp but discontinuous: penalty with total variation (TV ) Ω = arg min(h + TV ) Ω Heisenberg cell, i.e., minimum quantity of information necessary for coding (in both magnitude and phase) auto-terms

19 rationale Smoothing relationship C x (t, ω; ϕ) = W x (s, ξ) Φ(s t, ξ ω) dt dω 2π Key idea (Kodera et al., 76, Auger & F., 95) 1 replace the geometrical center of the smoothing TF domain (defined by Φ(t, ω)) by the center of mass of the Wigner distribution over this domain 2 reassign the value of the smoothed distribution to this local center of mass: Ĉ x (t, ω; ϕ) = C x (τ, ξ; ϕ)δ ( t ˆt x (τ, ξ), ω ˆω x (τ, ξ) ) dτ dξ 2π

20 spectrogram = smoothed Wigner Wigner-Ville spectrogram

21 spreading of auto-terms Wigner-Ville spectrogram

22 cancelling of cross-terms Wigner-Ville spectrogram

23 Wigner-Ville reassigned spectrogram

24 and localization Two examples For a Gaussian window h(t) = π 1/4 e t2 /2, reassigned spectrograms are asymptotically perfectly localized for 1 linear chirps c T (t) of infinite duration: lim Ŝc h T T (t, ω) = δ(ω at) 2 Hermite functions h M (t) of infinite order (F., JFAA 12): Ŝ h h M (t, ω) M δ(t 2 + ω 2 2(M 1))

25 Hermite functions M = 2 Wigner spectro. reass. spectro.

26 Hermite functions 1 order = 2 (theory) order = 2 (simulation)

27 Hermite functions M = 7 Wigner spectro. reass. spectro.

28 Hermite functions 1 order = 7 (theory) order = 7 (simulation)

29 Hermite functions M = 18 Wigner spectro. reass. spectro.

30 Hermite functions 1 order = 18 (theory) order = 18 (simulation)

31 and uncertainty Result The minimum uncertainty Gabor logon h(t) = π 1/4 e t2 /2 is such that W h (t, ω) = 2 e (t2 +ω 2) tω (W h ) = 1 S h h (t, ω) = e 1 2 (t2 +ω 2) tω (S h h ) = 2 Ŝ h h (t, ω) = 4 e 2(t2 +ω 2) tω (Ŝ h h ) = 1 2 Interpretation 1 2 < 1 Heisenberg defeated?

32 and uncertainty Resolving the logons paradox 1 General remark: as for Fourier, not to be confused with resolution (i.e., ability to separate closely spaced components) 2 Complete picture: = squeezed distribution + vector field r x (t, ω) = (ˆt x (t, ω) t, ˆω x (t, ω) ω) t such that r x (t, ω) = 1 2 log S h x (t, ω) 3 Interpretation: basins of attraction 4 Open question: geometry of such basins and relationship with Heisenberg?

33 factorization on extrema Voronoi and Delaunay a simplified model Bargmann With a circular Gaussian window, the STFT can be expressed as F h x (z) = F h x (z) e z 2 /4, with z = ω + jt and F h x (z) an entire function of order at most 2 Weierstrass-Hadamard F h x (z) = e Q(z) n (1 z n ) exp ( z n + z 2 n/2 ), where Q(z) is a quadratic polynomial and z n = z/z n Interpretation Characterization by zeroes and, by duality, by local maxima

34 more on extrema on extrema Voronoi and Delaunay a simplified model Motivation Better understanding of reassigned distributions: 1 Simplified modeling 2 Ultimate localization properties Proposed approach 1 Data: white Gaussian noise 2 Time- distribution: Gaussian spectrogram 3 Characterization: identification of local extrema (zeroes and maxima) + Voronoi tessellation and Delaunay triangulation 4 Analysis: distribution of cell areas, lengths between extrema and heights of local maxima

35 an example on extrema Voronoi and Delaunay a simplified model spectrogram (wgn) spectrogram (logon) Voronoi/Delaunay (min) Voronoi/Delaunay (max)

36 - patches on extrema Voronoi and Delaunay a simplified model Mean arrangement 1 Average connectivity 6 tiling with hexagonal cells Voronoi/Delaunay spectrogram (logon) 2 Maximum packing of circular patches

37 predictions on extrema Voronoi and Delaunay a simplified model Hexagonal tiling geometry D M /d m = 3; N M /N m = 1/3; A M /A m = 3

38 simulation results on extrema Voronoi and Delaunay a simplified model distance : 5.24 number: 24 area : 17.6 maxima distance : 3.17 number: 70 area : 6.3 minima

39 comparison with model on extrema Voronoi and Delaunay a simplified model (d M /d m )/sqrt(3) (A m /A M )/ mode theory median mean number of bins 0.5 mode theory median mean number of bins 1.5 (A M /d M 2 )/(sqrt(3)/2) 1.5 C/ mode theory median mean number of bins 0.5 mode theory median mean number of bins

40 analysis on extrema Voronoi and Delaunay a simplified model Distributions of lengths and areas 1 Ratios max/min: dispersion but reasonable agreement (in the mean) between experimental results and theoretical predictions 2 Ranges of values: if we call effective domain of the minimum uncertainty logon the circular domain which encompasses 95% of its energy, its radius and area are equal to 2.6 and 21.8, to be compared to the values d M / 3 3 and 2π/(3 3)A M 21.8 attached to the hexagonal tiling Interpretation Tiling cells minimum uncertainty logons

41 on extrema Voronoi and Delaunay a simplified model local maxima and Voronoi cells areas (1) Distributions 1 Heights: well-described by a Gamma distribution 2 Areas: idem and similar to the heights distribution for a proper renormalization areas of Voronoi cells (red) and local maxima of STFT magnitude (blue) p.d.f o : data + : Gamma fit joint p.d.f. ells

42 on extrema Voronoi and Delaunay a simplified model local maxima and Voronoi cells areas (2) Proposition p.d.f areas of Voronoi cells (red) and local maxima of STFT magnitude (blue) o : data + : Gamma fit 0.04 The value F of a local maximum of the STFT magnitude and 0.02 the area A of the associated Voronoi cell satisfy the uncertainty-type inequality A. F areas of Voronoi cells joint p.d.f local maxima of STFT magnitude Sketch of proof: 2 ( ) A (t2 + ω 2 ) Fx h (t, ω) 2 dt dω 2π F R 2 2 (A) 2

43 beyond the mean model on extrema Voronoi and Delaunay a simplified model Random Gabor expansion Previous results suggest a possible modeling of Gabor spectrograms as with Sx h (t, ω) = c mn Fh h (t t m, ω ω n ) m n 1 locations (t m, ω n ) distributed on some suitable randomized version of a triangular grid 2 magnitudes of the weights c mn Gamma-distributed 3 locations and weights partly correlated 2

44 Concluding remarks 1 Time- energy distributions Sparse representations Ultimate localization constrained by uncertainty 2 New insights on Complete characterization by local maxima in the Gabor case? 3 Analysis/synthesis Modeling sparse - distributions? New approaches to data driven representations?

45 some references Available at perso.ens-lyon.fr/patrick.flandrin/publis.html and/or upon request at with some Matlab codes at P. Flandrin, 2012: A note on reassigned Gabor spectrograms of Hermite functions, J. Fourier Anal. Appl. doi: /s P. Flandrin, E. Chassande-Mottin, F. Auger, 2012: Uncertainty and spectrogram geometry, in Proc. 20th European Signal Processing Conf. EUSIPCO-12, Bucharest (RO). P. Flandrin, P. Borgnat, 2010: Time- energy distributions meet compressed sensing, IEEE Trans. on Signal Proc. Vol. 58, No. 6, pp doi: /TSP P. Borgnat, P. Flandrin, 2008: Time- localization from constraints, in Proc. IEEE Int. Conf. on Acoust., Speech and Signal Proc. ICASSP-08, pp , Las Vegas (NV). J. Xiao, P. Flandrin, 2007 : Multitaper - for nonstationary spectrum estimation and chirp enhancement, IEEE Trans. on Signal Proc., Vol. 55, No. 6 (Part 2), pp P. Flandrin, F. Auger, E. Chassande-Mottin, 2003: Time- From principles to algorithms, in : Applications in Time-Frequency Signal Processing (A. Papandreou-Suppappola, ed.), Chap. 5, pp , CRC Press. P. Flandrin, P. Gonçalvès, 1996: Geometry of affine - distributions, Appl. Comp. Harm. Anal., Vol. 3, pp

Uncertainty and Spectrogram Geometry

Uncertainty and Spectrogram Geometry From uncertainty...... to localization Spectrogram geometry CNRS & École Normale Supérieure de Lyon, France Erwin Schrödinger Institute, December 2012 * based on joint work with François Auger and Éric