Uncertainty and Spectrogram Geometry

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1 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 Chassande-Mottin

2 From uncertainty to localization Spectrogram geometry from or to and Heisenberg refined Heisenberg (classical) 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 E x 2 ω(x) := 1 E x ω 2 X(ω) 2 dω 2π Theorem (Weyl, 27; Gabor, 46,... ) t (x) ω (X) 1 2, with lower bound attained for Gaussians x (t) = C e α t2, α < 0.

3 From uncertainty to localization Spectrogram geometry from or to and Heisenberg refined Heisenberg (-) Second-order (variance-type) measure for the joint - spreading of a signal x(t) L 2 (R) with energy distribution C x (t, ω; ϕ) within Cohen s class: 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 bound attained for Gaussian signals and matched Gaussian windows.

4 From uncertainty to localization Spectrogram geometry from or to and Heisenberg refined - covariance Definition (Cohen, 95) c(x) := t x(t) 2 d arg x(t) dt dt 1 Interpretation: covariance quantifies the coupling between and instantaneous c(x) = t ω x (t) 2 Intuition: covariance is zero in case of no coupling c(x) = t ω x (t) = t ω x (t) = t ω = 0

5 From uncertainty to localization Spectrogram geometry from or to and Heisenberg refined Schrödinger Theorem (Schrödinger, 30) t (x) ω (X) c 2 (x), with lower bound attained for Gaussians x (t) = e α t2 +β t+γ, Re{α} < 0. 1 Terminology: waveforms x attaining the lower bound correspond to squeezed states in quantum mechanics and linear chirps in signal theory 2 Interpretation: possibility of localization in the plane beyond pointwise energy concentration

6 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons chirp localization

7 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons how? 1 Effective localization: in the Wigner case, we have lim W x (t, ω) = δ (ω (β + 2 Im{α}t)) Re{α} 0 2 Generalization: localization on more arbitrary curves of the plane by modifying the symmetry rules underlying the Wigner distribution (Gonçalvès and F., 96) 3 Caveat: global localization holds for monocomponent signals only 4 Way out: reassignment (Kodera et al., 76)

8 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons rationale Starting with the smoothing relationship Sx h (t, ω) = W x (s, ξ) W h (s t, ξ ω) dt dω 2π, the key idea is 1 to replace the geometrical center of the smoothing - domain by the center of mass of the Wigner distribution over this domain, and 2 to reassign the value of the smoothed distribution to this local centroïd: ( ) Ŝx h (t, ω) = Sx h (τ, ξ)δ t ˆt x (τ, ξ), ω ˆω x (τ, ξ) dτ dξ 2π

9 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons spectrogram = smoothed Wigner Wigner-Ville spectrogram

10 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons spreading of auto-terms Wigner-Ville spectrogram

11 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons cancelling of cross-terms Wigner-Ville spectrogram

12 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons reassignment (Kodera et al., 76, Auger & F., 95) Wigner-Ville reassigned spectrogram

13 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons reassignment in action 1 Spectrogram: implicit computation of the local centroïds (Auger & F., 95) via two additional short- Fourier transforms (STFTs) with the companion windows (T h)(t) = t h(t) and (Dh)(t) = (dh/dt)(t) 2 Gaussian spectrogram: alternative computation of the local centroïds via partial derivatives of the magnitude of the STFT (Auger, Chassande-Mottin & F., 12) 3 Beyond spectrograms: possible generalizations to other smoothings (smoothed pseudo-wigner-ville, scalogram, etc.)

14 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons linear chirps Framework: Gaussian window h(t) = π 1/4 e t2 /2 Result: the linear chirp parameterized by α = 1 ( ) 1 2 T 2 i a ; β = 0 is such that lim Ŝx h (t, ω) = 1 δ(ω at) T 2π Interpretation: perfect localization

15 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions Framework: Gaussian window g(t) = π 1/4 e t2 /2 Hermite functions: with h k (t) = C k 1 1 T H k 1 ( 2πt/T )e π(t/t )2, H n (α) = ( 1) n e α2 dα n e α2, n = 0, 1 Radial reassignment: d n ˆω(t, ω) ˆt(t, ω) = ω t

16 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions Result (F., JFAA 12): Ŝ (g) h k (t, ω) = with ˆV δ and: 1 2 k 1 (k 1)! ˆV ) ( t 2 + ω 2 1 D (t, ω), D = {(t, ω) t 2 + ω 2 t 2 m + ω 2 m = 2(k 1)} Interpretation: clearance area and no perfect localization Asymptotic localization: Ŝ (g) h k (t, ω) k δ(t 2 + ω 2 2(k 1))

17 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions M = 2 Wigner spectro. reass. spectro.

18 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions order = 2 (theory) order = 2 (simulation)

19 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions M = 7 Wigner spectro. reass. spectro.

20 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions order = 7 (theory) order = 7 (simulation)

21 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions M = 18 Wigner spectro. reass. spectro.

22 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons Hermite functions order = 18 (theory) order = 18 (simulation)

23 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons logons Framework: Gaussian window h(t) = π 1/4 e t2 /2 Result: the Gabor logon, i.e., the previous linear chirp with a = 0, is such that S h x (t, ω) = e 1 2 (t2 +ω 2) tω (S h x ) = 2 W x (t, ω) = 2 e (t2 +ω 2) tω (W x ) = 1 Ŝ h x (t, ω) = 4 e 2(t2 +ω 2) tω (Ŝh x ) = 1 2 Interpretation: Heisenberg defeated?

24 From uncertainty to localization Spectrogram geometry sharp localization reassignment chirps, Hermite and logons resolving the paradox 1 General remark: as for Fourier, sharp localization not to be confused with resolution (i.e., ability to separate closely spaced components) 2 Complete picture: reassignment = 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 Sh x (t, ω) 3 Interpretation: basins of attraction 4 Open question: geometry of such basins and relationship with Heisenberg?

25 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model factorization Bargmann representation: with a circular Gaussian window, the STFT can be expressed as F h x (z) = F h x (z) e z 2 /4, where z = ω + jt and F h x (z) is an entire function of order at most 2 Weierstrass-Hadamard factorization: it follows that Fx h (z) = e ( ) Q(z) (1 z n ) exp z n + z n/2 2, n where Q(z) is a quadratic polynomial and z n = z/z n Interpretation: complete characterization by zeroes and, by duality, by local maxima

26 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model more on zeroes Result (Balasz et al., 11) Universal singularity of the phase derivative around zeroes of the STFT

27 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model phase-magnitude relationship In the Gaussian case, simple explanation based on: Proposition (Auger et al., 12) Phase and magnitude of a Gaussian STFT form a Cauchy pair, i.e., Φ h x (t, ω) = t ω log Mh x(t, ω) Φ h x ω (t, ω) = t log Mh x(t, ω), with F h x (t, ω) = M h x(t, ω) e jφh x (t,ω) e (t2 +ω 2 )/4

28 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model phase singularity 1 Magnitude: local behaviour in the vicinity of a zero (t 0, ω 0 ) M h x(t, ω) [(ω ω 0 ) 2 + (t t 0 ) 2] 1/2 2 Phase from magnitude: universal hyperbolic divergence of the phase derivative Φ h x (t 0, ω) t (ω ω 0 ) 1 ω ω0 Φ h x ω (t, ω 0) (t 0 t) 1 t t0

29 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model zeroes phase dislocations (Nye & Berry, 74) magnitude phase gradient in phase gradient in phase

30 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model phase dislocation ( domain)

31 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model phase dislocation (complex plane)

32 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model phase dislocation (zoom)

33 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model more on extrema Proposed approach 1 Data: white Gaussian noise 2 Time- representation: 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

34 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model an example spectrogram (wgn) spectrogram (logon) Voronoi/Delaunay (min) Voronoi/Delaunay (max)

35 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model mean arrangement Voronoi/Delaunay spectrogram (logon) 1 Observation: average connectivity 6 tiling with hexagonal cells 2 Interpretation: maximum packing of circular patches

36 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model predictions D M /d m = 3; N M /N m = 1/3; A M /A m = 3

37 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model simulation results distance : 5.24 number: 24 area : 17.6 maxima 0 10 distance : number: area : 6.3 minima

38 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model comparison with 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

39 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model analysis 1 Distributions: significant dispersion for lengths and areas 2 Ratios max/min: reasonable agreement between experimental results and theoretical predictions 3 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 4 Interpretation: tiling cells minimum uncertainty logons

40 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model local maxima and Voronoi cells areas (1) 1 Heights: well-described by a Gamma distribution 2 Areas: idem and similar to the heights distribution for a proper renormalization 2 (t 2 + ω 2 ) Fx h (t, ω) 2 dt dω ( 2π F R 2 (A) 2 Proposition A The value F of a local maximum of the STFT magnitude and the area A of the associated Voronoi cell satisfy the uncertainty-type inequality A. F 3 6 ) 2

41 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model local maxima and Voronoi cells areas (2) areas of Voronoi cells (red) and local maxima of STFT magnitude (blue) p.d.f o : data + : Gamma fit 0.02 areas of Voronoi cells joint p.d.f local maxima of STFT magnitude

42 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model towards a 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

43 From uncertainty to localization Spectrogram geometry on extrema Voronoi and Delaunay a simplified model concluding remarks 1 Spectrogram: geometry constrained by - uncertainty 2 Reassignment: access to simplified properties, both geometrical and statistical 3 Gabor case: complete characterization by zeroes or local maxima? 4 Modeling: new ways of sparsifying energy distributions?

On Time-Frequency Sparsity and Uncertainty

On Time-Frequency Sparsity and Uncertainty 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 Heisenberg (classical) from or to and Heisenberg refined