Traitement de signal Analyse spectrale : Fourier, temps...

Traitement de signalAnalyse spectrale : Fourier,

temps-frequence, temps-echelle

Laurent Duval

IFP Energies nouvelles, Rueil-Malmaison, [email protected]

September 23, 2015

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Motivation

• Standard framework• diversity and quantity of available data

• wide audience audio, image and video (main drive);exponentially increasing digital measures from different sensors(industrial, mobile devices)

• seismic data (petabytes)

• efficiency of standard (multi-dimensional) signal processingtechniques

• data acquisition and processing generally performed viastandard approaches (Nyquist-Shannon sampling, Fourieranalysis, parametric modeling, etc.), astonishingly efficient formost applications if properly used

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Motivation

• Present challenges• ”Infobesity” : managing huge data volumes (information

overload amplified by explosion in precision and sampling ofdigital data ”made” cheaper by sensors and storage)

• Complexity: achieve higher level tasks; more decision (less userdependent), more imitation; more and more complex,integrated environments, higher constraints

• Approaches• compression or sparsification aimed at high-fidelity rendering• attribute extraction aimed detection, diagnosis, classification,

inference

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Motivation

Figure 1: Example of a simple image ...

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Motivation

Figure 2: Example of a simple image, yet...

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Motivation

Figure 3: Example of an simple image, yet complex

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Motivation

• Doxa, common belief: more data more understanding

• Pitfall: less time to analyze

• Recent finding: from the observer point of view,representative or relevant information form a very tiny subsetof the set of n-sample, b-bit coded, m-dimensional data

• Thus, data could be faithfully represented (and processed) bya sparse combination of a few relevant coefficients. But whichone? And how?

• May (even) depend on data themselves

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Motivation

• Rediscovery: Ockham’s principle of parsimony, a.k.a Occam’srazor, law of parsimony, law of economy or law of succinctness

• William of Ockham (c. 1285-1349) : Numquam ponenda estpluralitas sine necessitate (Plurality must never be positedwithout necessity)

• Modern short version: the simplest explanation/theory is mostlikely the correct one

• Refutable principle, indirect, highly heuristic

• As every principle, should be use with parsimony

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Motivation

• Program: provide a panel of methods and tools aimed at moreefficient, sparser data representations

• Examples: time-scale and time-frequency (pyramids, wavelets,spectrograms, filter banks, lapped transforms), atomicrepresentation (bases, trames and dictionaries), higherdimension generalizations (geometric multiscale transforms)

• Not addressed: Fourier/Radon techniques extensions, modeling(AR, ARMA, ARMAX, Burg, Capon, Prony), PrincipalComponent Analysis (PCA), Independent Component Analysis(or Latent Variable Analysis), clustering, K-means, neuralnetworks

• Mathematical background (sic!)• linear algebra matrix analysis• selection, shrinkage & thresholding, non-linear filtering, robust

statistics• operator theory, approximation theory (Lp norms and spaces,

Sobolev, Besov)

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Examples of “stationary” signals and images

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Reminders on analog signals

• Integrable signals

s ∈ L1(R)←→∫ ∞−∞|s(t)|dt <∞

• Finite energy signals

s ∈ L2(R)←→ ||s||2 =

∫ ∞−∞|s(t)|2dt <∞

• Scalar product

〈s1, s2〉 =

∫ ∞−∞

s1(t)s∗2(t)dt

• Remark

L1(R) 6⊂ L2(R) and L2(R) 6⊂ L1(R)

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Reminders - Counter-example 1

• Signal in L2(R) but not in L1(R)

x =

{1 if |t| < 11|t| otherwise.

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Reminders - Counter-example 1

• Signal L1(R) but not in L2(R)

x =

{1√t

if |t| < 1

0 otherwise.

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History

A few names in the history:

• Joseph Fourier: 1807 (heat equation) [1768-1830]

• Gauß: 1805 (interpolation of orbits of celestial bodies)

• Runge: 1903

• Danielson-Lanczos: 1942

• Good: 1958

• Cooley-Tuckey: 1965 (FFT)

• Frigo-Johnson: 1998 (Fastest Fourier Transform in the West)

Source: Gauß and the history of the Fourier transform. Heideman,Johnson, Burrus, IEEE ASSP Magazine, 1984

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History

Lettre de C. Jacobi a A. Legendre, 1830

M. Fourier avait l’opinion que le but principal desmathematiques etait l’utilite publique et l’explication desphenomenes naturels; mais un philosophe comme luiaurait du savoir que le but unique de la science, c’estl’honneur de l’esprit humain, et que sous ce titre, unequestion de nombres vaut autant qu’une question dusysteme du monde.

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Continuous Fourier transform

• Definition

s(t)FT−→ S(f) =

∫ ∞−∞

s(t)e−ı2πftdt

function of the (dual) variable frequency f ∈ R• Existence 1: if s ∈ L1(R), then

• S(f) is continuous and bounded• lim|f |→∞ S(f) = 0

• Existence 2:s ∈ L2(R), iff S ∈ L2(R)

• Inversion:

s(t) =

∫ ∞−∞

S(f)eı2πftdf

for almost every t

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Fourier transform - properties 1

• Linearity

s1(t)FT−→ S1(f), s2(t)

FT−→ S2(f)

then

∀(λ, µ) ∈ C2 λs1(t) + µs2(t)FT−→ λS1(f) + µS2(f)

• Delay/translation

∀b ∈ R, s(t− b) FT−→ e−ı2πbfS(f)

delay/translation invariance in amplitude spectrum

• Modulation:

∀ν ∈ R, eı2πνts(t)FT−→ S(f − ν)

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• Scale change

∀α ∈ R∗, s(αt)FT−→ 1

|α|S(f

α)

turns dilatation onto contraction

• Time inverse: a special case

s(−t) FT−→ S(−f)

• Conjugation

s∗(t)FT−→ S∗(−f)

• Hermitian symmetry: if s is real, then:• Re(S(f)) even, Im(S(f)) odd• |S(f)| even, argS(f) odd (mod 2π)

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• Convolution

(s1 ∗ s2)(t) =

∫ ∞−∞

s1(u)s2(t− u)du = (s2 ∗ s1)(t)

• Conditions• s1 ∈ L1(R), s2 ∈ L1(R)⇒ s1 ∗ s2 ∈ L1(R)• s1 ∈ L2(R), s2 ∈ L2(R)⇒ s1 ∗ s2 ∈ L2(R)

(s1 ∗ s2)(t)FT−→ S1(f)S2(f)

• Parseval-Plancherel equalities: if s1, s2 ∈ L1(R):• 〈s1, s2〉 = 〈S1, S2〉• ‖s‖2 = ‖S‖2

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Fourier transform - examples

• s(t) = e−atu(t),<(a) > 0:S(f) = 1

a+2ıπf

• s(t) = e−at cos(2πf0t)u(t):

S(f) = 12

[1

a+2πı(f−f0) + 1a+2πı(f+f0)

]• s(t) = e−a|t|:

S(f) = 2|a|a2+(2πf)2

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Digital signals - sampling

• For a signal s(t), regularly sampled:

s[k] = s(kT )

with T : sampling period; fs = 1/T : sampling frequency

• Sampling theorem: if S(f) = 0 for |f | ≥ B and fs ≥ 2B, thethe signal is sampled without information loss (theoretically),with Shannon-Nyquist formula:

s(t) =

∞∑k=−∞

s[k]sinc

(π(t− kT )

T

), sinc(t) =

{sin(t)t if t 6= 01 otherwise.

• cardinal theorem of interpolation theory: (Cauchy)-Whittaker,1915; Nyquist, 1928; Kotelnikov, 1933; Whittaker, 1935;Raabe, 1938; Gabor, 1946; Shannon, 1948; Someya, 1949

• Note: Balian-Low theorem for time-frequency analysis

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Figure 4: Example of aliasing

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• Comments• sufficient but non-necessary condition• two-part theorem: (1) sampling, (2) reconstruction• caution: jitter, amplitude quantization, noise• slow convergence of the sinc, instability• signals cannot be time-limited and frequency-bounded• extensions to band-limited signals exist (iterative

Papoulis-Gerchberg); optimally 2(fu − fl). Aliases in ± freq.content do not overlap for integer k:

2fuk≤ fs ≤

2flk − 1

• alternatives: non-regularly sampled data (Lomb-Scargleperiodogram); sparse/finite-rate-of-innovation signals (no morethan n events per unit of time): compressive sensing, sparsesampling

• Beware of weak analogies between continuous/discrete

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Digital signals

• Absolutely convergent sequences

(s[k])k∈Z ∈ l1(R)⇔∞∑

k=−∞|s[k]| <∞

• Square-summable sequences

(s[k])k∈Z ∈ l2(R)⇔∞∑

k=−∞|s[k]|2 <∞

• Remarkl1(R) ⊂ l2(R)

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Discrete-Time Fourier transform (DTFT)

• Definition 1: z-transform

(s[k])k∈ZZT−→ S(z) =

∞∑k=−∞

s[k]z−k

• Definition 2: Discrete-Time Fourier transform

(s[k])k∈ZDTFT−→ S(f) =

∞∑k=−∞

s[k]z−k, z = eı2πf

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Discrete-Time Fourier transform (DTFT)

• Normalized frequency

f = Tfphys

• Periodicityf = 1 : S(f + 1) = S(f)

• Existence(s[k])k∈Z ∈ l2(R)

• Special case: if (s[k])k∈Z ∈ l1(R), then S(f) is continuousand bounded

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DTFT - properties 1

• Linearity

λs1[k] + µs2[k]DTFT−→ λS1(f) + µS2(f)

• Integer delay/translation

s[k − n]DTFT−→ e−ı2πnfS(f)

• Modulationeı2πνs[k]

DTFT−→ S(f − ν)

• Time inversions[−k]

DTFT−→ S(−f)

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DTFT - properties 2

• Conjugaison

s∗[k]DTFT−→ S∗(−f)

• Parseval-Plancherel equalities

∞∑k=−∞

s1[k]s∗2[k] =

∫ 1/2

−1/2S1(f)S∗2(f)df

∞∑k=−∞

|s[k]|2 =

∫ 1/2

−1/2|S(f)|2df

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DTFT - properties 3

• Convolution

(s1 ∗ s2)[k] =

∞∑l=−∞

s1[l]s2[k − l]DTFT−→ S1(f)S2(f)

• Inversion

s[k] =

∫ 1/2

−1/2S(f)eı2πfkdf

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Fourier series

• If s is periodic (and continuous), s(t+ 2π) = s(t) define:

a0 =1

π

∫ −π−π

s(t)

ak =1

π

∫ −π−π

s(t) cos(kt)dt

bk =1

π

∫ −π−π

s(t) sin(kt)dt

then the infinite Fourier series is:

s(t) ' a02

+

∞∑k=1

[ak cos(kt) + bk sin(kt)]

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Discrete Fourier transform

• Definition

(s[k])0≤k≤K−1DFTK−→ s[p]0≤p≤K−1

with

s[p]0≤p≤K−1 =

K−1∑k=0

s[k]e−ı2πkpK

• Link to the DTFT : if s[k] = 0 pour k < 0 et k ≥ K, then

s[p] = S( pK

)i.e. K-sample sampling of DTFT S(f) on [0, 1]

• Inversion

s[k] =1

K

K−1∑p=0

s[p]eı2πkpK

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All Fourier transforms unite: Pontryagin duality

Transform Original domain Transform domain

Fourier transform R RFourier series T ZDiscrete-time Fourier transform (DTFT) Z TDiscrete Fourier transform (DFT) Z/nZ Z/nZ

Figure 5: Signals in the different domains

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Fast Fourier transform - FFT

• Fast algorithms exist (since Gauss)

FFTK ⇒ complexity of O(K log2(K)) operations

• Cyclic or periodic convolution: Let (s1[k])0≤k≤K−1 and(s2[k])0≤k≤K−1

(s1 ∗©s2)[k]DFTK−→ s1[p] · s2[p]

where (s1 ∗©s2)[k] represents the K-point convolution of theperiodized sequences:

(s1 ∗©s2)[k] =

k∑l=0

s1[l]s2[k − l] +

K−1∑l=k+1

s1[l]s2[K + k − l]

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Reminders - Key message

• Nature of the data and the transform• Continuous and discrete natures ARE different• Generally stuff works• Intuition may be misleading (ex.: FFT on 8-sample signals,

non proper windows)• Sometimes special care is needed: re-interpolation,

pre-processing to avoid edge effects, instabilities, outliers

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Windows• Several uses

• apodization, tapering (edges, jumps)• ”stationarizing”• spectral estimation, filter design

Figure 6: Windows in the original domain

• Many (parametric) designs:• Rect., Bartlett, Hann, Hamming, Kayser, Chebychev,

Blackman-Harris,...• Gen. cosine: a0 − a1 cos( 2πn

M−1 ) + a2 cos( 4πnM−1 )− a3 cos( 6πn

M−1 )

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Windows• Several uses

• apodization, tapering (edges, jumps)• ”stationarizing”• spectral estimation, filter design

Figure 6: Windows in the frequency domain

• Many (parametric) designs:• Rect., Bartlett, Hann, Hamming, Kayser, Chebychev,

Blackman-Harris,...• Gen. cosine: a0 − a1 cos( 2πn

M−1 ) + a2 cos( 4πnM−1 )− a3 cos( 6πn

M−1 )

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Side dishes - Hilbert transform

• Linear operator (Cauchy kernel)

• Important tool in signal processing, mechanics, waves

Hf(t) =1

πp.v.

∫ +∞

−∞

f(s)

t− sds

F(Hf(t))(ω) = (−ısign(ω)).F(f)(ω)

• Analytic signal: f(t) = f(t) + ıHf(t)

• Examples of Hilbert pairs:• cos(t)→ sin(t) (eıt) (cisoid)• 1

1+t2 →t

1+t2

• Useful for envelope extraction, inst. freq. estimation,time-frequency processing

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Side dishes - Hilbert transform and pairsReminders [Gabor-1946][Ville-1948]

H{f}(ω) = −ısign(ω)f(ω)

Figure 7: Hilbert pair 1

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Reminders: set averages

• s(n): discrete time random process (stationary stochasticprocess)

• expectation:µs(n) = E{s(n)}

• variance:σ2s(n) = E{|s(n)− µs(n)|2}

• autocovariance:

cs(k, l) = E{(s(k)− µs(k))(s(l)− µs(l))∗}

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Reminders: power spectral densityFor an autocorrelation ergodic process:

limN→∞

1

2N + 1

N∑n=−N

s(n+ k)s(n) = rss(k)

• if s(n) is known for every n, power spectrum estimation• caveat 1: samples are not unlimited [0, . . . , N − 1], sometimes

small• caveat 2: corruption (noise, interfering signals)

Recast the problem: from the biased estimator of the ACF

rss(k) =

N−1−k∑n=0

s(n+ k)s(n)

estimate power spectrum (periodogram)

Px(eıω) =

N−1∑k=−N+1

rss(k)eıkω

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Time and frequency resolution

• Energy

E =

∫|s(t)|2dt =

∫|S(f)|2df

• Time or frequency location

t = 1/E

∫t|s(t)|2dt f = 1/E

∫f |S(f)|2df

• Energy dispersion

∆t =

√1/E

∫(t− t)2|s(t)|2dt

∆f =

√1/E

∫(f − f)2|S(f)|2df

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Heisenberg-Gabor inequality

• Theorem (Weyl, 1931)If s(t), ts(t), s′(t) ∈ L2 then

‖s(t)‖2 ≤ 2‖ts(t)‖‖s′(t)‖

• Equality:Iff s(t) is a modulated Gaussian/Gabor elementary function:

s′(t)/s(t) ∝ t

s(t) = C exp[−α(t− tm)2 + ı2πνm(t− tm)]

• ProofIntegration by part + Cauchy-Schwarz

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Uncertainty principle (UP)

• Time-frequency UPFor finite-energy every signal s(t), with ∆t and ∆f finite:

∆t∆f ≥ 1

4π

with equality for the modulated Gaussian only

• Principles‖s′(t)‖2 = |ı2π|2‖fS(f)‖2

• Observations• Fourier (continuous) fundamental limit: arbitrary ”location”

cannot be attained both in time and frequency• have to choose between time and frequency locations• Gaussians are ”the best”

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Uncertainty principle (UP) for project management

Applies to other domains

Figure 8: Dilbert

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Uncertainty principle - time

• One may write

s(t) =

∫s(u)δ(t− u)du

• δ(t) is neutral w.r.t. convolution• interpreted as a decomposition of s(t) onto a basis of shiftedδ(t− u): ∆t = 0 at

• FT of basis functions: e−ı2πft: ∆f =∞

UP: as a limit of 0×∞

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Uncertainty principle - frequency

• One may write

s(t) =

∫S(f)eı2πftdf

• interpreted as a decomposition on pure waves eı2πft: ∆t =∞• FT of basis functions: δ(f − t) : ∆f = 0

UP: as a limit of ∞× 0

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Uncertainty principle - illustration

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Basis formalism interpretation

• Orthonormality

〈eı2πft, eı2πgt〉 = δ(f − g)

〈δ(f), δ(g)〉 = δ(f − g)

• Scalar productS(f) = 〈s(t), eı2πft〉

s(t) = 〈s(t), δ(f)〉• Matching of a signal with a vector, a basis function (pure

wave, Dirac)• Synthesis: continuous sum of orthogonal projection onto basis

functions• Relative interest of the two bases? Other bases?

(Walsh-Hadamard, DCT, eigenbase)• How to cope with mixed resolution?

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Sliding window Fourier transform

• PrinciplesFourier analysis on time-space slices of the continuous s(t)with a sliding window h(t− τ)

• Short-term/short-time Fourier transform (STFT)

Ss(τ, f ;h) =

∫s(t)h∗(t− τ)e−ı2πftdt

• Wider domain of applications than FT• depend on h• FT as a peculiar instance (valid for other transforms: not a

new tool, only a more versatile ”leatherman”-like multi-tool)

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Sliding window Fourier transform

Figure 9: Leatherman wave black

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Sliding window Fourier transform - illustration

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Sliding window Fourier transform - time-freq. completude

Notion of a ”complete” descrition (i.e. somehow invertible)

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Sliding window Fourier transform - windows

• Related to frequency analysisDepend on the window choice h (shape, length)

• Continuous time windows• rectangular: poor frequency resolution• gaussian: best time-frequency trade-off? (Gabor, 1946)

• Discrete time windows• τ discretized (jumps vs. redundancy)• different criteria: side lobes, equiripple, apodizing; Bartlett,

Hamming, Hann, Blackmann-Harris, Blackmann-Nutall,Kaiser, Chebychev, Bessel, Generalized raised cosine, Lanczos,Flat-top,...

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Sliding window Fourier transform - paving

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Sliding window Fourier transform - reconstruction

• Simple analogy• synthesis: what two numbers add to result 3• a+ b = 3: infinite number of solutions, e.g.

2945.75 + (−2942.75) but irrelevant• assume they are integers?• assume they are positive? 1 + 2 = 3 or 2 + 1 = 3• aim: increase interpretability, information compaction

(0 + 3 = 3), reduce overshoot

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Sliding window Fourier transform - reconstruction

• Redundant transformation!

• Inversion

s(t) =

∫ +∞

−∞

∫ +∞

−∞Ss(u, ξ;h)g(t− u)eı2πtududξ,

provided that ∫ +∞

−∞g(t)h∗(t)dt = 1.

(perfect reconstruction, no information loss)• special case: admissible normalized window h

g(t) = h(t)

but not the only solution (truncated sin)• a bit more involved for discrete time, less if only approximate

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Sliding window Fourier transform - spectrogram

• Definition|Ss(τ, f ;h)|2

• The spectrogram is a (bilinear) time-frequency distribution

E =

∫∫|Ss(τ, f ;h)|2dτdf

for normalized admissible window h

• Parseval formula

〈s1, s2〉 =

∫∫Ss1(τ, f ;h)Ss2(τ, f ;h)dτdf

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Sliding window Fourier transform - monoresolution

• Reason: basis functions

h(t− τ)eı2πft

all possess similar resolution

• Examples:• s(t) = δ(t− t0) −→ |Ss(τ, f ;h)|2 = |h(t0 − τ)|2• s(t) = eı2πf0t −→ |Ss(τ, f ;h)|2 = |H(f0 − f)|2

• Uses• for long range oscillatory signals, long windows are necessary• for short range transient, short windows needed• possibility to use several in parallel• incentive to use several ones simultaneously

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Sliding window Fourier transform - illustration

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Other time frequency distributions

• Quadratic or bilinear distributions• Wigner-Ville and avatars (smoothed, pseudo-, reweighted)• Cohen class (WV, Rihaczek, Born-Jordan, Choi-Williams)• property based: covariance, unitarity, inst. freq. & group delay,

localization (for specific signals), support preservation,positivity, stability, interferences

• Bertrand class, fractional Fourier transforms, linear canonicaltransformation (4 param.: FT, fFT, Laplace,Gauss-Weierstrass, Segal-Bargmann, Fresnel transforms)

• generally not applied in more than 1-D

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Conclusions and perpectives

• Time-scale/time-frequency

• tools adapted to non-stationarity• may be adapted in scale/frequency• flexible choice of tools• flexible choice of sampling/redundancy

• deterministic processing• 1D: trends, oscillations, singularities• nD: low-dimensional singularities, edges, textures

• non-deterministic processing• noise, fractals, brownian processes

• high hybridization potential• parametric modeling, statistics, heuristic, local learning

• Very active research

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Traitement de signal Analyse spectrale : Fourier, temps...

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