Introduction to time-frequency analysis. From linear to energy-based representations

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1 Introduction to time-frequency analysis. From linear to energy-based representations Rosario Ceravolo Politecnico di Torino Dep. Structural Engineering UNIVERSITA DI TRENTO Course on «Identification and Control of Dynamical Systems» July -6, 01

2 Time-frequency analysis: historical notes Extension of the classical Fourier analysis Born in the 1940s (Gabor s pioneering studies) Mathematical time-frequency analysis emerged during the 1970s (de Bruijn and other scientists) In the 1980s time frequency analysis was brought to the attention of a larger scientific community

3 Classical Fourier analysis Detects frequencies contained in a signal

4 In the real world, signals have time-varying frequencies: Example: Spectrogram of a whale whistle Jeffrey C. O'Neill (jco8 at cornell.edu) More generally, interest in time-frequency representation is motivated by the limitations of classical spectral estimation techniques in analyzing strongly non-stationary behaviours.

5 Mono-component signals: the Hilbert transform Time domain Frequency domain h(t) +90 t f f -90

6 Analytic signal The analytic signal of a real-valued signal x(t) is defined as the sum of the signal and its Hilbert transform: Analytic signals are particularly useful in dealing with bandlimited signals, e.g. amplitude modulated signals:

7 Analytic signal of an amplitude modulated signal z x t a t exp j f0t xh t a t sin f0t Im Re t x t a t cos f0t

8 Instantaneous frequency With signals that features a time localization of spectral components, a quantity referred to as instantaneous frequency may be obtained as the derivative of the phase of the analytic signal. z x t x t jx H t a t exp j t 1 d arg z x t 1 d f0 t dt d t Such a definition is capable of describing the time localization of a specific class of signals, but proves to be unsuitable for multi-component ones.

9 Instantaneous frequency

10 Instantaneous frequency Two components, what to do?

11 In all cases where mono-dimensional representations are inadequate one can turn to bi-dimensional (joint) functions of the variables time and frequency, which are referred to as time frequency representations (TFRs) of the signal. Several strategies: enhancements of Hilbert transform, e.g. Hilbert-Huang transform linear-transforms, based on time or frequency local windowing: Gabor, Wavelet energy/correlation based distributions: Wigner-Ville, quadratic transforms, Cohen class of transforms etc. other

12 Though a huge variety of plausible theories and perspectives has been proposed for time-frequency representation, one method cannot be claimed to be superior to the others under all conditions. The benefits of each time-frequency approach should be highlighted and demonstrated by referring to specific applications.

13 Hilbert-Huang transform (HT + EMD) Example: El Centro earthquake accelerogram N. E. Huang et al 1996

14 Time-frequency analysis in dynamics : any method providing an information on the temporal behaviour of vibrations. For example : wavelet transform, Gabor transform, Wigner transform etc Choi-Williams representation of an acceleration signal measured on the Queensborough bridge, Vancouver (Ceravolo et al 1996)

15 Mono-component signals Localization in time of frequencies, i.e. chasing instantaneous frequencies

16 Mono-component signals Music is a time-frequency representation

17 Mono-component signals

18 Mono-component signals

19 Two-component signals

20 Multi-component signals

21 Linear transforms: Short-Time Fourier transform (STFT) Moving window Spectrum γ(t-t ) x(t ) STFTX t,f x t ' * t t ' e j ft ' dt '

22 Linear transforms: Short-Time Fourier transform (STFT) The STFT may also be expressed in terms of signal and window spectra: STFTx t,f X f ' * f ' f e j ( f ' f )t df ' where X and Г are respectively the Fourier transform of x and γ. Accordingly, the STFT can be interpreted as the result of passing the signal through a filter translating in frequency.

23 STFT: time and frequency resolution x t t t0 x t e j f t 0 STFTx t,f e j ft t t 0 0 STFTx t,f e j f t f f0 0 f Heisenberg-Gabor inequality: T B 1 Frequency resolution: B f t t T Time resolution t

24 Linear transforms: Wavelet Transform (WT) Analyzing Wavelet (t) WTx (t, f ) x(t ') ( ) t' f * f t ' t dt ' f fc c t Large a : Low frequency (t/a) (t/a) t Bad time resolution Good frequency resolution Small a : High frequency t Good time resolution Bad frequency resolution

25 Linear transforms: time and frequency resolution STFT WT f f t t

26 Wavelet (Morlet window) Example: El Centro earthquake accelerogram N. E. Huang et al 1996

27 Quadratic transforms and marginals Quadratic TFRs allow for interpreting the distributions from an energy point of view. This interpretation is expressed by the so-called marginal properties : Tx t,f df x t Tx t,f dt X f instantaneous power spectral energy density Consequently : Ex x t dt Tx t,f dtdf X f df signal energy

28 Spectrogram (SPEC) and Scalogram (SCAL) The marginal properties are not sufficient to identify an energy density at every point in the time-frequency plane, since the uncertainty principle does not allow such a notion. Vice-versa, many quadratic TFRs may loosely support an energetic interpretation even if they do not satisfy the marginal properties, among them the SPEC and the SCAL: SPECx SCALx t,f STFT t,f x t,f WT t,f x

29 Spectrogram (SPEC), effect of window length

30 Positivity and marginals Tx t,f 0 Marginal properties not satisfied

31 Quadratic superposition principle In SPEC the linearity structure of the STFT is violated, and in fact any quadratic TFR satisfies the quadratic superposition principle : x t c1x1 t c x t Tx t,f c1 Tx t,f c Tx t,f 1 c1ctx x t,f c c1tx x t,f 1 cross terms 1

32 The correlation form and the Wigner-Ville transform Stationary signal Nonstationary signal r ( t, ) x ( t * r ( ) x(t ) x (t ) F.T. Energy spectrum )x* ( t F.T. w.r.t. Wigner-Ville distribution W ( t, ) r ( t, ) e j d 1 A (, ) e j( t ) A (, ) d d e j ( t )W ( t, ) d td ) F.T. w.r.t. t Ambiguity function A (, ) Relation between WD and AF: double Fourier transforms 1 W ( t, ) r ( t, ) e j t d t

33 Wigner-Ville transform Example: musical sound

34 Wigner-Ville transform Marginal properties satisfied, ergo positivity not satisfied

35 Wigner-Ville transform: example (after Galleani & Cohen 000)

36 Wigner-Ville transform: example, exact solution

37 Shift-invariant class (Cohen class of transforms) Among quadratic transforms, those belonging to the shiftinvariant class are characterized by the invariance of its members to time and frequency shifts. Cohen demonstrated that every member of the shiftinvariant class are filtered versions of the WD, and that it is possible to use a general formula for describing all of them: Tx t,f rx t ', t t ', e j f dt ' d t, g, e j t d Where g is the time kernel that uniquely identifies the specific TFR.

38 Shift-invariant class (Cohen class of transforms) Indeed, equivalent formulas can be written in four different domains: temporal correlation domain, time-frequency domain, ambiguity function domain, spectral correlation domain. F.T. w.r.t. F.T. w.r.t. t F.T. w.r.t. F.T. w.r.t. R(, ) W( t, ) t A(, F.T. w.r.t. ) r( t, ) F.T. w.r.t. F.T. w.r.t. F.T. w.r.t. t

39 Cross terms in the Wigner-Ville distribution Bilinear structure of Wigner-Ville distribution Cross-terms if the signal has multiple components A signal with two components x (t ) f (t ) g (t ) Wigner-Ville Distribution W x ( t, ) W f ( t, ) W g ( t, ) Re W f, g ( t, ) where, W f, g (t, ) f (t ) g * (t )e j d : Cross Wigner-Ville distribution

40 Cross terms in the Wigner-Ville distribution: two parallel chirp signals Cross Talk Time signalk x( t ) A1e j ( t 1t ) Freq. = - 1 A e k j ( t t ) 1 ( 1+ )/ 0 Wigner-Ville distribution k W ( t, ) { A1 ( kt 1 ) A ( kt ) A1 A ( kt 1 ) cos[( 1 )t ]} t A. Wigner-Ville distribution Cross Talk Freq. = ( 1+ )/ ( - 1) Ambiguity function Frequency = 1, A(, ) { ( k )[ A1 e A1 A e j 1 j 1 A e j ] [ ( k 1 ) ( k 1 )]} 0 -( - 1) 1 k B. Ambiguity function

41 Cross terms in the Wigner-Ville distribution: two parallel chirp signals WVD and AF are parallel to each other Signal components cross the origin in the AF plane Characteristics of A( 0, 0 ) A(, ) x( t ) x* ( t )e j t dt A( 0,0 ) x( t ) x* ( t )dt x( t ) dt A( 0, 0 ) represents signal s energy Cross-talk appears with some distance from the origin in the AF Cross-talk can be sorted out in the AF

42 Cross term filtering in the AF domain Cross-terms Signal Components AF Domain

43 Cross term filtering in the AF domain A (, ) A Signal (, ) A Cross (, ) Signal components : Must be maintained Cross-talk : Must be eliminated Locations in the AF Close to the origin Some distance from the origin Introduction of a window Higher weight Lower weight

44 Cross term filtering: kernels

45 Cross term filtering: effect of the kernel

46 Desirable properties of the TFRs P0 P1 P PR O PE R T Y N on-negativity: T t,f 0 t f R ealness: T t,f T t,f T im e-frequency shift: y t x t t T t,f T t t,f y t x t e T t,f T t,f f T im e m arginal: x x T t,f d f x x ft x t, f d f T t,f d f y x t x g, independent of t and f X f x tt x t, f d t T x t,f d t 0 g, 0 1 g 0, 1 g, Finite tim e support: x t 0 if t T T t,f 0 p e r t T Finite frequency support: X f 0 if f B T t, f 0 if f B R educed interference g, d a rg X f dt d a r g x t dt G roup delay: P7 g *, 0 x Instantaneous frequency: P6 y f0 t Frequency m arginal: T t,f d t P5 x 0 P4 g, * P3 C O N D IT IO N O N T H E K E R N E L g, is the A F of som e f t 0 t, 0 t x P8 g, e j f d f x P9 g, low pass filter type in, plane

47 Desirable properties of the TFRs Transforms P0 P1 P P3 P4 P5 P6 P7 P8 P9 Spectrogram (SPEC) Wigner (WD) "Alias-Free" Wigner Pseudo-Wigner Smoothed-Pseudo-Wigner Cone-Kernel Reduced Interference Choi-Williams (CWD) (*) (*)

48 T-F representation of three harmonics + two impulses Spectrogram smoothed pseudo Wigner cone-kernel Wigner-Ville Choi-Williams reduced interference distribution

49 T-F representation of a signal measured on an alloy beam Wigner-Ville Choi-Williams

Lecture 3 Kernel properties and design in Cohen s class time-frequency distributions

Lecture 3 Kernel properties and design in Cohen s class time-frequency distributions Time-frequency analysis, adaptive filtering and source separation José Biurrun Manresa 22.02.2011 Time-Frequency representations