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

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

2 Time-Frequency representations Linear STFT Wavelet Bilinear or Quadratic Cohen s class Spectrogram Wigner-Ville Choi-Williams... Affine distributions 2

3 Cohen s class The Cohen s class is the family of time-frequency energy distributions covariant by translations in time and frequency,,,,, 3

4 Cohen s class The approach characterizes time-frequency distributions by an auxiliary function called the kernel function The properties of a particular distribution are reflected by simple constraints on the kernel Therefore, it is possible to choose those kernels with prescribed, desirable properties This general class can be described in a number of different ways 4

5 General description of Cohen s class All time-frequency representations can be obtained from, 1 4, 2 2 or equivalently, 1 4, 2 2 where, is the kernel function 5

6 Alternate forms of description I Relationship with the WVD, 1 4 Φ,, where Φ, is the 2-D Fourier transform of the kernel, Φ,, 6

7 Relationship with the WVD The WVD is the element of the Cohen s class for which or equivalently Φ,, 1 Φ, can be chosen as a smoothing function, and as such,, will be a smoothed vesion of the WVD that attenuates in particular ways the interference terms of the WVD 7

8 Relationship between the WVD and the spectrogram Considering the unitarity property,, The spectrogram can be expressed as a smoothing of the WVD, 1 2,, 8

9 Smoothing of the WVD The smoothing function Φ,, is controlled only by the short-time window We can add another degree of freedom Φ, where is the Fourier transform of It allows a progressive, independent control in both time and frequency of the smoothing applied to the WVD 9

10 Smoothed-Pseudo-WVD The obtained distribution is expressed as, and it is called the smoothed-pseudo-wvd (SPWVD) The compromise of the spectrogram between time and frequency resolution is replaced by a compromise between joint time-frequency resolution and the level of interference terms 10

11 Example of WVD 11

12 Example of PWVD 12

13 Example of SPWVD 13

14 From the spectrogram to the WVD 14

15 The ambiguity function The symmetrical ambiguity function (AF), 2 2 The AF is a measure of the time-frequency correlation of the signal It is usually complex and satisfies the Hermitian even symmetry,, 15

16 The AF and the WVD The ambiguity function is the 2-D Fourier transform of the WVD,, Consequently for the AF, a dual property corresponds to nearly all the properties of the WVD 16

17 Properties of the AF Marginal properties The temporal and spectral autocorrelations are the cuts of the AF along the and axis, respectively 0, and,0 The energy of is the value of the AF at the origin of the, plane, which corresponds to its maximum value, 0,0, 17

18 Properties of the AF Time-frequency shift invariance Shifting a signal in the time-frequency plane leaves its Afinvariant apart from a phase factor,, Interference geometry: the AF-signal terms are mainly located around the origin, whereas the AF-interference terms appear at a distance from the origin which is proportional to the time-frequency distance between the involved components 18

19 Properties of the AF 19

20 Properties of the AF 20

21 Alternate forms of description II Characteristic function formulation, 1 4, where, is the characteristic function formulation,, 2 2,,, 21

22 Basic properties related to the kernel Marginals: instantaneous energy and energy density spectrum Time marginal,,0 1 Frequency marginal, 0, 1 22

23 Basic properties related to the kernel Total energy: if the marginals are correctly given, the total energy will be the energy of the signal (although we can retain total energy conservation without complying with the marginals), 1 0,0 1 Uncertainty principle: the condition for the uncertainty principle is that both the marginals are correctly given 23

24 Basic properties related to the kernel Reality: the distribution must satisfy,, since the AF already satisfies the reality property, the only way for the distribution to satisfy this property is for the kernel to also satisfy the identical condition,, 24

25 Basic properties related to the kernel Time and frequency shifts: the distribution must satisfy,, For this property, it needs to be assumed that the kernel is not a function of time and frequency. Therefore,, is time-shift invariant if is independent of, is frequency-shift invariant if is independent of 25

26 Basic properties related to the kernel Scaling invariance: the distribution must satisfy ; 0,, The only way that a function of two variables can satisfy this property is if it is a product of the two variables. Therefore,, This is also referred to as product kernel 26

27 Basic properties related to the kernel Weak finite support: for a finite duration signal the distribution is zero before the signal starts and after the signal ends. Similarly, for a bandlimited signal the distribution is zero outside the bands, 0 for 2, 0 for 2 27

28 Basic properties related to the kernel Strong finite support: the distribution is zero whenever the signal or the spectrum are zero, 0 for 2, 0 for 2 28

29 Other important energy distributions The Rihaczek distribution: consider the interaction energy between a signal restricted to an infinitesimal interval centered on, and passed through an infinitesimal bandpass filter centered on, approximated by the expression leading to the actual distribution,

30 Other important energy distributions The Rihaczek distribution corresponds to the element of the Cohen s class for which, Verfies almost all properties mentioned for the WVD, except for reality Since it is complex valued, interpretation can be akward in practice 30

31 Other important energy distributions The Margenau-Hill distribution: it corresponds to the real part of the Rihaczek distribution The kernel for this distribution is therefore given by, cos 2 Verfies most properties mentioned for the WVD, except for compatibility with filterings and modulations and unitarity 31

32 Other important energy distributions The interference structure of the Rihaczek and Margenau-Hill distributions is different from the WVD: the interference terms corresponding to two points located on, and, are positioned at the coordinates, and, Thus, the use of the Rihaczek (or Margenau-Hill) distribution for signals composed of multi-components located at the same position in time or in frequency is not advised, since the interference terms will then be superposed to the signal terms 32

33 Other important energy distributions 33

34 Other important energy distributions The Page distribution: it was motivated by the construction of a casual distribution, 1 2 The kernel for this distribution is given by, 34

35 Other important energy distributions The Page distribution verfies most properties mentioned for the WVD, except for compatibility with filterings and group delay Actually, it is the only distribution of the Cohen s class which is simultaneously causal, unitary, compatible with modulations, and preserves time-support There is a frequency-smoothed version of the Page distribution, called the pseudo Page distribution 35

36 Other important energy distributions Joint-smoothings of the WVD: the following distributions correspond to particular cases of the Cohen s class for which the parameterization function depends only on the product of the variables and, where is a decreasing function such that 0 1 A direct consequence of this definition is that the marginal properties will be respected 36

37 Other important energy distributions Since is a decreasing function, is a low-pass function, and thus, this parameterization function will reduce the interferences. That is why these distributions are also known as the Reduced Interference Distributions (RID) Some examples are: Choi-Williams, Born-Jordan and Zhao- Atlas-Marks distributions 37

38 Other important energy distributions The Choi-Williams distribution: a natural choice for kernel is to consider a gaussian function The correspondent CWD is, 1 4 1,

39 Other important energy distributions When, it is obtained the WVD. Inversely, the smaller, the better the reduction of the interferences This distribution verifies many properties of the WVD, except for compatibility with filterings and modulations, finite support and unitarity The cross-shape of the parameterization function implies that the efficiency of this distribution strongly depends on the nature of the signal; if the signal is composed of synchronized components in time or in frequency, the CWD will present strong interferences 39

40 Other important energy distributions 40

41 Other important energy distributions The Born-Jordan distribution: if it is imposed to the CWD the further condition to preserve time- and frequencysupports, the simplest choice for is then, sin 2 2 The Zhao-Atlas-Marks distribution: it is the Born-Jordan distibution, but smoothed along the frequency axis, sin 41

42 Other important energy distributions 42

43 Conclusions The Cohen s class gather all the quadratic time-frequency distributions covariant by shifts in time and in frequency It offers a wide set of powerful tools to analyze non-stationary signals. The basic idea is to devise a joint function of time and frequency that describes the energy density or intensity of a signal simultaneously in time and in frequency The most important element of this class is probably the Wigner-Ville distribution, which satisfies many desirable properties 43

44 Conclusions Since these distributions are quadratic, they introduce crossterms in the time-frequency plane which can disturb the readability of the representation One way to attenuate these interferences is to smooth the distribution in time and in frequency, according to their structure The consequence of this is a decrease of the time and frequency resolutions, and more generally a loss of theoretical properties 44

45 References and further reading Time Frequency Analysis: Theory and Applications by Leon Cohen. Prentice Hall; Chapters 9, 11 and 12 Biosignal and Medical Image Processing, Second Edition by John L. Semmlow. CRC press; chapter 6 pp The Time Frequency Toolbox tutorial ( 45

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