An Introduction to Time-Frequency Analysis Speaker: Po-Hong Wu Advisor: Jian-Jung Ding Digital Image and Signal Processing Lab GICE, National Taiwan University 1NTU, GICE, MD531, DISP Lab

Outline Introduction Short-Time Fourier Transform Gabor Transform Wigner Distribution Function Spectrogram S Tranform Cohen’s Class Time-Frequency Distribution Fractional Fourier Transform Motion on Time-Frequency Distributions Hilbert-Huang Transform Conclusion Reference 2NTU, GICE, MD531, DISP Lab

Introduction Fourier transform (FT) t varies from  ∞~∞ Time-Domain  Frequency Domain [A1] Why do we need time-frequency transform? 3NTU, GICE, MD531, DISP Lab

Example: x(t) = cos(  t) when t < 10, x(t) = cos(3  t) when 10  t < 20, x(t) = cos(2  t) when t  20 [B2] 4NTU, GICE, MD531, DISP Lab

Short Time Fourier Transform w(t): mask function 也稱作 windowed Fourier transform or time-dependent Fourier transform 5NTU, GICE, MD531, DISP Lab

When w(t) is a rectangular function w(t) = 1 for |t|  B, w(t) = 0, otherwise [B3] 6NTU, GICE, MD531, DISP Lab

 Advantage: less computation time  Disadvantage: worse representaion  Application: deal with large data Ex: real time processing 7NTU, GICE, MD531, DISP Lab

Gabor Transform A specail case of the STFT where Other definition [B4] 8NTU, GICE, MD531, DISP Lab

Why do we choose the Guassian function?  Among all functions of w(t), the Gaussian function has area in time-frequency distribution is minimal than other STFT.  Gaussian function is an eigenfunction of Fourier transform, so the Gabor transform has the same properties in time domain and in frequency domain. 9NTU, GICE, MD531, DISP Lab

Approximation of the Gabor Transform Because of when |a|> Because of when |a|> NTU, GICE, MD531, DISP Lab

Generalization of the Gabor Transform  For larger σ: higher resolution in the time domain but lower resolution in the frequency domain  For smaller σ: higher resolution in the frequency domain but lower resolution in the time domain 11NTU, GICE, MD531, DISP Lab

Resolution Using the generalized Gabor transform with larger σ Using other time unit instead of second 12NTU, GICE, MD531, DISP Lab

Wigner Distribution Function Other definition [B5] 13NTU, GICE, MD531, DISP Lab

Signal auto-correlation function Spectrum auto-correlation function Ambiguity function (AF) [B6] A x ( ,  ) IFT f  FT t  IFT f  FT t  S x ( , f ) FT t  IFT f  C x (t,  ) W x (t, f ) 14NTU, GICE, MD531, DISP Lab

Modified Wigner Distribution  Wigner Ville Distribution For compressing inner interference Analytic signal 15NTU, GICE, MD531, DISP Lab

 Pseudo Wigner Distribution For surpressing outer interference where [B7] 16NTU, GICE, MD531, DISP Lab

 Gabor-Wigner Distribution  [B8] 17NTU, GICE, MD531, DISP Lab

Spectrogram Another form [B9] 18NTU, GICE, MD531, DISP Lab

S-Transform  Original S-Transform Where w(t)= [B10] 19NTU, GICE, MD531, DISP Lab

 Generalized S-Transform Another definition Ristriction 20NTU, GICE, MD531, DISP Lab

 Novel S-Transform with the Special Varying Window Restriction When, it becomes the Gabor transform. When, it becomes the original S-trnasform. 21NTU, GICE, MD531, DISP Lab

Cohen’s Class Time-Frequency Distribution Ambiguity function [B11] IFT f  FT t  IFT f  FT t  IFT f  FT t  22NTU, GICE, MD531, DISP Lab

For the ambiguity function The auto terms are always near to the origin. The cross terms are always from the origin. [ B12] 23NTU, GICE, MD531, DISP Lab

Kernel function Choi-Williams Distribution [B13] 24 NTU, GICE, MD531, DISP Lab

Cone-Shape Distribution 25NTU, GICE, MD531, DISP Lab

Fractional Fourier Transform How to rotate the time-frequency distribution by the angle other than  /2, , and 3  /2? 26 NTU, GICE, MD531, DISP Lab

Zero rotation: Consistency with Fourier transform: = FT Additivity of rotation: rotation : 27NTU, GICE, MD531, DISP Lab

[A3] 28NTU, GICE, MD531, DISP Lab

 Application Decomposition in the time-frequency distribution 29NTU, GICE, MD531, DISP Lab

f-axis Signal noise t-axis FRFT  FRFT  noiseSigna l cutoff line Signa l cutoff line noise 30NTU, GICE, MD531, DISP Lab

Modulation and Multiplexing 31NTU, GICE, MD531, DISP Lab

Time domain Frequency domain fractional domain Modulation Shifting Modulation + Shifting Shifting Modulation Modulation + Shifting Differentiation  j2  f Differentiation and  j2  f  −j2  f Differentiation Differentiation and  −j2  f 32NTU, GICE, MD531, DISP Lab

Motion on Time-Frequency Distributions  Horizontal Shifting  Vertical Shifting 33NTU, GICE, MD531, DISP Lab

 Dilation  Shearing 34NTU, GICE, MD531, DISP Lab

 Rotation If F{x(t)}=X(f), then F{X(t)}=x(-f). We can derive: 35NTU, GICE, MD531, DISP Lab

Hilbert-Huang Transform  Introduction Most of distribution are designed for stationary and linear signals, but, In the real world, most of signals are non-stationary and non-linear. HHT consists two parts: empirical mode decomposition (EMD) Hilbert spectral analysis (HSA) 36NTU, GICE, MD531, DISP Lab

 Empirical decomposition function Any complicated data can be decomposed into a finite and small number of intrinsic mode functions (IMF) by sifting processing. Intrinsic mode function (1)In the whole data set, the number of extrema and the number of zero-crossing must either equal or differ at most by one. (2)At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. 37NTU, GICE, MD531, DISP Lab

Sifting Process (1)First, find all the local maxima extrema of x(t). (2) Interpolate (cubic spline fitting) between all the maxima extrema ending up with some upper envelope IMF 1; iteration IMF 1; iteration 0 38NTU, GICE, MD531, DISP Lab

(3) Find all the local minima extrema. (4) Interpolate (cubic spline fitting) between all the minima extrema ending up with some lower envelope. 39NTU, GICE, MD531, DISP Lab

(5) Compute the mean envelope between upper envelope and lower envelope. (6) Compute the residue residue 40NTU, GICE, MD531, DISP Lab

(7) Repeat the above procedure (step (1) ~ step (6)) on the residue until the residue is a monotonic function or constant. The original signal equals the sum of the various IMFs plus the residual trend. 41NTU, GICE, MD531, DISP Lab

EX: 42NTU, GICE, MD531, DISP Lab

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 Hilbert Spectral Anaysis 46NTU, GICE, MD531, DISP Lab

47NTU, GICE, MD531, DISP Lab AdvantageDisadvantage STFT and Gabor transform 1.Low computation 2.The range of the integration is limited 3.No cross term 4.Linear operation 1.Complex value 2.Low resolution Wigner distribution function 1.Real 2.High resolution 3.If the time/frequency limited, time/frequency of the WDF is limited with the same range 1.High computation 2.Cross term 3.Non-linear operation Cohen’s class distribution 1.Avoid the cross term 2.Higher clarity 1.High computation 2.Lack of well mathematical properties Gabor-Wigner distribution function 1.Combine the advantage of the WDF and the Gabor transform 2.Higher clarity 3.No cross-term 1.High computation

Conclusion  We introduce many distributions here and put most attention on computation time and representations. We can find that the representation with higher clarity cost more computation time for all methods. Resolution Computation time  The Hilbert-Huang transform is the most power method to deal with non-linear and non-stationary signals but lacks of physical background. 48NTU, GICE, MD531, DISP Lab

Reference [1][A]J. J. Ding, “Time-Frequency Analysis and Wavelet Transform,” National Taiwan University, [Online].Available: [2][B]W. F. Wang, “Time-Frequency Analyses and Their Fast Implementation Algorithm,” Master Thesis, National Taiwan University, June, [3]Luis B. Almeida, Member, IEEE, “The Fractional Fourier Transform and Time-Frequency Representations,” IEEE Transaction On Signal Processing, vol. 42, no. 11, November [4]M. R. Spiegel, Mathematical Handbook of Formulas and Tables, McGraw- Hill, [5]N. E. Huang, Z. Shen and S. R. Long, et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time Series Analysis ＂, Proc. Royal Society, vol. 454, pp , London, NTU, GICE, MD531, DISP Lab