Sep.2008DISP Time-Frequency Analysis 時頻分析  Speaker: Wen-Fu Wang 王文阜  Advisor: Jian-Jiun Ding 丁建均 教授   Graduate Institute of Communication Engineering  National Taiwan University, Taipei, Taiwan, ROC

Sep.2008DISP Outline  Introduction  Short Time Fourier Transform  Gabor Transform  Wigner Distribution  Gabor-Wigner Transform  Cohen’s Class Time-Frequency Distribution

Sep.2008DISP Outline  S Transform  Hilbert-Huang Transform  Applications of Time-Frequency Analysis  Conclusions  References

Sep.2008DISP Introduction  Fourier Transform (FT)  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

Sep.2008DISP Introduction  Instantaneous Frequency If then the instantaneous frequency of f (t) are If order of >1, then instantaneous frequency varies with time

Sep.2008DISP Introduction  Example : → chirp function Instantaneous frequency =

Sep.2008DISP Introduction  Time-Frequency analysis  Example : t -axis f -axis amplitude

Sep.2008DISP Short Time Fourier Transform  The earliest Time-Frequency representation was the short time Fourier transform (STFT)  This scheme divides the temporal signal into a series of small overlapping pieces.  The STFT of a function is defined by

Sep.2008DISP Short Time Fourier Transform  is the window function.  The principle of STFT time s(t) h(t) FT time frequency STFT

Sep.2008DISP Short Time Fourier Transform  Example :

Sep.2008DISP Short Time Fourier Transform  Advantage: (1) Least computation time for digital implementation compared with other (2) Its ability to avoid cross-term problem

Sep.2008DISP Gabor Transform  In fact, Gabor transform is a special case of STFT.  When the of STFT, it can be rewritten as  Why does it choose the Gaussian function as a window?

Sep.2008DISP Gabor Transform  The principle of Gabor Transform time s(t) h(t) FT time frequency STFT

Sep.2008DISP Gabor Transform  Example:

Sep.2008DISP Gabor Transform  Advantage: (1) Its ability to avoid cross-term problem (2) The resolution is better than STFT

Sep.2008DISP Wigner Distribution  The Wigner distribution is defined as  In terms of the spectrum, it is  is the Fourier transform of and * means the complex conjugate.

Sep.2008DISP Wigner Distribution  WD has much better time-frequency resolution  WD is not a linear distribution  WD has more computation time  If the signal is composed by several time-frequency components, additional interference will be produced.

Sep.2008DISP Wigner Distribution  Example:

Sep.2008DISP Wigner Distribution  The inner interference is caused by interference between positive and negative frequency of the signal itself  In order to reduce to inner interference problem, Wigner Ville Distribution can be used

Sep.2008DISP Wigner Distribution  The outer interference is caused by mutual interference of multi- component in signal  In order to reduce to outer interference problem, Modified Wigner Distribution can be used

Sep.2008DISP Wigner Ville Distribution  Due the inner interference is caused by interference between positive and negative frequency of the signal itself  We can use analytic version signal to replace the original signal for filtering out negative frequency

Sep.2008DISP Wigner Ville Distribution  We denote the analytic signal of real valued signal by  is Hilbert transform of  We can redefine WD by analytic signal

Sep.2008DISP Modified Wigner Distribution  Due outer interference is caused by mutual interference of multi- component in signal  We can select suitable window function is a way to suppress the outer interference, but retain the sharpness of auto terms.

Sep.2008DISP Modified Wigner Distribution  The Modified Wigner Distribution is defined as  When the window function, it also become Wigner distribution.

Sep.2008DISP Modified Wigner Distribution  Example: WD MWD

Sep.2008DISP Modified Wigner Ville Distribution  Combined form of the WVD and MWD  The advantage of WVD is filtering out inner interference  The advantage of MWD is suppressing the outer interference, but retain the sharpness of auto terms  The MWVD can avoid the inner and outer interference at the same time

Sep.2008DISP Gabor-Wigner Transform  We have compared the properties of the WD and Gabor transform in previous section  The advantage of WD is its high clarity, and the disadvantage of WD is it’s the cross-term problem

Sep.2008DISP Gabor-Wigner Transform  In contrast, the advantage of Gabor transform is its ability to avoid cross- term problem, but its clarity is not as good as that of the WDF  The Gabor-Wigner transform (GWT) can achieve the higher clarity and avoiding cross-term problem at the same time.

Sep.2008DISP Gabor-Wigner Transform  Combined form of the WDF and the Gabor transform (a) (b) (c) (d)

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  The definition of the Cohen class is  is the ambiguity function  How does the Cohen’s class distribution avoid the cross term?

Sep.2008DISP  For the ambiguity function: (1) The auto term is always near to the origin (2) The cross-term is always far from the origin AF WD Cohen’s Class Time-Frequency Distribution

IFT f  FT t  IFT f  FT t  IFT f  FT t  Sep.2008DISP Cohen’s Class Time-Frequency Distribution  Relationship between Wigner distribution and ambiguity function

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  How does the Cohen’s class distribution avoid the cross term?  Choi-Williams Distribution  Cone-Shape Distribution

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  Example: Ambiguity Function Choi-Williams Distribution

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  Example: Ambiguity Function Wigner Distribution

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  Some popular distributions and their kernels TFDKernelFormulation Page Levin Kirkwood Spectrogram Wigner-Ville 1 Choi-Williams Cone shape

Sep.2008DISP Cohen’s Class Time-Frequency Distribution  Advantage: The Cohen’s class distribution may avoid the cross term and has higher clarity  Disadvantage: It requires more computation time and lacks of well mathematical properties

Sep.2008DISP S Transform  Unlike STFT, the width of S transform’s window changes with frequency.  Closely related to the wavelet transform

Sep.2008DISP S Transform Example:

Sep.2008DISP Hilbert-Huang Transform  Traditional data analysis methods are all based on linear and stationary assumptions  The HHT consists of two parts: empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA).

Sep.2008DISP Hilbert-Huang Transform---EMD  Intrinsic Mode Functions (IMF)

Sep.2008DISP Hilbert-Huang Transform--EMD  The Sifting Process IMF 1; iteration IMF 1; iteration 0

Sep.2008DISP Hilbert-Huang Transform--EMD

Sep.2008DISP Hilbert-Huang Transform--EMD residue

Sep.2008DISP Hilbert-Huang Transform--EMD

Sep.2008DISP Hilbert-Huang Transform--HSA  We have obtained the intrinsic mode function components by EMD process method  Then we will do the Hilbert transform to each IMF component.

Sep.2008DISP Applications of Time-Frequency Analysis  (1) Finding the Instantaneous Frequency  (2) Sampling Theory  (3) Modulation and Multiplexing  (4) Filter Design  (5) Signal Representation  (6) Random Process Analysis

Sep.2008DISP Applications of Time-Frequency Analysis  (7) Acoustics  (8) Data Compression  (9) Spread Spectrum Analysis  (10) Radar Signal Analysis  (11) Biomedical Engineering  (12) Economic Data Analysis

Sep.2008DISP Conclusions 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

Sep.2008DISP Conclusions  Compare with Fourier, wavelet and HHT analyses FourierWaveletHilbert Basis A priori Adaptive Frequency Convolution: globalConvolution: regionalDifferentiation: local Presentation Energy-frequencyEnergy-time-frequency Non-linear No Yes Non-stationary NoYes Uncertainty Yes No Harmonics Yes No Feature extraction No Discrete: no Continuous: yes Yes Theoretical base Theory complete Empirical

Sep.2008DISP Conclusions  Advantage: The instantaneous frequency can be observed  Disadvantage: Higher complexity for computation  Which method is better?

Sep.2008DISP References  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, 1998 ＂ The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Non- Stationary Time Series Analysis ＂  N. E. Huang, S. Shen, ＂ Hilbert-Huang Transform and its Applications ＂, World scientific, Singapore, Hilbert-Huang Transform and its Applications

Sep.2008DISP References  S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Processing, vol. 49, no. 8, pp , Aug Relations between the fractional operations and the Wigner distribution, ambiguity function  L. Cohen, ＂ Time-Frequency distributions-A review” Proc. IEEE, Vol. 77, No. 7, pp , July Time-Frequency distributions-A review  C. H. Page, “Instantaneous Power Spectra,” National Bureau of Standards, Washington, D. C., 1951Instantaneous Power Spectra

Sep.2008DISP References  S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans. Signal Processing, vol. 55, no. 10, pp , Oct  R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr