Presenter : 施信毓 Date : 2007/11/29

ACCESS IC LAB

Graduate Institute of Electronics Engineering, NTU

Presenter : 施信毓 Date : 2007/11/29

Another Point of View : FT - JTFA

From Fourier Transform to Joint Time-Frequency

Analysis

Graduate Institute of Electronics Engineering,National Taiwan University, Taipei 106, Taiwan

ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU

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Xin-Yu Shih

Outline

Fourier TransformJoint Time-Frequency Analysis Linear Time-Frequency Method Quadratic Time-Frequency Method

Short-time Fourier Transform & Spectrogram Wavelet Transform & ScalogramConclusionReference


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Fourier Transform (FT) : Convert the time-domain signals into frequency-domain spectrum.

Example : Linear chirp signal

Fourier Transform

2j f tX f x t e dt

Time Signal Frequency Spectrum


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Zoom FT :

Concentrates (“zooms”) FFT on a narrow band of

frequencies.

Pros :

Improves frequency resolution

Distinguishes between closely-spaced frequencies

Cons :

Baseband analysis requires longer acquisition time for

better resolution – requires more computation

“Zoom” Fourier Transform Analysis (1/3)


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“Zoom” Fourier Transform Analysis (2/3)Baseband FT Analysis

“Zoom” FT Analysis


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“Zoom” Fourier Transform Analysis (3/3)

How to implement ?

LPF


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It’s not suitable for time-varying signals. Example : non-stationary signals

Limitation of FT (1/2)T = 0.0 ~ 0.4s : Freq = 2 HzT = 0.4 ~ 0.7s : Freq = 10 HzT = 0.7 ~ 1.0s : Freq = 20 Hz

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

50

100

150

200

250

Time (s)

Ma

gn

itud

e

Ma

gn

itud

e

Frequency (Hz)

Do not appear at all times


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Different time-domain signals Identical frequency spectrum

Example : original v.s reversed signals

Limitation of FT (2/2)

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

50

100

150

Time (s)

Ma

gn

itud

e

Ma

gn

itud

e

Frequency (Hz)0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 250

50

100

150

Time (s)

Ma

gn

itud

e

Ma

gn

itud

e

Frequency (Hz)

Frequency = 2Hz 20Hz Frequency = 20Hz 2Hz


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In real world, most interesting signals contain numerous non-stationary or transitory characteristics.

Examples : Non-stationary Signals


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Joint Time-Frequency Analysis (JTFA) : Give a good time-

frequency representation of the non-stationary signal.

Joint Time-Frequency Analysis


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Different Analysis Tools : JTFA v.s FT


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Visualize time-frequency location/concentration of time-

domain signal x(t)

Time-Frequency Plane


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Linear TF analysis :

Measure contribution of TF point to signal x(t)

General approach : Inner product of x(t) with “test signal”

or “sounding signal” located about

Linear TF Representation (LTFR) :

Linear Time-Frequency Method (1/2)


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Linear TF synthesis :

Recover or synthesize signal x(t) from

General approach :

where x(t) is represented as superposition of TF localized

signal

components, weighted by “TF coefficient function”

Problem : How to construct test (analysis) functions

and synthesis functions ?

Linear Time-Frequency Method (2/2)


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Quadratic TF analysis :

Measure “energy contribution” of TF point to

signal x(t)

Simple Approach :

Calculate the square function of the LTFR magnitude

Quadratic TF Representation (QTFR) :

Quadratic Time-Frequency Method (1/2)


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TF energy distribution :

Use QTFR to distribute signal energy over TF plane.

Problem : How to construct test (analysis)

functions ?

Quadratic Time-Frequency Method (2/2)


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Problem : Construct family of analysis functions

such that is localized about TF point .

Systematic approach : derived from “prototype

function”

via unitary “TF displacement operator” :

Same for synthesis functions :

Construction of Analysis/Synthesis Functions


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TF shift : (STFT)

TF scaling + time shift : (WT)

Two Classical Definitions of Operator U


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Short-Time Fourier Transform (STFT)


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STFT analysis as convolution:

Filter-bank interpretation/implementation:

STFT and Constant-BW Filter-bank : Analysis


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STFT synthesis as convolution:


STFT and Constant-BW Filter-bank : Synthesis


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Spectrogram analysis as convolution:


Spectrogram Analysis as Constant-BW Filter-bank


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STFT / Spectrogram : Example


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Wavelet Transform (WT)


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WT analysis as convolution:


WT and Constant-Q Filter-bank : Analysis


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WT synthesis as convolution:


WT and Constant-Q Filter-bank : Synthesis


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Scalogram analysis as convolution:


Scalogram Analysis as Constant-Q Filter-bank


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WT / Scalogram : Example


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Spectrogram v.s Scalogram


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Comparison of Different Analysis Tools


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Fourier Transform : (stationary signals) Analyzes the frequency components in the time-domain signals.

Joint Time-Frequency Analysis : (non-stationary signals)

Short-Time Fourier Transform (STFT) : Maps a signal into a two-dimensional function of time and frequency.

Precision is determined by the size of the window.

Window is always the same for all frequencies.

Wavelet Transform (WT) : Uses a windowing technique with variable-sized regions.

Does not use a time-frequency region, but rather a time-scale region.

Higher computation complexity

Conclusion

Documents

Presenter : 施 信 毓 Date : 2007/11/29

Presenter : 施信毓 Date : 2007/11/29