An Introduction to S-Transform for Time-Frequency Analysis S.K. Steve Chang SKC-2009.

An Introduction to S-Transform for

Time-Frequency Analysis

S.K. Steve Chang

SKC-2009

Question: Do we live in…

Time-Domain

or

Frequency-Domain?

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Time Domain vs Frequency Domain

Time information is lost.

2000 4000 6000Frequency (Hz)

Spectrum

0.1 0.9Time (Seconds)

Waveform

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Spectrogram (time-Frequency)

Time (Seconds)

0.1 0.3 0.5 0.7 0.9

Fre

qu

ency

(H

Z)

2000

4000

6000

8000

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Fourier Transform & Short Time Fourier Transform

Window

€

FourierTransform{x(t)} =

X( f ) = x(t)e−i2πftdt−∞

∞

∫

€

STFT{x(t)} =

X(τ , f ) = x(t)W (t − τ )e−i2πftdt−∞

∞

∫

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STFT Fourier Transform

€

Spectrogram{x(t)} = X(τ , f )2

€

FourierTransform{x(t)} =

X( f ) = X(τ , f )−∞

∞

∫ dτ

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STFT: Fixed Window, Fixed Resolution

Window Length

Frequency Resolution

Time Resolution

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Example (400 msec sample)

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(T=25 ms) Poor Frequency Resolution

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(T=125 ms) Fixed Resolution

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(T=375 ms) Fixed Resolution

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(T=1000 ms) Poor High Frequency Time Resolution

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S-Transform & Short Time Fourier Transform

€

STFT{x(t)} =

X(τ , f ) = x(t)W (t − τ )e−i2πftdt−∞

∞

∫€

S −Transform{x(t)} =

S(τ , f ) =f

2πx(t)e

−(t−τ )2 f 2

2−∞

∞

∫ e−i2πftdt

SKC-2009

Gaussian Window for S-Transform

Function of Frequency

€

W (t, f ) =f

2πe−t 2 f 2

2

€

S(τ , f ) = x(t)W (t − τ , f )−∞

∞

∫ e−i2πftdt

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Gaussian Window for S-Transform

High Frequency

Low Frequency

Time Shifted

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and from

€

X( f )

€

x(t)

€

S(τ , f )

€

X( f ) = S(τ , f )dτ−∞

∞

∫

€

x(t) = S(τ , f )dτ−∞

∞

∫{ }−∞

∞

∫ e i2πftdf

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Two Chirps and Two Bursts

S-T

STFT Wigner

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S-Transform and Wavelet Transform

€

M(t, f ) =f

2πe−t 2 f 2

2 e−i2πftMother Wavelet:

€

S(τ , f ) = e−i2πfτC(τ , f )€

Continuous−Wavelet −Transform{x(t)}

=C(τ ,d) = x(t)M(t − τ

d)dt

−∞

∞

∫

=C(τ , f ) = x(t)M((t − τ ) f )−∞

∞

∫ dt

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S-Transform is not Wavelet Transform

Mother Wavelet does not have zero mean:

€

S(τ , f ) = e−i2πfτC(τ , f )€

M(t, f )dt−∞

∞

∫ ≠ 0

Morlet’s Wavelet

€

=M(t, f ) +K( f )

Phase Correction

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Generalized S-Transform

€

W (t, f , p) =f

2π pe−t 2 f 2

2p 2

€

S(τ , f , p) = x(t)W (t − τ , f , p)−∞

∞

∫ e−i2πftdt

Resolutionp Time Frequency

€

Δt( )p =1

2

p

f

Δf( )p =1

2

f

πp

Standard Deviation

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Generalized S-Transform: Bi-Gaussian

€

W (t, f , p) =f

2π

2

(P1 + P2)e−

t 2 f 2

2p( t )2

where

p(t)= P1, t ≥ 0

= P2, t < 0

€

S(τ , f , p) = x(t)W (t − τ , f , p)−∞

∞

∫ e−i2πftdt

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Generalized S-Transform: Bi-Gaussian

SharperFront EdgeFor ArrivalTime

Sacrifice FrequencyResolution

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Generalized S-Transform: Bi-Gaussian

Sharper Arrival Time

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Generalized S-Transform: Your Own

€

W (t, f , p) =f

2π

2

(P1 + P2)e−f 2p(t )2

2

where

p(t) =P1 + P22P1P2

(t − k) +P1 − P22P1P2

(t − k)2 + P32

k =(P1 − P2)

2P32

4P1P2

€

S(τ , f , p) = x(t)W (t − τ , f , p)−∞

∞

∫ e−i2πftdt

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Generalized S-Transform: Your Own

€

W (t, f , p)−∞

∞

∫ dt =1 Or

W (t − τ , f , p)−∞

∞

∫ dt =1 for all τ

€

S(τ , f , p) = x(t)W (t − τ , f , p)−∞

∞

∫ e−i2πftdt

Constraint:

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Examples - ApplicationsEngine Knock Pressure STFT

S-Transform with New Window S-Transform

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Power Disturbance DetectionPower Disturbance

ContourContour

Stationary Phase Stationary Phase

Power Disturbance

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Earthquake DetectionNoisy Signal Time-Frequency Filter

Inverse S-Transform

Earthquake SignalEarthquake Signal

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Earthquake SeismologyOriginal Data

Filter 1

Filter 2

S-FiltersSKC-2009

Seismic Data

STFTS-Transform

Seismic Trace

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Medical: Forearm Blood FlowRest

ActivitiesOcclusion

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Geomagnetic StormInduced Current

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Gear Vibration Signal Decoposition

Helicopter Rotor under Fatigue Test

Transient Components

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Image Analysis

Original Image Dominant Frequency

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S-Transform

We live in Time-Frequency World.

• Similar to Short-Time Fourier Transform• Similar to Continuous Wavelet Transform• Generalized S-Transform: Many Different

Window Choices• Time-Frequency Analysis: Many Applications

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