An Introduction to S-Transform for Time-Frequency Analysis S.K. Steve Chang SKC-2009.
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Transcript of An Introduction to S-Transform for Time-Frequency Analysis S.K. Steve Chang SKC-2009.
Time Domain vs Frequency Domain
Time information is lost.
2000 4000 6000Frequency (Hz)
Spectrum
0.1 0.9Time (Seconds)
Waveform
SKC-2009
Spectrogram (time-Frequency)
Time (Seconds)
0.1 0.3 0.5 0.7 0.9
Fre
qu
ency
(H
Z)
2000
4000
6000
8000
SKC-2009
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−∞
∞
∫
SKC-2009
STFT Fourier Transform
€
Spectrogram{x(t)} = X(τ , f )2
€
FourierTransform{x(t)} =
X( f ) = X(τ , f )−∞
∞
∫ dτ
SKC-2009
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
SKC-2009
and from
€
X( f )
€
x(t)
€
S(τ , f )
€
X( f ) = S(τ , f )dτ−∞
∞
∫
€
x(t) = S(τ , f )dτ−∞
∞
∫{ }−∞
∞
∫ e i2πftdf
SKC-2009
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
SKC-2009
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
SKC-2009
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
SKC-2009
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
SKC-2009
Generalized S-Transform: Bi-Gaussian
SharperFront EdgeFor ArrivalTime
Sacrifice FrequencyResolution
SKC-2009
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
SKC-2009
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:
SKC-2009
Power Disturbance DetectionPower Disturbance
ContourContour
Stationary Phase Stationary Phase
Power Disturbance
SKC-2009
Earthquake DetectionNoisy Signal Time-Frequency Filter
Inverse S-Transform
Earthquake SignalEarthquake Signal
SKC-2009
Gear Vibration Signal Decoposition
Helicopter Rotor under Fatigue Test
Transient Components
SKC-2009