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STFT as Filter Bank
Introduction to Wavelet Transform
Yen-Ming LaiDoo-hyun Sung
November 15, 2010ENEE630, Project 1
Wavelet Tutorial Overview
• DFT as filter bank
• STFT as filter bank
• Wavelet transform as filter bank
Discrete Fourier Transform
n
jwnjw enxeX )()(
DFT for fixed w_0
n
njwjw oo enxeX )()(
DFT for fixed w_0
n
njwjw oo enxeX )()(
fix specific frequency w_0
DFT for fixed w_0
n
njwjw oo enxeX )()(
pass in input signal x(n)
DFT for fixed w_0
n
njwjw oo enxeX )()(
modulate by complex exponential of frequency w_0
DFT for fixed w_0
n
njwjw oo enxeX )()(
summation = convolve result with “1”
Why is summation convolution?
n
nkgnfkgf )()()(
Why is summation convolution?
n
nkhnfkhf )()()(
start with definition
Why is summation convolution?
nnh 1)(
Let
Why is summation convolution?
n
nff 1)(1
Why is summation convolution?
n
nff 1)(1
convolution with 1 equivalent to summation
DFT for fixed w_0
n
njwjw oo enxeX )()(
summation = convolve result with “1”
DFT for fixed w_0
n
njwjw oo enxeX )()(
output X(e^jw_0) is constant
DFT for fixed w_0
DFT for fixed w_0
input signal x(n)
DFT for fixed w_0
fix specific frequency w_0
DFT for fixed w_0
modulate by complex exponential of frequency w_0
DFT for fixed w_0
summation = convolve with “1”
DFT for fixed w_0
Transfer function H(e^jw)
DFT for fixed w_0
summation = convolution with 1
DFT for fixed w_0
i.e. impulse response h(n) = 1 for all n
DFT for fixed w_0
)(21)( weeHn
jwnjw
DFT for fixed w_0
output X(e^jw_0) is constant
Frequency Example
Frequency Example
Arbitrary example
Frequency Example
modulation = shift
Frequency Example
convolution by 1 = multiplication by delta
DFT as filter bank
n
jwnjw enxeX )()(
DFT as filter bank
n
njwjw oo enxeX )()(
fix specific frequency w_0
DFT as filter bank
one filter bank
DFT as filter bank
n
jwnjw enxeX )()(
w continuous between [0,2pi)
DFT as filter bank
njwoe
njwe 2
)(nx
…njwe 1
)( ojweX
)( 1jweX
)( 2jweX
uncountably many filter banks
DFT as filter bank
njwoe
njwe 2
)(nx
…njwe 1
)( ojweX
)( 1jweX
)( 2jweX
Uncountable cannot enumerate all (even with infinite number of terms)
DFT as filter bank
njwoe
njwe 2
)(nx
…njwe 1
)( ojweX
)( 1jweX
)( 2jweX
bank of modulators of all frequencies between [0, 2pi)
DFT as filter bank
njwoe
njwe 2
)(nx
…njwe 1
)( ojweX
)( 1jweX
)( 2jweX
bank of identical filters with impulse response of h(n) = 1
Short-Time Fourier Transform
k
jwkjwSTFT enkvkxneX )()(),(
k
jwkjwSTFT enkvkxneX )()(),(
Short-Time Fourier Transform
two variables
k
jwkjwSTFT enkvkxneX )()(),(
Short-Time Fourier Transform
frequency
k
jwkjwSTFT enkvkxneX )()(),(
Short-Time Fourier Transform
shift
k
jwkjwSTFT enkvkxneX )()(),(
Short-Time Fourier Transform
shifted window function v(k)
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
let dummy variable be n instead of k
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
fix frequency w_0 and shift m
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
pass in input x(n)
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
multiply by shifted window and complex exponential
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
summation = convolve with 1
Short-Time Fourier Transform
n
njwjwSTFT
oo emnvnxmeX )()(),(
output constant determined by frequency w_0 and shift m
Short-Time Fourier Transform
Short-Time Fourier Transform
fix frequency w_0 and shift m
Short-Time Fourier Transform
pass in input x(n)
Short-Time Fourier Transform
multiply by shifted window and complex exponential
Short-Time Fourier Transform
summation = convolve with 1
Short-Time Fourier Transform
output constant determined by frequency w_0 and shift m
Short-Time Fourier Transform
k
jwkjwSTFT enkvkxneX )()(),(
Short-Time Fourier Transform
k
jwkjwSTFT enkvkxneX )()(),(
dummy variable k instead of n
Short-Time Fourier Transform
k
jwkjwSTFT enkvkxneX )()(),(
shift is n (previously m)
Short-Time Fourier Transform
k
jwkjwSTFT enkvkxneX )()(),(
rewrite
k
knjwjwnjwSTFT enkvkxeneX )()()(),(
Short-Time Fourier Transform
multiply by e^-jwn and e^jwn
k
knjwjwnjwSTFT enkvkxeneX )()()(),(
Short-Time Fourier Transform
n is shift variable
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
Short-Time Fourier Transform
LTI system ))(( nhx
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
Short-Time Fourier Transform
Impulse response
jwnenv )(
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
Short-Time Fourier Transform
flipped window modulated by +w
jwnenv )(
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
Short-Time Fourier Transform
modulation by -w
Short-Time Fourier Transform
Short-Time Fourier Transform
fixed shift n
Short-Time Fourier Transform
fixed frequency w_0
Short-Time Fourier Transform
convolution with modulated
window
Multiplication by shifted window
transform
freq domain
Short-Time Fourier Transform
modulation by –w_0 shift by –w_0
freq domain
input X(e^jw)
transfer function (window transform shifted by +w_0)
LTI system output
final output after shift by –w_0
STFT as filter bank
k kh
knjwjwnjwSTFT
oo enkvkxeneX )(
)()()(),(
k kh
knjwjwnjwSTFT
oo enkvkxeneX )(
)()()(),(
STFT as filter bank
fixed shift n
k kh
knjwjwnjwSTFT
oo enkvkxeneX )(
)()()(),(
STFT as filter bank
fixed frequency w_0
STFT as filter bank
fixed shift n, fixed shift w_0 = one filter bank
STFT as filter bank
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
STFT as filter bank
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
fixed shift n
STFT as filter bank
k kh
knjwjwnjwSTFT enkvkxeneX
)(
)()()(),(
let w vary between [0, 2pi)
STFT as filter bank
njwenv 1)(
njwenv 2)(
njwenv 0)(njwe 0
njwe 1
njwe 2
),( 0 neX jw
),( 1 neX jw
),( 2 neX jw
)(nx
uncountably many filters since w in [0, 2 pi)
STFT as filter bank
njwenv 1)(
njwenv 2)(
njwenv 0)(njwe 0
njwe 1
njwe 2
),( 0 neX jw
),( 1 neX jw
),( 2 neX jw
)(nx
bandpass filters separated by infinitely small shifts
transfer function (window transform shifted by +w_0)
STFT as filter bank
njwenv 1)(
njwenv 2)(
njwenv 0)(njwe 0
njwe 1
njwe 2
),( 0 neX jw
),( 1 neX jw
),( 2 neX jw
)(nx
demodulators
STFT as filter bank
njwenv 1)(
njwenv 2)(
njwenv 0)(njwe 0
njwe 1
njwe 2
),( 0 neX jw
),( 1 neX jw
),( 2 neX jw
)(nx
segments of X(e^jw)
final output after shift by –w_0
Fix number of frequencies
2...0 110 Mwww
STFT as filter bank
njwenv 1)(
njwMenv 1)(
njwenv 0)(njwe 0
njwe 1
njwMe 1
),( 0 neX jw
),( 1 neX jw
),( 1 neX Mjw
)(nx
bank of M filters
STFT as filter bank
njwenv 1)(
njwMenv 1)(
njwenv 0)(njwe 0
njwe 1
njwMe 1
),( 0 neX jw
),( 1 neX jw
),( 1 neX Mjw
)(nx
M band pass filters
M band pass filters
Uniformly spaced frequencies
2...0 110 Mwww
2...0 1210 Mwwww
STFT as filter bank
njwenv 1)(
njwMenv 1)(
njwenv 0)(njwe 0
njwe 1
njwMe 1
),( 0 neX jw
),( 1 neX jw
),( 1 neX Mjw
)(nx
bank of M filters becomes…
STFT as filter bank
uniform DFT bank
M band pass filters
…
STFT as filter bank
Let E_k(z)=1 for all k
STFT as filter bank
Let E_k(z)=1 for all k
STFT as filter bank
window becomes rectangle M samples long
… …
Uncertainty principle
Uncertainty principle
wide window
Uncertainty principle
narrow bandpass
Uncertainty principle
wide window
poor time resolution
Uncertainty principle
narrow bandpass
good frequency resolution
Uncertainty principle
narrow window
Uncertainty principle
wide bandpass
Uncertainty principle
narrow window
good time resolution
Uncertainty principle
wide bandpass
poor frequency resolution
Special case
narrowest window
Special case
widest bandpass
Special case
narrowest window
perfect time resolution
Special case
widest bandpass
no frequency resolution
v(n)=delta(n)
v(n)=delta(n)
k
jwkjwSTFT enkvkxneX )()(),(
v(n)=delta(n)
v(n)=delta(n)=1 if n=0, 0 otherwise
k
jwkjwSTFT ekvkxeX )0()()0,(
v(n)=delta(n)
k
jwkjwSTFT ekvkxeX )0()()0,(
v(n)=delta(n)
STFT DFT
v(m)=delta(m)
v(0)=delta(0)=1
STFT DFT
LTI system output band limited
LTI system output band limited
can decimate in time domain
Decimation in time domain
)(zHo M
Decimation in time domain
)(zHo M
LTI system output band limited
Decimation in time domain
)(zHo M
maximal decimation (total of M samples across M channels)
Copies in frequency domain
ow
ow
Copies in frequency domain
ow
ow
LTI system output band limited
Copies in frequency domain
ow
ow
copies after maximal decimation
Decimated STFT
M
M
M
Decimated STFT
M
M
M
uniformly spaced versions of same window filter
Decimated STFT
M
M
M
constant maximal decimation
Decimated STFT
M
M
M
decimation by M samples window shift of M
Decimated STFT (sliding window)
Decimated STFT (sliding window)
time axis
Decimated STFT (sliding window)
frequency axis
Decimated STFT (sliding window)
shift window by integer multiples of M
Decimated STFT
M
M
M
decimation by M samples window shift of M
Decimated STFT (sliding window)
calculate M uniformly spaced samples of DFT
Decimated STFT
M
M
M
uniformly spaced versions of same window filter
Decimated STFT (sliding window)
Decimated STFT (sliding window)
time axis
Decimated STFT (sliding window)
frequency axis
Decimated STFT (sliding window)
shift window by integer multiples of M
Decimated STFT
M
M
M
decimation by M samples window shift of M
Decimated STFT (sliding window)
calculate M uniformly spaces samples of DFT
Decimated STFT
M
M
M
uniformly spaced versions of same window filter
Decimated STFT (sliding window)
uniform sampling of time/frequency
Decimated STFT (sliding window)
M fixes sampling
Decimated STFT
M
M
M
decimation by M samples
M versions of same window filter
Decimated STFT
1n
1Mn
let decimation vary
on
Decimated STFT (sliding window)
let window shifts vary
Decimated STFT
1n
1Mn
let window transforms vary
on
Decimated STFT (sliding window)
let window transforms vary
Wavelet Transform
Wavelet Transform
Wavelet Transform
let decimation vary
Wavelet Transform
let window shifts vary
Wavelet Transform
let window transforms vary
Wavelet Transform
let window transforms vary
Wavelet Transform
let window transforms vary
Summary
Decimated STFT
M
M
M
decimation by M samples
M versions of same window filter
Decimated STFT (sliding window)
M fixes sampling
Decimated STFT (sliding window)
uniform sampling of time/frequency
Wavelet Transform
let decimation varylet window
transforms vary
Wavelet Transform
let window shifts vary
let window transforms vary
Wavelet Transform
non-uniform sampling of
time/frequency grid
Reference
• Multirate Systems and Filter Banks by P.P. Vaidyanthan, pp.457-486