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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 )()()(


n

nkhnfkhf )()()(

start with definition


nnh 1)(

Let


n

nff 1)(1


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


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 )()(


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



two variables

k



frequency

k



shift

k



shifted window function v(k)


n

njwjwSTFT

oo emnvnxmeX )()(),(

let dummy variable be n instead of k


n

njwjwSTFT


fix frequency w_0 and shift m


n

njwjwSTFT


pass in input x(n)


n

njwjwSTFT


multiply by shifted window and complex exponential


n

njwjwSTFT


summation = convolve with 1


n

njwjwSTFT


output constant determined by frequency w_0 and shift m


fix frequency w_0 and shift m


pass in input x(n)


multiply by shifted window and complex exponential


summation = convolve with 1


output constant determined by frequency w_0 and shift m


k



k


dummy variable k instead of n


k


shift is n (previously m)


k


rewrite

k

knjwjwnjwSTFT enkvkxeneX )()()(),(


multiply by e^-jwn and e^jwn

k

knjwjwnjwSTFT enkvkxeneX )()()(),(


n is shift variable

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(


LTI system ))(( nhx

k kh


)(

)()()(),(


Impulse response

jwnenv )(

k kh


)(

)()()(),(


flipped window modulated by +w

jwnenv )(

k kh


)(

)()()(),(


modulation by -w


fixed shift n


fixed frequency w_0


convolution with modulated

window

Multiplication by shifted window

transform

freq domain


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


)(

)()()(),(

STFT as filter bank

k kh


)(

)()()(),(

fixed shift n

STFT as filter bank

k kh


)(

)()()(),(

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

window becomes rectangle M samples long

… …

Uncertainty principle


wide window


narrow bandpass


wide window

poor time resolution


narrow bandpass

good frequency resolution


narrow window


wide bandpass


narrow window

good time resolution


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


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


can decimate in time domain

Decimation in time domain

)(zHo M


)(zHo M



)(zHo M

maximal decimation (total of M samples across M channels)

Copies in frequency domain

ow

ow


ow

ow



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)


time axis


frequency axis


shift window by integer multiples of M

Decimated STFT

M

M

M



calculate M uniformly spaced samples of DFT

Decimated STFT

M

M

M



time axis


frequency axis


shift window by integer multiples of M

Decimated STFT

M

M

M



calculate M uniformly spaces samples of DFT

Decimated STFT

M

M

M



uniform sampling of time/frequency


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


let window shifts vary

Decimated STFT

1n

1Mn

let window transforms vary

on

Wavelet Transform

Wavelet Transform

let decimation vary

Wavelet Transform


Summary

Decimated STFT

M

M

M

decimation by M samples

M versions of same window filter


M fixes sampling


uniform sampling of time/frequency

Wavelet Transform

let decimation varylet window

transforms vary

Wavelet Transform



Wavelet Transform

non-uniform sampling of

time/frequency grid

Reference

• Multirate Systems and Filter Banks by P.P. Vaidyanthan, pp.457-486

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