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README • Lecture notes will be animated by clicks. • Each click will indicate pause for audience to observe slide. • On further click, the lecturer will explain the slide with highlighted notes.

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Page 1: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

README

• Lecture notes will be animated by clicks.

• Each click will indicate pause for audience to observe slide.

• On further click, the lecturer will explain the slide with highlighted notes.

Page 2: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as Filter Bank

Introduction to Wavelet Transform

Yen-Ming LaiDoo-hyun Sung

November 15, 2010ENEE630, Project 1

Page 3: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Tutorial Overview

• DFT as filter bank

• STFT as filter bank

• Wavelet transform as filter bank

Page 4: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Discrete Fourier Transform

n

jwnjw enxeX )()(

Page 5: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

Page 6: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

fix specific frequency w_0

Page 7: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

pass in input signal x(n)

Page 8: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

modulate by complex exponential of frequency w_0

Page 9: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

summation = convolve result with “1”

Page 10: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Why is summation convolution?

n

nkgnfkgf )()()(

Page 11: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Why is summation convolution?

n

nkhnfkhf )()()(

start with definition

Page 12: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Why is summation convolution?

nnh 1)(

Let

Page 13: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Why is summation convolution?

n

nff 1)(1

Page 14: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Why is summation convolution?

n

nff 1)(1

convolution with 1 equivalent to summation

Page 15: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

summation = convolve result with “1”

Page 16: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

n

njwjw oo enxeX )()(

output X(e^jw_0) is constant

Page 17: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

Page 18: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

input signal x(n)

Page 19: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

fix specific frequency w_0

Page 20: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

modulate by complex exponential of frequency w_0

Page 21: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

summation = convolve with “1”

Page 22: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

Transfer function H(e^jw)

Page 23: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

summation = convolution with 1

Page 24: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

i.e. impulse response h(n) = 1 for all n

Page 25: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

)(21)( weeHn

jwnjw

Page 26: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT for fixed w_0

output X(e^jw_0) is constant

Page 27: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Frequency Example

Page 28: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Frequency Example

Arbitrary example

Page 29: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Frequency Example

modulation = shift

Page 30: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Frequency Example

convolution by 1 = multiplication by delta

Page 31: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

n

jwnjw enxeX )()(

Page 32: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

n

njwjw oo enxeX )()(

fix specific frequency w_0

Page 33: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

one filter bank

Page 34: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

n

jwnjw enxeX )()(

w continuous between [0,2pi)

Page 35: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

njwoe

njwe 2

)(nx

…njwe 1

)( ojweX

)( 1jweX

)( 2jweX

uncountably many filter banks

Page 36: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

njwoe

njwe 2

)(nx

…njwe 1

)( ojweX

)( 1jweX

)( 2jweX

Uncountable cannot enumerate all (even with infinite number of terms)

Page 37: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

njwoe

njwe 2

)(nx

…njwe 1

)( ojweX

)( 1jweX

)( 2jweX

bank of modulators of all frequencies between [0, 2pi)

Page 38: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

DFT as filter bank

njwoe

njwe 2

)(nx

…njwe 1

)( ojweX

)( 1jweX

)( 2jweX

bank of identical filters with impulse response of h(n) = 1

Page 39: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

k

jwkjwSTFT enkvkxneX )()(),(

Page 40: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT enkvkxneX )()(),(

Short-Time Fourier Transform

two variables

Page 41: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT enkvkxneX )()(),(

Short-Time Fourier Transform

frequency

Page 42: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT enkvkxneX )()(),(

Short-Time Fourier Transform

shift

Page 43: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT enkvkxneX )()(),(

Short-Time Fourier Transform

shifted window function v(k)

Page 44: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

let dummy variable be n instead of k

Page 45: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

fix frequency w_0 and shift m

Page 46: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

pass in input x(n)

Page 47: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

multiply by shifted window and complex exponential

Page 48: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

summation = convolve with 1

Page 49: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

n

njwjwSTFT

oo emnvnxmeX )()(),(

output constant determined by frequency w_0 and shift m

Page 50: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

Page 51: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

fix frequency w_0 and shift m

Page 52: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

pass in input x(n)

Page 53: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

multiply by shifted window and complex exponential

Page 54: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

summation = convolve with 1

Page 55: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

output constant determined by frequency w_0 and shift m

Page 56: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

k

jwkjwSTFT enkvkxneX )()(),(

Page 57: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

k

jwkjwSTFT enkvkxneX )()(),(

dummy variable k instead of n

Page 58: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

k

jwkjwSTFT enkvkxneX )()(),(

shift is n (previously m)

Page 59: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

k

jwkjwSTFT enkvkxneX )()(),(

rewrite

Page 60: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

knjwjwnjwSTFT enkvkxeneX )()()(),(

Short-Time Fourier Transform

multiply by e^-jwn and e^jwn

Page 61: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

knjwjwnjwSTFT enkvkxeneX )()()(),(

Short-Time Fourier Transform

n is shift variable

Page 62: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

Short-Time Fourier Transform

LTI system ))(( nhx

Page 63: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

Short-Time Fourier Transform

Impulse response

jwnenv )(

Page 64: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

Short-Time Fourier Transform

flipped window modulated by +w

jwnenv )(

Page 65: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

Short-Time Fourier Transform

modulation by -w

Page 66: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

Page 67: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

fixed shift n

Page 68: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

fixed frequency w_0

Page 69: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

convolution with modulated

window

Multiplication by shifted window

transform

freq domain

Page 70: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Short-Time Fourier Transform

modulation by –w_0 shift by –w_0

freq domain

Page 71: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

input X(e^jw)

Page 72: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

transfer function (window transform shifted by +w_0)

Page 73: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

LTI system output

Page 74: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

final output after shift by –w_0

Page 75: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

k kh

knjwjwnjwSTFT

oo enkvkxeneX )(

)()()(),(

Page 76: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT

oo enkvkxeneX )(

)()()(),(

STFT as filter bank

fixed shift n

Page 77: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k kh

knjwjwnjwSTFT

oo enkvkxeneX )(

)()()(),(

STFT as filter bank

fixed frequency w_0

Page 78: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

fixed shift n, fixed shift w_0 = one filter bank

Page 79: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

Page 80: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

fixed shift n

Page 81: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

k kh

knjwjwnjwSTFT enkvkxeneX

)(

)()()(),(

let w vary between [0, 2pi)

Page 82: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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)

Page 83: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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

Page 84: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

transfer function (window transform shifted by +w_0)

Page 85: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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

Page 86: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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)

Page 87: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

final output after shift by –w_0

Page 88: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Fix number of frequencies

2...0 110 Mwww

Page 89: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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

Page 90: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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

Page 91: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

M band pass filters

Page 92: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uniformly spaced frequencies

2...0 110 Mwww

2...0 1210 Mwwww

Page 93: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

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…

Page 94: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

uniform DFT bank

Page 95: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

M band pass filters

…

Page 96: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

Let E_k(z)=1 for all k

Page 97: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

Let E_k(z)=1 for all k

Page 98: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

STFT as filter bank

window becomes rectangle M samples long

… …

Page 99: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

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Uncertainty principle

wide window

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Uncertainty principle

narrow bandpass

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Uncertainty principle

wide window

poor time resolution

Page 103: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

narrow bandpass

good frequency resolution

Page 104: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

narrow window

Page 105: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

wide bandpass

Page 106: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

narrow window

good time resolution

Page 107: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Uncertainty principle

wide bandpass

poor frequency resolution

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Special case

narrowest window

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Special case

widest bandpass

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Special case

narrowest window

perfect time resolution

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Special case

widest bandpass

no frequency resolution

Page 112: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

v(n)=delta(n)

v(n)=delta(n)

Page 113: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT enkvkxneX )()(),(

v(n)=delta(n)

v(n)=delta(n)=1 if n=0, 0 otherwise

Page 114: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT ekvkxeX )0()()0,(

v(n)=delta(n)

Page 115: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

k

jwkjwSTFT ekvkxeX )0()()0,(

v(n)=delta(n)

STFT DFT

Page 116: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

v(m)=delta(m)

Page 117: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

v(0)=delta(0)=1

STFT DFT

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LTI system output band limited

Page 119: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

LTI system output band limited

can decimate in time domain

Page 120: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimation in time domain

)(zHo M

Page 121: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimation in time domain

)(zHo M

LTI system output band limited

Page 122: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimation in time domain

)(zHo M

maximal decimation (total of M samples across M channels)

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Copies in frequency domain

ow

ow

Page 124: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Copies in frequency domain

ow

ow

LTI system output band limited

Page 125: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Copies in frequency domain

ow

ow

copies after maximal decimation

Page 126: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

Page 127: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

uniformly spaced versions of same window filter

Page 128: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

constant maximal decimation

Page 129: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

decimation by M samples window shift of M

Page 130: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

Page 131: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

time axis

Page 132: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

frequency axis

Page 133: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

shift window by integer multiples of M

Page 134: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

decimation by M samples window shift of M

Page 135: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

calculate M uniformly spaced samples of DFT

Page 136: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

uniformly spaced versions of same window filter

Page 137: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

Page 138: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

time axis

Page 139: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

frequency axis

Page 140: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

shift window by integer multiples of M

Page 141: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

decimation by M samples window shift of M

Page 142: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

calculate M uniformly spaces samples of DFT

Page 143: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

uniformly spaced versions of same window filter

Page 144: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

uniform sampling of time/frequency

Page 145: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

M fixes sampling

Page 146: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

decimation by M samples

M versions of same window filter

Page 147: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

1n

1Mn

let decimation vary

on

Page 148: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

let window shifts vary

Page 149: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

1n

1Mn

let window transforms vary

on

Page 150: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

let window transforms vary

Page 151: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

Page 152: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

Page 153: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let decimation vary

Page 154: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let window shifts vary

Page 155: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let window transforms vary

Page 156: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let window transforms vary

Page 157: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let window transforms vary

Page 158: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Summary

Page 159: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT

M

M

M

decimation by M samples

M versions of same window filter

Page 160: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

M fixes sampling

Page 161: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Decimated STFT (sliding window)

uniform sampling of time/frequency

Page 162: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let decimation varylet window

transforms vary

Page 163: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

let window shifts vary

let window transforms vary

Page 164: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Wavelet Transform

non-uniform sampling of

time/frequency grid

Page 165: README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain

Reference

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


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