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Page 1: Digital Speech Processing— Lecture 10 Short-Time … speech...Short-Time Fourier Analysis Methods - Filter Bank Design 2 Review of STFT 1 123 0 2 1 ˆˆ ˆ ˆ ˆ ˆ ˆ.() [][]ˆ

1

Digital Speech Processing—Lecture 10

Short-Time Fourier Analysis Methods - Filter

Bank Design

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2

Review of STFT1

1 2 3 0

2

1

ˆ ˆˆ

ˆˆ

ˆˆ

ˆ. ( ) [ ] [ ]

ˆ function of (looks like a time sequence)ˆ function of (looks like a transform)

ˆ ˆ( ) defined for , , ,...;

. Interpretations of ( )ˆ. fix

ω ω

ω

ω

ω

ω π

∞−

=−∞

= −

= ≤ ≤

∑j j mn

m

jn

jn

X e x m w n m e

n

X e n

X e

n ˆ

ˆ ˆ

ˆˆed, variable; ( ) DTFT [ ] [ ]

DFT View OLA implementationˆ ˆ 2. variable, fixed; ( ) [ ] [ ]

Linear Filtering view filter bank implementation FBS i

ω

ω ω

ω ω

ω −

= = −⎡ ⎤⎣ ⎦⇒ ⇒

= = ∗

⇒ ⇒⇒

jn

j j nn

X e x m w n m

n n X e x n e w n

mplementation

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3

Review of STFT

3

22

ˆˆ

. Sampling Rates in Time and Frequency ˆrecover [ ] from ( )

1. time: ( ) has bandwidth of Hertz samples/sec rate

Hamming Window: (Hz) sample

ω

ω

= ⇒

jn

j

S

x n X e

W e BB

FBL

exactly

4

at

(Hz) or every L/4 samples

2. frequency: [ ] is time limited to samples need at least frequency samples to avoid time aliasing

SFL

w n LL

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4

Review of STFT with OLA method can recover ( ) using lower

sampling rates in either time or frequency, e.g., can sample every samples (and divide by window), or can use fewer than frequency sam

x n

LL

exactly

ples (filter bank channels); but these methods are highly subject to aliasing errors with any modifications to STFT can use windows (LPF) that are longer than samples and

still recover with L

frequency channels; e.g., ideal LPF is infinite in time duration, but with zeros spaced samples apart where / is the BW of the ideal LPF

< ⇒

S

N LN

F N

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5

Review of STFTH0(ejω)

H1(ejω)

HN-1(ejω)

+…

X(ejω) Y(ejω)

1

0

1

0

1

0

[ ] [ ]

( ) ( )

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ] [ ] ...

need to design digital filters that match criteria for exact reconstr

ω

ω ω

δ

δ

δ

=

=

=−∞

=−∞

=

= =

= = =

= −

= − = + +

%

%

%

kj nk

Nj j

kk

N

kk

r

r

h n w n e

H e H e

h n h n n w n p n

p n N n rN

h n N w rN n rN w w N

uction of [ ] and which still work with modifications to STFTx n

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Tree-Decimated Filter Banks

6

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Tree-Decimated Filter Banks• can sample STFT in time and frequency using lowpass

filter (window) which is moved in jumps of R<L samples if L≤N where:– L is the window length, – R is the window shift, and – N is the number of frequency channels

• for a given channel at ω=ωk, the sampling rate of the STFT need only be twice the bandwidth of the window Fourier transform– down-sample STFT estimates by a factor of R at the transmitter– up-sample back to original sampling rate at the receiver– final output formed by convolution of the up-sampled STFT with

an appropriate lowpass filter, f [n]7

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Filter Bank Channels

8

Fully decimated and interpolated filter bank channels; (a) analysis with bandpass filter, down-shifting frequency and down-sampling; (b) synthesis with up-sampler followed by lowpassinterpolation filter and frequency up-shift; (c) analysis with frequency down-shift followed by lowpass filter followed by down-sampling; (d) synthesis with up-sampling followed by frequency up-shift followed by bandpass filter.

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Full Implementation of Analysis-Synthesis System

9

Full implementation of STFT analysis/synthesis system with channel decimation by a factor of R, and channel interpolation by the same factor R (including a box for short-time modifications.

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10

Review of Down and Up-Sampling

R↓][nx [ ] [ ]dx n x nR=

11 2

0

( )/( ) ( )

aliasing addition

ω ω π−

=

= ∑R

j j r Rd R

rX e X e

R↑][nx[ / ] 0, 1, 2,...

[ ]0 otherwiseu

x n R nx n

= ± ±⎧= ⎨⎩

( ) ( ) imaging removal

ω ω=j j RuX e X e

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11

[ ]

1

0

1

0

1

1

1 ( )

assuming no short time modifications ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) [ ] [ ]

( ) ( ) ( ) ( )

k k

rR

k k

k

k

rR

j j mrR rR

m W

Nj j n

rRkN

j nrR

k

j n m

m W r k

X k X e w rR m x m e

y n P k Y e eN

P k X k f n rR eN

x m w rR m f n rR P k eN

ω ω

ω ω

ω

ω

=

=

∞−

∈ =−∞ =

= = −

=

= −

⎡ ⎤= − −⎢ ⎥

⎣ ⎦

∑ ∑1

0

1

0

11 ( )

recognizing that when ( ) ,

( )

can show that the condition for ( ) ( ), is

( ) ( ) ( )

k

N

Nj n m

k q

r

P k k

e n m qNN

y n x n n

w rR n qN f n rR q m n

ω δ

δ

− ∞−

= =−∞

=−∞

• = ∀

= − −

• = ∀

− + − = ⇒ =

∑ ∑

Filter Bank Reconstruction

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12

Filter Bank Reconstruction

( ) ( )( )

1

0

2 21

0

1

1

1

( ) ( )

( ) ( )

frequency domain equations, assuming: [ ] [ ] ; [ ] [ ] ;

( ); ( ) ;

( ) ( )

( ) ( )

( )

k k

k k k k

k

j n j nk k

j j j jk k

Njj

kk

R j jR R

k

j

h n w n e g n f n e

H e W e G e F e

Y e P k G eN

H e X eR

X e

ω ω

ω ω ω ω ω ω

ωω

π πω ω

ω

− −

=

− − −

=

= =

= =

= ⋅

⎡ ⎤⎢ ⎥⎣ ⎦

=

∑l l

l

( )1

0

2 21 1

1 0

1( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) aliasing terms

k

k

Nj j

k kk

R Nj jjR Rk k

k

j j

P k G e H eRN

X e P k G e H eRN

H e X e

ω ω

π πω ωω

ω ω

=

− −− −

= =

+ ⋅

= ⋅ +

∑ ∑l l

l

%

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Filter Bank Reconstruction

1

0

21

0

1 1

1 0 1 2( )

ˆ conditions for perfect reconstruction ( ( ) ( ))

( ) ( ) ( ) ( )

and

( ) ( ) ( ) for , ,...,

k

k

j j

Njj j

k kk

N jj Rk k

k

Y e X e

H e P k G e H eRN

P k G e H e RRN

ω ω

ωω ω

πωω

=

− −

=

• =

= =

= =

∑l

%

l

Flat Gain

Alias Cancellation

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Filter Bank Reconstruction

14

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15

Decimated Analysis-Synthesis Filter Bank

2

1 0 1 14

1 04

( ) ( )

( )

practical solution for the case ( ) ( ), (2-band solution)where the conditions for exact reconstruction become

( ) ( ) ( ) ( ) ( ) ( )

and

( ) ( ) (

ω ω ω π ω π

ω ω π

− −

• = = =

⎡ ⎤+ =⎣ ⎦j j j j

j j

w n f n N R

P F e W e P F e W e

P F e W e

( ) ( )

2

2 2

1 0

0 1 114

( ) ( )

( )

) ( ) ( ) ( )

aliasing cancels exactly if ( ) ( ) (with ( ) ( ))giving

( ) ( )

ˆ( ) ( )

ω π ω π

ω ω

ω ω π ω

− −

− −

⎡ ⎤+ =⎣ ⎦

• = − = =

⎡ ⎤− =⎢ ⎥⎣ ⎦= −

j j

j j

j j j M

P F e W e

P P F e W e

W e W e e

y n x n M

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Decimated Analysis-Synthesis Filter Bank

16

1( )

0

1( ) [ ] ( )

( ) ( ) ( )

when aliasing is completely eliminated (by suitable choice of filters),the overall analysis-synthesis system has frequency response:

where

ω ωω

ω ω ω

−−

=

=

=

∑%

%

k

Njj

ek

j j je

H e P k W eRN

W e W e F e

h

1

0

[ ] [ ] [ ]

[ ] [ ] [ ]1[ ] [ ]

where

ω−

=

=

= ∗

= ∑ k

e

eN

j n

k

n w n p n

w n w n f n

p n P k eRN

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Decimated Analysis-Synthesis Filter Bank

• To design a decimated analysis/synthesis filter bank system, need to determine the following:1. the number of channels, N, and the decimation/interpolation

ratio R. (Usually determined by the desired frequency resolution)

2. the window, w[n], and the interpolation filter impulse response, f [n]. (These are lowpass filters that should have good frequency-selective properties such that the bandpass channel responses do not overlap into more than one band on either side of the channel).

3. the complex gain factors, P[k], k=0,1,…,N-1. (These constants are important for achieving flat overall response and alias cancellation.)

17

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Maximally Decimated Filter Banks

• we show here that it is possible to obtain exact reconstruction with R=N and N<L.– R=N is termed maximal decimation because this is the

largest decimation factor that can be used with an N-channel filter bank and still achieve exact reconstruction.

– nominal bandwidth of the bandpass channel signal is increased from ±π/N to ±π

– down-sampling by N accomplishes frequency down-shifting normally accomplished by modulation

18

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Maximally Decimated Filter Banks

19

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Analysis-Synthesis Filter Bank (all modulators removed)

20

Page 21: Digital Speech Processing— Lecture 10 Short-Time … speech...Short-Time Fourier Analysis Methods - Filter Bank Design 2 Review of STFT 1 123 0 2 1 ˆˆ ˆ ˆ ˆ ˆ ˆ.() [][]ˆ

Two Channel Filter Banks

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Two Channel Filter Banks

22

0 0 1 1

( ) ( )0 0 1 1

2; , 0,1

1( ) ( ) ( ) ( ) ( ) ( )21 ( ) ( ) ( ) ( )2

For this two-band system, we have: Applying the frequency domain analysis we get:

ω ω ω ω ω ω

ω ω π ω ω π

ω π

− −

= = = =

⎡ ⎤= +⎣ ⎦

⎡+ +

k

j j j j j j

j j j j

R N k k

Y e G e H e G e H e X e

G e H e G e H e ( )

( ) ( )0 0 1 1

( )

1/ [0], [1]

( ) ( ) ( ) ( ) 0

not including multiplier or the gain factors Aliasing can be eliminated completely by choosing the filters such that:

(ali

ω π

ω ω π ω ω π

− −

⎤⎣ ⎦

⎡ ⎤+ =⎣ ⎦

j

j j j j

X e

N P P

G e H e G e H e

0 0 1 11 ( ) ( ) ( ) ( ) 1 (2

asing cancelation)

Perfect reconstruction requires:

flat gain condition)ω ω ω ω⎡ ⎤+ =⎣ ⎦j j j jG e H e G e H e

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Quadrature Mirror Filter (QMF) Banks

23

( )1 0 1 0

0 0 0 0

( )1 1 1 0

[ ] [ ] ( ) ( )

[ ] 2 [ ] ( ) 2 ( )

[ ] 2 [ ] ( ) 2 ( )

Alias cancellation achieved if the filters are chosen to satisfythe conditions:

Note that the

j n j j

j j

j j

h n e h n H e H e

g n h n G e H e

g n h n G e H e

π ω ω π

ω ω

ω ω π

= ⇔ =

= ⇔ =

= − ⇔ = −

0[ ]re is only a single lowpass filter, with impulse response

on which all other filters are based; filter has nominal cutoff of /2 rad with a narrow transition to a stopband with adequate attenuat

h nπ

1 0

0 1

[ ] [ ][ ] [ ]

ionto isolate the two bands The filter is the complementary highpass filter The filters and are called Quadrature Mirror Filters (QMF)

j nh n e h nh n h n

π=

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Quadrature Mirror Filter (QMF) Banks

24

( ) ( )

( ) ( ) ( 2 )0 0 0 0

2 2( )0 0

2 ( ) ( ) 2 ( ) ( ) 0

( ) ( ) ( )

Substituting for the filter relationships we get: i.e., perfect alias cancellation for any choice of lowpass filter

ω ω π ω π ω π

ω ω ω π

− − −

− =

= −

j j j j

j j j

H e H e H e H e

Y e H e H e

( ) ( )2 2( )0 0

( ) ( ) ( )

( ) ( )

For perfect reconstruction we require that:

i.e., flat magnitude with delay of samples, due to the delayof the causal filters of the filte

ω ω ω

ω ω π ω− −

⎡ ⎤ =⎢ ⎥⎣ ⎦

⎡ ⎤− =⎢ ⎥⎣ ⎦

%j j j

j j j M

X e H e X e

H e H e e

Mr bank

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Quadrature Mirror Filters

25

0 0

0 0

[ ] [( 1) ], 0 1;( 1) / 2

( ) (

Assume FIR lowpass filter of length samplessuch that thisfilter has linear phase with a delay of samplesand a frequency response of the form: j j

Lh n h L n n L

L

H e A eω

= − − ≤ ≤ −−

=

( ) ( )

( 1)/2

0

2 2( 1) ( )0 0

) ( 1)

( )

( ) ( ) ( )

must be oddwhere is real. Using this lowpass filter in thefilter bank, the condition for perfect reconstruction becomes:

j L

j

j j j L j j

e L

A e

H e A e e A e e

ω ω

ω

ω ω π ω π ω

− −

− − −

⎡ ⎤= −⎢ ⎥⎣ ⎦%

( ) ( )

( 1)

2 2( ) ( 1)0 0

1

( ) ( ) ( )

If is odd, we get:

linear phase is guaranteed, but not perfect reconstruction

L

j j j j L

L

H e A e A e eω ω ω π ω

− − −

⎡ ⎤= +⎢ ⎥⎣ ⎦%

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Quadrature Mirror Filters

26

( ) ( )2 2( )0 0( ) ( ) 1

For flat gain we need to satisfy the condition:

Solution is to use high order linear phase QMF filters Johnston proposed an algorithm for design of the basic

lowpass fi

j jA e A eω ω π−⎡ ⎤+ =⎢ ⎥⎣ ⎦

lter that minimized the deviation from unity while providing a desired level of stopband attenuation

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Quadrature Mirror Filters

0 0[ ] [ ] [ ]= =h n w n g n 1 1[ ] [ ] [ ]π= =j nh n w n e g n

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28

Quadrature Mirror Filters

• practical realization of QMF filters

• note that aliasing is cancelled, but the overall frequency response is not perfectly flat

Lowpass filter designed by windowing

Highpass filter is mirror image of lowpass filter

Page 29: Digital Speech Processing— Lecture 10 Short-Time … speech...Short-Time Fourier Analysis Methods - Filter Bank Design 2 Review of STFT 1 123 0 2 1 ˆˆ ˆ ˆ ˆ ˆ ˆ.() [][]ˆ

Quadrature Mirror Filters

29

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30

• The motivation for the subband approach is to process different speech properties independently in each band and thus being able to localize the band (compact bandwidth) is important.

• Also,based on the original motivation of short-time analysis, temporal focus (within a limited time duration) is also important.

• Filter bank design is thus result of a trade-off between perfect reconstruction and bandwidth concentration in a joint criterion:

• When processing (e.g., quantization, a non-linear operation) is involved, harmonic distortions are inevitable and perfect reconstruction cannot be achieved, as seen in the simple example.

• There exists a design procedure (Smith-Barnwell) for QMF filters that attempts to meet the design constraints as best as possible

Issues with Perfect Reconstruction

∫∫ −−+=π ωπ

ω

ω ωαωα0

22 )|)(|1()1(|)(| deTdeWJ jj

s

Stop-band leakage Deviation from PR

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31

1. Start with a sequence of length, say, 2L-1 (L even) that satisfiesthat is, even samples=0, except at n=0.

, the value of the sequence at n=0, needs to be large enough to satisfy the positivity condition (see below).

2. Factorsuch that all roots of are inside the unit circle.

3. Positivity condition:4. The mirror channel:

5. To eliminate aliasing:

Smith-Barnwell QMF Design

{ } )()()()( 100

−== zWzWngzG Z

0C);(2)()1()( 0 nCngng n δ=−+

)( zW

0)()()()(2

000 >== − ωωωω jjjj eWeWeWeG

1,)()( 01 −=−−= −− LNeeWeW Njjj ωωω

)()1()(ly equivalentor )()( 011

01 nNwnwzzWzW nN −−=−−= −−

)()1()(ly equivalentor )()( 1010 nwnfzWzF n−=−=)()1()(ly equivalentor )()( 0101 nwnfzWzF n−−=−−=

)()( and )()( 0110 zWzFzWzF −−=−=

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32

Smith-Barnwell QMF Design - Example

condition positivity esatisfy th toadded is 0.1 0,at =n

12length has )( then ,)( ofLength 0 −= LngLnw7,,2,1,0,)2/sin()( ±±±== Ln

nnng

ππ

0.0455]- 0.0000 0.0637 0.0000- 0.1061- 0.0000 0.3183 0.6 0.3183 0.0000 0.1061- 0.0000- 0.0637 0.0000 -0.0455[)( =ng

L = 8, N = L – 1 = 7

choose

{ } )()()()( 100

−== zWzWngzG Z

Factorize )()()( 100

−= zWzWzG

0.1461]- 0.1182 0.1590 0.2032- 0.2332- 0.3426 0.8091 1[)(0 =nw

circleunit theinside roots all with length :(z)0 LW

1]- 0.8091 0.3426- 0.2332- 0.2032 0.1590 0.1182- -0.1461[)()1()( 01 =−−= nNwnw n

1] 0.8091 0.3426 0.2332- 0.2032- 0.1590 0.1182 -0.1461[)()1()( 10 =−= nwnf n

0.1461]- 0.1182- 0.1590 0.2032 0.2332- 0.3426- 0.8091 1[)()1()( 01 −=−−= nwnf n

all arrays start with n=0

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33

QMF Frequency Responses (Example)

Normalized frequency (xπ)

1050

-5-10-15-20

20

-2-4-6-8

-10-12

Mag

nitu

de (d

B)

Pha

se(d

egre

e, x

100)

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Tree-Structured Filter Banks

34

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Tree-Structured Filter Banks

35

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Sampling Pattern for Tree-Structured Filter Banks

36

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Tree-Structured Filter Banks

37

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38

Tree-Structured QMF Filterbank

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39

Quantization noise is concentrated in the band that it is generated in.

Quantized Channel Signals

][ˆ0 ny

][ˆ1 ny

][ˆ ny

]1[ −nδ

]1[ −nδ][nδ

][nδ

][0 nh 2↓ 2↑ ][0 ng

][1 nh 2↓ 2↑ ][1 ng

][0 ne

][1 ne

0 1

2 20 1ˆ ( ) ( ) ( )ω ω ωσ σ= +j j j

e eyP e G e G e

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40

Subband Coding

• advantages of subband coding

• each subband can be encoded according to the perceptual criterion appropriate to that band

• quantization noise can be confined to the band that produces it

• low energy bands can be encoded so as to produce less noise

• applicable to coding in the 9.6-16 Kbps range

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41

Practical Example of Subband Coder

• subband coder using tree-structured filer bank (Crochiere, 1981)

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42

Subband Coder Subjective Quality

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43

Subband Coder Waveforms

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44

Concept of Time-Frequency Resolution• When interested in the changing characteristics of the signal,

short time Fourier transform or short time spectral analysis is used.

12222

12222

|)(||)(|)(

|)(||)(|)(

−∞

∞−

∞−

−∞

∞−

∞−

⎥⎦⎤

⎢⎣⎡•−=

⎥⎦⎤

⎢⎣⎡•−=

∫∫

∫∫

ωωωωωωσ

σ

ω dGdG

dttgdttgtt

c

ct

:spreadFrequency

:spread Temporal

τττω ωτdetgftF j∫∞

∞−

−−= )()(),(time. in focus providing function window the is )(tg

Continuous prolate spheroidal wave function: a bandlimited signal that have the highest energy concentration in a specified time interval.

.5.0)(=ωσσ t

tg product bandwidth-time minimum the achieves it Gaussian, is If

• To maintain a similar level of uncertainty, the window function should be short to provide time resolution for high frequency components and long to provide frequency resolution for low frequency components.

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45

Wavelet-Based MethodsAnalysis Filters Synthesis Filters

Band 1 ↓ R Q Q-1 ↑ R Band 1

Band 2 ↓ R Q Q-1 ↑ R Band 2

Band N ↓ R Q Q-1 ↑ R Band N

. . . . . . . .

Subband method

Analysis Filters Synthesis Filters

↓ R Q Q-1 ↑ R

↓ R Q Q-1 ↑ R

↓ R Q Q-1 ↑ R

Wavelet method

WaveletTransform

InverseTransform

What is in the Wavelet Transform block? What is the difference?

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46

Wavelet Transform• Built on set of expansion (or basis) functions that are

derived from a mother wavelet through scaling and translation:

∞<Ψ

= ∫∞

0

20 |)(| ωωω

ψ dC

)}({)}({ ,, tt aa ττ ψωψω F)( and F)( Let =Ψ=Ψ

: waveletmother a be Let )(tψ ⎟⎠⎞

⎜⎝⎛ −

=a

ta

taτψψ τ

1)(,

Continuous Wavelet Transform:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎞⎜⎜⎝

⎛ −== ∫∫

∞−

∞− af

adt

attf

adtttfaF aw

τψττψψτ τ***

, *)(1)(1)()(),(

Inverse transform: ∫ ∫∞

∞−

∞−= 2, )(),(1)(

adadtaF

Ctf aw

τψτ τψ

00 =Ψ )(

analysis for signal the compress or stretch∫∞

∞−⎟⎠⎞

⎜⎝⎛ −= dt

atatf

aaFw

τψτ *)(1),(

mspectrogra scalogram; == 22 |),(||),(| τωτ FaFw

Admissibility condition:

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47

Wavelet and Fourier Transform

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −

= ∫∞

∞− af

adt

attf

aaFw

τψττψτ ** *)(1)(1),(

)}({)}({ ,, tt aa ττ ψωψω F)( and F)( =Ψ=Ψ )}({ tfF F)( =ω

τωττ ωωτψψ j

aa eaaa

ta

t −Ψ=Ψ⇔⎟⎠⎞

⎜⎝⎛ −

= )()(1)( ,,

)()(*)(1)),((),( * ωωτψττω aFaa

fa

aFaW w Ψ=⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛−== FF

transform. waveletthe of tionrepresenta domainfrequency the is ),( ωaW

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48

Wavelet and Its Spectrum

)/( atΨ

σ

aa <1,σ

10, << aaσ0ω

a/0ω

a/0ω

a/ωσ

)(tΨ

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49

Scaling and Translation• For discrete wavelet transform,

mm anaa −− == 000 ττ and

,12 00 == τ and If a

and

m, n are integers

Znmntaatt mmnma ∈−== ,)()()( 00

2/0,, τψψψ τ

Znmntt mmnm ∈−= ,)2(2)( 2/

, ψψ

wavelets.affine called is it complete, is set the If )}({ , tnmψ

∫∞

∞−−== dtntatfawaF mm

nmw )()(),( 002/

0, τψτ

∑∑=m n

nmnm twtf )()( ,, ψ

*, ,( ) ( ) ( ) ( )Orthogonality condition : m n p qt t dt m p n qψ ψ δ δ

−∞= − −∫

The orthogonality condition may not be easy to satisfy for a general set of “wavelets;” but it is still possible to invert discrete wavelet transform, sometimes.

dyadic or octave sampling

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50

STFT and CWT

0ω 05ω03ω 08ω04ω0ω 02ω

time

frequ

ency

timefre

quen

cy

Constant-Q bandwidthConstant bandwidth

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51

Discrete Wavelet Transforms

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52

Other Simpler Transforms or Filter Banks

• Short-time analysis can also be viewed as data transformation, executed in a block-by-block manner, using for example:– Discrete Cosine Transform– Discrete Sine Transform– Filter bank based on DFT/FFT

• General remarks:– Discussion so far aims at a filter bank framework that allows perfect

reconstruction (i.e., no loss of information in the original signal if desired);

– But many speech processing systems do not require perfect reconstruction; rather, one may want to focus on “extraction” of key parameters or features from the speech signal;

– Therefore, other analysis methods (e.g., transformations that do not have inverse) may still apply.

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53

DFT and DCTN-point DFT (Discrete Fourier Transform)

Relates to real part of 2N-point DFT when the signal is symmetric

implies

2N-pointsymmetric

2N-point DFTN-point DCT

Possible excessive discontinuity at edges

∑∑−

=

=

− =→−==1

0

12

0

)/2( )/2cos()(2)()2()( and )()(N

n

N

n

knNj NknnxkXnNxnxenxkX ππ

∑−

=

−=1

0

)/2()()(N

n

knNjenxkX π

But, be careful about the point of symmetry!!

∑∑∞

−∞=

=

+==r

N

k

knNjkNj rNnxeeXN

nx )()(1)(~ 1

0

)/2()/2( ππ

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54

even around and odd around ; is even around and odd around .

Discrete Cosine Transforms

This image cannot currently be displayed.

∑−

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +=

1

0 21cos)(

N

nk n

Nknxc π

∑−

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++=

1

1 21cos)()0(

21 N

nk k

Nnnxxc π

DCT-I ( )2

1

1 (0) ( 1) ( 1) ( ) cos2 1

Nk

kn

knc x x N x nNπ−

=

⎛ ⎞= + − − + ⎜ ⎟−⎝ ⎠∑

DCT-II

Most common DCT; equivalent (up to a scale factor of 2) to a DFT of 4N real inputs of even symmetry with the even-indexed elements set to zero; i.e., an artificial shift of half sample in the original sampling setup.

DCT-III(IDCT)

0x Nx21

−=kkc21

−= Nk

Still more variants.

Extensively used in audio, image and video

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55

Discrete Sine Transform

1,,2,1,0,,1

)1)(1(sin1

2−=

+++

+= Nji

Nij

Nsij L

π

[ ] 10,

−== N

jiijsS

Relates to imaginary part of ~2N-point DFT

Sequence correspondsTo 8-point DFT

3-point sequence 8-point anti-

symmetric sequence

2-D case; eliminate one dimension for 1-D

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56

Discrete (Time) Fourier Transform - Revisited• The DTFT (discrete time Fourier transform) of an N-

point sequence is

• Sample the DTFT at

• The result is the DFT (Discrete Fourier Transform)

• If we compute the inverse DFT, we obtain

ω k = (2π/N)k, k = 0,1,K, N −1.

.continuous is Frequency ω∑−

=

−=1

0)()(

N

n

njj enxeX ωω

∑∑∞

−∞=

=

+==r

N

k

knNjkNj rNnxeeXN

nx )()(1)(~ 1

0

)/2()/2( ππ

)()()(1

0

)/2(/2 kXenxeX

N

n

knNjNk

j == ∑−

=

−=

ππω

ω

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57

DFT for Power Spectral Density Estimation

∑−

=

−=1

0)()(

N

n

njj enxeX ωω

ω

)(ωkP

2)()( ωω jeXS =

: kth channel processing (weighting) function

kPSS kk ∀= ),()()( ωωω

ωω

Examples of non-uniform filter banks

Perfect reconstruction may not be a requirement.