1

Digital Speech ProcessingLecture 10

Short-Time Fourier Analysis Methods - Filter

Bank Design

2

Review of STFT1

1 2 3 0

2

1

. ( ) [ ] [ ]

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

( ) defined for , , ,...;

. Interpretations of ( ). fix

=

=

=

j j mnm

jn

jn

X e x m w n m e

n

X e n

X e

n

ed, variable; ( ) DTFT [ ] [ ]

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

Linear Filtering view filter bank implementation FBS i

= =

= =

jn

j j nn

X e x m w n m

n n X e x n e w n

mplementation

3

Review of STFT

3

22

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

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

Hamming Window: (Hz) sample

=

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

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

5

Review of STFTH0(ej)

H1(ej)

HN-1(ej)

+

X(ej) Y(ej)

1

0

1

0

1

0

[ ] [ ]

( ) ( )

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

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

need to design digital filters that match criteria for exact reconstr

=

=

=

=

=

= =

= = =

=

= = + +

%

%

%

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

Tree-Decimated Filter Banks

6

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

filter (window) which is moved in jumps of R

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.

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.

10

Review of Down and Up-Sampling

R][nx[ ] [ ]dx n x nR=

11 2

0

( )/( ) ( )

aliasing addition

=

= R

j j r Rd R

rX e X e

R][nx[ / ] 0, 1, 2,...

[ ]0 otherwiseu

x n R nx n

= =

( ) ( ) imaging removal

=j j RuX e X e

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

=

=

= =

= =

=

=

=

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

= =

=

=

=

=

+ = =

Filter Bank Reconstruction

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

=

=

= =

= =

=

=

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

=

= =

+

= +

l l

l

%

13

Filter Bank Reconstruction

1

0

21

0

1 1

1 0 1 2( )

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

=

=

=

= =

= =

l

%

l

Flat Gain

Alias Cancellation

Filter Bank Reconstruction

14

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

( ) ( ) (

= = =

+ = 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

( ) ( )

( ) ( )

+ =

= = =

= =

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

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

=

=

=

%

%

k

Njj

ek

j j je

H e P k W eRN

W e W e F e

h

1

0

[ ] [ ] [ ]

[ ] [ ] [ ]1[ ] [ ]

where

=

=

=

= k

e

eN

j n

k

n w n p n

w n w n f n

p n P k eRN

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

Maximally Decimated Filter Banks

we show here that it is possible to obtain exact reconstruction with R=N and N

Maximally Decimated Filter Banks

19

Analysis-Synthesis Filter Bank (all modulators removed)

20

Two Channel Filter Banks

21

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:

= = = =

= +

+ +

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

+ =

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) + = j j j jG e H e G e H e

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

= =

= =

= =

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

on which all other filters are based; fi