1

Digital Speech Processing— Lecture 10

Short-Time Fourier Analysis Methods - Filter

Bank Design

2

Review of STFT 1

1 2 3 0

2

1

ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

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

j n

j n

X e x m w n m e

n

X e n

X e

n ˆ

ˆ ˆ

ˆˆed, variable; ( ) DTFT [ ] [ ]

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

Linear Filtering view filter bank implementation FBS i

ω

ω ω

ω ω

ω −

= = −⎡ ⎤⎣ ⎦ ⇒ ⇒

= = ∗

⇒ ⇒ ⇒

j n

j j n n

X e x m w n m

n n X e x n e w n

mplementation

3

Review of STFT

3

2 2

ˆ ˆ

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

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

Hamming Window: (Hz) sample

ω

ω

⇒

⇒

= ⇒

j n

j

S

x n X e

W e B B

FB L

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

⇒

SF L

w n L L

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

L L

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 L N

F N

5

Review of STFT H0(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 n k

N j j

k k

N

k k

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 lowpass interpolation 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=

1 1 2

0

( )/( ) ( )

aliasing addition

ω ω π −

−

=

= ∑ R

j j r R d R

r X e X e

R↑][nx [ / ] 0, 1, 2,...

[ ] 0 otherwiseu

x n R n x n

= ± ±⎧ = ⎨ ⎩

( ) ( ) imaging removal

ω ω=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 m rR rR

m W

N j j n

rR k N

j n rR

k

j n m

m W r k

X k X e w rR m x m e

y n P k Y e e N

P k X k f n rR e N

x m w rR m f n rR P k e N

ω ω

ω ω

ω

ω

−

∈

−

=

−

=

∞ −

∈ =−∞ =

•

= = −

=

= −

⎡ ⎤ = − −⎢ ⎥

⎣ ⎦

∑

∑

∑

∑ ∑ 1

0

1

0

1 1 ( )

recognizing that when ( ) ,

( )

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

( ) ( ) ( )

k

N

N j n m

k q

r

P k k

e n m qN N

y n x n n

w rR n qN f n rR q m n

ω δ

δ

−

− ∞ −

= =−∞

∞

=−∞

• = ∀

= − −

• = ∀

− + − = ⇒ =

∑

∑ ∑

∑

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 n k k

j j j j k k

N jj

k k

R j j R 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 e N

H e X e R

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

N j j

k k k

R Nj jjR R k k

k

j j

P k G e H e RN

X e P k G e H e RN

H e X e

ω ω

π πω ωω

ω ω

−

=

− −− −

= =

+ ⋅

= ⋅ +

∑

∑ ∑ l l

l

%

13

Filter Bank Reconstruction

1

0

21

0

1 1

1 0 1 2 ( )

ˆ conditions for perfect reconstruction ( ( ) ( ))

( ) ( ) ( ) ( )

and

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

k

k

j j

N jj j

k k k

N jj R k k

k

Y e X e

H e P k G e H e RN

P k G e H e R RN

ω ω

ωω ω

πωω

−

=

− −

=

• =

= =

= =

∑

∑ l

%

l

Flat Gain

Alias Cancellation

Filter Bank Reconstruction

14

15

Decimated Analysis- Synthesis Filter Bank

2

1 0 1 1 4

1 0 4

( ) ( )

( )

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 1 1 4

( ) ( )

( )

) ( ) ( ) ( )

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

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

N jj

e k

j j j e

H e P k W e RN

W e W e F e

h

1

0

[ ] [ ] [ ]