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U. Glaeser

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FIGURE 27.22 Filter banks: (a) analysis filter bank, (b) synthesis filter bank.<br />

where T s is the sampling period. The respective transformation of the analog frequency ω a into the digital<br />

frequency ω d is given by<br />

Filter Banks<br />

© 2002 by CRC Press LLC<br />

x(n)<br />

H0<br />

( z)<br />

1( ) z H<br />

...<br />

HM − 1( z)<br />

M = 32<br />

M<br />

M<br />

...<br />

M<br />

G0<br />

( z)<br />

G1(<br />

z)<br />

1( ) z GM −<br />

(27.47)<br />

A filter bank is a collection of digital filters with a multiple input and/or a multiple output [52]. The<br />

filter bank with one input and M outputs is referred to as the analysis filter bank. On the other hand, the<br />

synthesis filter bank consists of M inputs and one output (cf., Fig. 27.22). Splitting of the input signal<br />

into decimated subbands via an analysis filter bank and then reconstructing the initial signal from<br />

subband signals with a respective synthesis filter bank is referred to as the subband coding (SBC) technique<br />

commonly used for lossless or nearly lossless data compression.<br />

A filter bank in the main path of the MPEG-1 audio coder [39] consists of 32 subband filters with a<br />

normalized bandwidth of π/(32T s), where T s is the input audio signal sampling period. The impulse<br />

responses of particular filters in this filter bank are defined as<br />

(27.48)<br />

where h(n) is an impulse response of the prototype lowpass filter. In the analysis filter bank case, the<br />

output signal in ith subband is defined as a convolution<br />

...<br />

(a) (b)<br />

π<br />

−<br />

2M<br />

(c)<br />

j<br />

2<br />

( e )<br />

Ω<br />

H<br />

ω d<br />

π<br />

2M<br />

π<br />

M<br />

M<br />

M<br />

...<br />

Ω<br />

M<br />

H0 1 H H M −1<br />

=<br />

2π π<br />

M<br />

2<br />

---- arctan<br />

Ts T ⎛ s<br />

----ω ⎞<br />

⎝ a<br />

2 ⎠<br />

( 2i + 1)<br />

( n – 16)π<br />

Hi( n)<br />

= h( n)<br />

cos----------------------------------------- 64<br />

511<br />

∑<br />

Si( m)<br />

=<br />

x( m– n)<br />

∗ Hi( n)<br />

n=0<br />

+<br />

+<br />

x( ) ^ n<br />

Ω<br />

(27.49)

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