Module_3_1.pdf

Subband Coding and Concept of
Filterbank
Module-III
1
Dr. Anil Kumar
Assistant Professor, Electronic & Communication Engineering

• Introduced by R. Crochiere in 1976 (speech coding)
• Frequency domain technique
• Used in speech coding in 1977 for speech coder
• Block Diagram
Subband Coding
Block diagram of Subband Coding system

• Filter bank: Subband coding applications such as
speech coding, Image compression
• Transmultiplexer System: Communication system
Types of Subband Coding
Transmultiplexer System

• Advantage of subband coding
 Quantization noise is localized
 Quantization step size vary independently
in subbands.
 Bit allocation can be done on the basis of
energy content of signal due to subband
coding.
Subband Coding (Cont…)
Fourier Transform a signal x(n)

• Disadvantage of subband coding
 Aliasing Distortion: Due to sub sampling
operations
 Phase Distortion: Due to imperfect phase
responses of the analysis /synthesis filters
Quantization noise is localized
 Amplitude distortion: Due to imperfect
frequency responses of analysis / synthesis
filters
 Concept of Quadrature Mirror filter (QMF) bank
(By A. Croisier in 1976)
 First QMF bank used for speech coding in 1977
( Esteban and Galand)
Subband Coding (Cont…)

Multirate Systems
• A multirate system is a bank of low pass,
bandpass and high ass filters which covers a
band in the frequency spectrum.
• The possible components of multirate system
include down sampler, up sampler and delay
elements.
• These systems operate in two modes either in
analysis /synthesis mode or synthesis /analysis
mode
6

Digital Filter Banks
• The digital filter bank is set of bandpass
filters with either a common input or a
summed output
• An M-band analysis filter bank is shown
below
7

8
• The subfilters Hk (z) in the analysis filter
bank are known as analysis filters
• The analysis filter bank is used to
decompose the input signal x[n] into a set of
subband signals vk[n] with each subband
signal occupying a portion of the original
frequency band

• An L-band synthesis filter bank is shown
below
• It performs the dual operation to that of the
analysis filter bank
9

1
0
• The subfilters Fk (z) in the synthesis filter
bank are known as synthesis filters
• The synthesis filter bank is used to combine
k
belonging to contiguous frequency bands)
into one signal y[n] at its output
a set of subband signals v
^ [n] (typically

Uniform Digital Filter Banks
1
1
n0 h0[n]zn
H0(z)  
• A simple technique to design a class of
filter banks with equal passband widths
• Let H0 (z) represent a causal lowpassdigital
filter with a real impulse responseh0[n]:
• The filter H0 (z) is assumed to be anIIR
filter without any loss of generality



0 2
s
p 
• Assume that H0 (z) has its passband edge
p and stopband edge s around /M,
where M is some arbitrary integer, as
indicated below
M
1
2

1
3
• Now, consider the transfer function Hk (z)
whose impulse response hk[n] is givenby
hk[n]  h0[n]ej2kn/M  h0[n]Wkn,
M
0  k  M 1
where we have used the notationWM  e j2/M
• Thus,

 n
k
n
Hk (z)  n hk[n]z  n h0[n]zWM  ,
0  k  M

1
4
• i.e.,
Hk (z)  H0 (zW k ), 0  k  M 1
M
• The corresponding frequency response is
given by
Hk (e j)  H0 (e j(2k/M )), 0  k  M 1
• Thus, the frequency response of Hk (z) is
obtained by shifting the response of H0 (z)
to the right by an amount 2k/M

• The responses of Hk (z) , Hk (z) , . . . , Hk (z)
are shown below
1
5

10
• Note: The impulse responses hk[n] are, in
general complex, and hence |Hk (e j)| does
not necessarily exhibit symmetry with
respect to  = 0
• The responses shown in the figure of the
previous slide can be seen to be uniformly
shifted version of the response of the basic
prototype filter H0 (z)

obtained is called a uniform filter bank
11
• The M filters defined by
M
could be used as the analysis filters in the
analysis filter bank or as the synthesis filters
in the synthesis filter bank
• Since the magnitude responses of all M
filters are uniformly shifted version of that
of the prototype filter, the filter bank
Hk (z)  H0 (zW k ), 0  k  M 1

12
Uniform DFT Filter Banks
Polyphase Implementation
• Let the prototype lowpass transfer function
be represented in its M-band polyphase
form:

0 
M 1  M
0 z E (z )
H (z) 


n0
n0
n
h0[  nM ]zn,
e[n]z 
E(z) 
where E(z) is the -th polyphase
component of H0 (z):
0    M

19
• In deriving the last expression we have used
• Substituting z with zW k in the expression
 M
k z W
0  M
1  k M kM
E (z W )
H (z) 
M
for H0 (z)we arrive at the M-band polyphase
decomposition of Hk (z):
M 
0 M 
 M 1zW kE (zM ), 0  k  M 1
M
the identity W kM 1

20
• The equation on the previous slide can be
written in matrix form as
....
M
M
M
k
W k
W 2k
H (z) [1




 

W (M 1)k
]


z(M 1)EM 1(zM )
2 M
1
z 1
E (z M
)
E0(zM )
0k M 1
z E
.2(z )
..

21
• All M equations on the previous slide can
be combined into one matrix equation as
M D
 1
• In the above D is the M  M DFT matrix






(zM )
M 1
2 M
1
z 1
E (z M
)
E0(zM )
z E
.2(z )
..
 


z(M1)E
 









2  
1
1
1 1 1
1)
M M
M
M
M M
M 1
W 2(M 
W4
W2
W ( M 1)  
W 1 W 2

(z)
H (z)   1
H1(z)
H0 (z) 
M
.
.
.
M
.
.
M
.
.
.
.
...
...
...
1 W (M 1) W 2(M 1)... W (M 1)
...
2
.
 .
H

22
• An efficient implementation of the M-band
uniform analysis filter bank, more
commonly known as the uniform DFT
analysis filter bank, is then as shown below

23
• The computational complexity of an M-band
uniform DFT filter bank is much smaller than
that of a direct implementation as shown
below

24
• Following a similar development, we can
derive the structure for a uniform DFT
synthesis filter bank as shown below
Type I uniform DFT
synthesis filter bank
Type II uniform DFT

25
IIR transfer function H0(z)
• The above equation can be used to
determine the polyphase components of an










M
(z )

M 1
z(M1)E
z E1(z )
1 M
E0(zM )
1


 
 H (z) 
HM 1(z)
H1(z)
0


M
D  H
.2(z) 
..
z2E
.2(zM )
..
• Now Ei (zM ) can be expressed in terms of

Reference:
S. K. Mitra, “Digital Signal Processing: A
Computer Based Approach” Mc Graw Hill

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