7/24/2019 D- Filter Bank
1/40
Copyright 2001, S. K. Mitra1
Digital Filter BanksDigital Filter Banks The digital filter bank is set of bandpass
filters with either a common input or a
summed output
AnM-band analysis filter bankis shownbelow
7/24/2019 D- Filter Bank
2/40
Copyright 2001, S. K. Mitra2
Digital Filter BanksDigital Filter Banks
The subfilters in the analysis filterbank are known as analysis filters
The analysis filter bank is used to
decompose the input signalx[n] into a set of
subband signals with each subband
signal occupying a portion of the originalfrequency band
)(zk
][nvk
7/24/2019 D- Filter Bank
3/40
Copyright 2001, S. K. Mitra3
Digital Filter BanksDigital Filter Banks
AnL-band synthesis filter bankis shownbelow
It performs the dual operation to that of the
analysis filter bank
7/24/2019 D- Filter Bank
4/40Copyright 2001, S. K. Mitra4
Digital Filter BanksDigital Filter Banks
The subfilters in the synthesis filterbank are known as synthesis filters
The synthesis filter bank is used to combine
a set of subband signals (typically
belonging to contiguous frequency bands)
into one signaly[n]at its output
)(zFk
][nvk^
7/24/2019 D- Filter Bank
5/40Copyright 2001, S. K. Mitra
5
Uniform Digital Filter BanksUniform Digital Filter Banks
A simple technique to design a class offilter banks with equal passband widths is
outlined next
Let represent a causal lowpass digital
filter with a real impulse response :
The filter is assumed to be an IIR
filter without any loss of generality
)(0 zH
!" #"$ #$ n nznhzH ][)( 00
][0nh
)(0 z
7/24/2019 D- Filter Bank
6/40Copyright 2001, S. K. Mitra
6
Uniform Digital Filter BanksUniform Digital Filter Banks
Assume that has its passband edgeand stopband edge around%/M,whereMis some arbitrary integer, as indicated below
)(0 z
&%0 2%
p&
s&
%
p&s&
7/24/2019 D- Filter Bank
7/40Copyright 2001, S. K. Mitra
7
Uniform Digital Filter BanksUniform Digital Filter Banks
Now, consider the transfer functionwhose impulse response is given by
where we have used the notation
Thus,
)(zk][nhk
,][][][0
/2
0
knnkj
k
Wnhenhnh #% $$
jeW /2%#$
10 #''k
( ) ,][][)( 0!! " #"$#"
#"$ # $$
n
nkn
nkk zWnhznhzH
10 #''k
7/24/2019 D- Filter Bank
8/40
Copyright 2001, S. K. Mitra8
Uniform Digital Filter BanksUniform Digital Filter Banks
i.e.,
The corresponding frequency response isgiven by
Thus, the frequency response of is
obtained by shifting the response of
to the right by an amount2%k/M
),()( 0k
k zWHzH $ 10 #''k
),()( )/2(0kjj
k eHeH %#&& $ 10 #''k
)(zk)(0 zH
7/24/2019 D- Filter Bank
9/40
Copyright 2001, S. K. Mitra9
Uniform Digital Filter BanksUniform Digital Filter Banks The responses of , , . . . ,
are shown below
)(zHk )(zHk )(zHk
7/24/2019 D- Filter Bank
10/40
Copyright 2001, S. K. Mitra10
Uniform Digital Filter BanksUniform Digital Filter Banks
Note:The impulse responses are, ingeneral complex, and hence does
not necessarily exhibit symmetry with
respect to&= 0
The responses shown in the figure of the
previous slide can be seen to be uniformlyshifted version of the response of the basic
prototype filter
][nhk|)(| &jk eH
)(0 zH
7/24/2019 D- Filter Bank
11/40
Copyright 2001, S. K. Mitra11
Uniform Digital Filter BanksUniform Digital Filter Banks
TheMfilters defined by
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 allM
filters are uniformly shifted version of that
of the prototype filter, the filter bank
obtained is called a uniform filter bank
),()( 0k
k zWHzH $ 10 #''k
7/24/2019 D- Filter Bank
12/40
Copyright 2001, S. K. Mitra12
Uniform DFT Filter BanksUniform DFT Filter BanksPolyphase Implementation
Let the prototype lowpass transfer functionbe represented in itsM-band polyphase
form:
where is the -th polyphase
component of :
! #$ #$ 100 MzEzzH ! !! )()(
)(z0
!
,][][)( !! "$#"
$# *$$
0 00 nn
nn znMhznezE !!!
)(zE!
10 #'' !
7/24/2019 D- Filter Bank
13/40
Copyright 2001, S. K. Mitra13
Uniform DFT Filter BanksUniform DFT Filter Banks
Substitutingzwith in the expressionfor we arrive at theM-band polyphase
decomposition of :
In deriving the last expression we have used
the identity
)(zH0
k
zW
! #$ ##$1
0kMM
MkMk WzEWzzH ! !
!! )()(
)(zk
101
0 #''$! #
$
##
MkzEWz Mk
M! !
!!
,)(
1$kMW
7/24/2019 D- Filter Bank
14/40
Copyright 2001, S. K. Mitra14
Uniform DFT Filter BanksUniform DFT Filter Banks
The equation on the previous slide can bewritten in matrix form as
].... )(
[)( kkk
k WWWzH12
1 ####
$
++
+++
,
-
..
.
..
/
0
###
#
#
)(
)(
)(
)(
)(
MM
M
M
M
M
zEz
zEz
zEz
zE
11
22
11
0
10 #''k
..
.
7/24/2019 D- Filter Bank
15/40
Copyright 2001, S. K. Mitra15
Uniform DFT Filter BanksUniform DFT Filter Banks AllMequations on the previous slide can
be combined into one matrix equation as
In the aboveDis the DFT matrix
M D1#
1
++++
+
,
-
.
.
..
.
/
0
#
##
#
#
)(
)(
)(
)(
)(
M
M
M
M
M
M
zEz
zEz
zEz
zE
1
1
22
11
0
...++++++
,
-
.
.
.
.
.
.
/
0
$
++++
+
,
-
.
.
.
.
.
/
0
######
####
####
# 21121
1242
121
1
21
0
1
1
1
1111
)()()(
)(
)(
)(
)( )(
)(
MMM
MMMM
MMMM
M WWW
WWW
WWW
zH
zH zH
zH
......
......
...
...
...
...
......
7/24/2019 D- Filter Bank
16/40
Copyright 2001, S. K. Mitra16
Uniform DFT Filter BanksUniform DFT Filter Banks An efficient implementation of theM-band
uniform analysis filter bank, morecommonly known as the uniform DFT
analysis filter bank, is then as shown below
7/24/2019 D- Filter Bank
17/40
Copyright 2001, S. K. Mitra17
Uniform DFT Filter BanksUniform DFT Filter Banks
The computational complexity of anM-banduniform DFT filter bank is much smaller than
that of a direct implementation as shown
below
7/24/2019 D- Filter Bank
18/40
Copyright 2001, S. K. Mitra18
Uniform DFT Filter BanksUniform DFT Filter Banks
For example, anM-band uniform DFTanalysis filter bank based on anN-tap
prototype lowpass filter requires a total of
multipliers
On the other hand, a direct implementation
requiresNMmultipliers
NMM *22 log
7/24/2019 D- Filter Bank
19/40
Copyright 2001, S. K. Mitra19
Uniform DFT Filter BanksUniform DFT Filter Banks Following a similar development, we can
derive the structure for a uniform DFTsynthesis filter bankas shown below
Type I uniform DFT Type II uniform DFTsynthesis filter bank synthesis filter bank
7/24/2019 D- Filter Bank
20/40
Copyright 2001, S. K. Mitra20
Uniform DFT Filter BanksUniform DFT Filter Banks Now can be expressed in terms of
The above equation can be used todetermine the polyphase components of an
IIR transfer function
+++
++
,
-
.
..
.
.
/
0
###
#
#
)(
)(
)(
)(
)(
MM
M
M
M
M
zEz
zEz
zEz
zE
11
2
21
10
1$
++
+
,
-
..
.
/
0
# )(
)(
)(
)(
zH
zH
zH
zH
1
2
1
0
D
......
)(i zE
)(zH0
7/24/2019 D- Filter Bank
21/40
Copyright 2001, S. K. Mitra21
Nyquist FiltrsNyquist Filtrs Under certain conditions, a lowpass filter
can be designed to have a number of zero-valued coefficients
When used as interpolation filters these
filters preserve the nonzero samples of theup-sampler output at the interpolator output
Moreover, due to the presence of thesezero-valued coefficients, these filters arecomputationally more efficient than other
lowpass filters of same order
7/24/2019 D- Filter Bank
22/40
Copyright 2001, S. K. Mitra22
LLthth-Band Filters-Band Filters
These filters, called theNyquist filtersor
Lth-band filters, are often used in single-rate
and multi-rate signal processing
Consider the factor-of-Linterpolator shown
below
The input-output relation of the interpolatorin thez-domain is given by
L][nx ][ny)(zH][nxu
)()()( L
zXzHzY $
7/24/2019 D- Filter Bank
23/40
Copyright 2001, S. K. Mitra23
LLthth-Band Filters-Band Filters
IfH(z) is realized in theL-band polyphaseform, then we have
Assume that thek-th polyphase component
ofH(z) is a constant, i.e., :
! #$ #$ 1
0
)()( L
i
Li
i zEzzH
2$)(zEk
)(...)()()( 1)1(
11
0L
kkLL zEzzEzzEzH #### ***$
)(...)( 1)1(
1)1( L
LLL
kk zEzzEz #
##*
*# ***
7/24/2019 D- Filter Bank
24/40
Copyright 2001, S. K. Mitra24
LLthth-Band Filters-Band Filters
Then we can expressY(z) as
As a result,
Thus, the input samples appear at the outputwithout any distortion for all values ofn,
whereas, in-between output samples
are determined by interpolation
!
#
$## *2$
1
0)()()()(
L
LLLk zXzEzzXzzY!
!!
k3!
][][ nxkLny 2$*
)1( #L
7/24/2019 D- Filter Bank
25/40
Copyright 2001, S. K. Mitra25
LLthth-Band Filters-Band Filters
A filter with the above property is called a
Nyquist filteror anLth-band filter
Its impulse response has many zero-valued
samples, making it computationally
attractive
For example, the impulse response of an
Lth-bandfilter fork= 0 satisfies thefollowing condition
456$][Lnh otherwise,0 0, $2 n
7/24/2019 D- Filter Bank
26/40
Copyright 2001, S. K. Mitra26
LLthth-Band Filters-Band Filters Figure below shows a typical impulse
response of athird-band filter(L= 3)
Lth-band filterscan be either FIR or IIR
filters
7/24/2019 D- Filter Bank
27/40
Copyright 2001, S. K. Mitra27
LLthth-Band Filters-Band Filters
If the0-thpolyphase component ofH(z) is a
constant, i.e., then it can be shown
that
(assuming2= 1/L)
Since the frequency response of isthe shifted version of ,
the sum of all of theseLuniformly shiftedversions of add up to a constant
2$)(0 zE
! #$ $2$10
1)(Lk
kL LzWH
)( kLzWH)( )/2( LkjeH %#& )( &jeH
)( &jeH
7/24/2019 D- Filter Bank
28/40
Copyright 2001, S. K. Mitra28
Half-Band FiltersHalf-Band Filters
AnLth-bandfilter forL= 2 is called ahalf-band filter
The transfer function of a half-band filter is
thus given by
with its impulse response satisfying
)()( 211 zEzzH #*2$
456$]2[ nh
otherwise,0
0, $2 n
7/24/2019 D- Filter Bank
29/40
Copyright 2001, S. K. Mitra29
Half-Band FiltersHalf-Band Filters
The condition
reduces to
(assuming2= 0.5)
IfH(z)has real coefficients, then
Hence
)()( 211 zEzzH #*2$
1)()( $#* zzH
)()( )( %& $# jj eHeH
1)()( )( $* %& jj eHeH
7/24/2019 D- Filter Bank
30/40
Copyright 2001, S. K. Mitra30
Half-Band FiltersHalf-Band Filters
and add upto1 for all7
Or, in other words, exhibits a
symmetry with respect to the half-band
frequency%/2, hence the name half-band
filter
)(
)2/( 7#%j
eH )(
)2/( 7*%j
eH
)( &jeH
7/24/2019 D- Filter Bank
31/40
Copyright 2001, S. K. Mitra31
Half-Band FiltersHalf-Band Filters Figure below illustrates this symmetry for ahalf-band lowpass filter for which passband
and stopband ripples are equal, i.e.,
and passband and stopband edges are
symmetric with respect to%/2, i.e.,
sp 8$8
%$&*& sp
7/24/2019 D- Filter Bank
32/40
Copyright 2001, S. K. Mitra32
Half-Band FiltersHalf-Band Filters
Attractive property:About50% of thecoefficients ofh[n] are zero
This reduces the number of multiplications
required in its implementation significantly
For example, ifN= 101,an arbitrary Type 1
FIR transfer function requires about50multipliers, whereas, aType 1half-band
filter requires only about25 multipliers
7/24/2019 D- Filter Bank
33/40
Copyright 2001, S. K. Mitra33
Half-Band FiltersHalf-Band Filters
An FIR half-band filter can be designedwith linear phase
However, there is a constraint on its length
Consider a zero-phase half-band FIR filter
for which , with
Let the highest nonzero coefficient beh[R]
][*][ nhnh #2$ 1|| $2
7/24/2019 D- Filter Bank
34/40
Copyright 2001, S. K. Mitra34
Half-Band FiltersHalf-Band Filters
ThenRis odd as a result of the condition
ThereforeR= 2K+1 for some integerK
Thus the length ofh[n] is restricted to be of
the form2R+1 = 4K+3 [unlessH(z) is a
constant]
456$]2[ nh
otherwise,00, $2 n
7/24/2019 D- Filter Bank
35/40
Copyright 2001, S. K. Mitra35
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
A lowpass linear-phaseLth-band FIR filter
can be readily designed via the windowed
Fourier series approach
In this approach, the impulse response
coefficients of the lowpass filter are chosen
as where is theimpulse response of an ideal lowpass filter
with a cutoff at %/Land w[n]is a suitablewindow function
][][][ nwnhnh LP 9$ ][nhLP
7/24/2019 D- Filter Bank
36/40
Copyright 2001, S. K. Mitra36
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
Now, the impulse response of an idealLth-
band lowpass filterwith a cutoff at
is given by
It can be seen from the above that
"''"#%%$ nn
LnnhLP ,)/sin(][
Lc /%$&
...,2,for0][ LLnnhLP ::$$
7/24/2019 D- Filter Bank
37/40
Copyright 2001, S. K. Mitra37
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
Hence, the coefficient condition of theLth-
band filter
is indeed satisfied
Hence, anLth-band FIR filtercan be
designed by applying a suitable window
w[n]to
45
6
$][Lnh otherwise,00, $2 n
][nhLP
7/24/2019 D- Filter Bank
38/40
Copyright 2001, S. K. Mitra38
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
There are many other candidates forLth-
band FIR filters
Program 10_8can be used to design an Lth-
band FIR filter using the windowed Fourierseries approach
The program employs the Hammingwindow
However, other windows can also be used
D i f Li Ph
7/24/2019 D- Filter Bank
39/40
Copyright 2001, S. K. Mitra39
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
Figure below shows the gain response of a
half-band filter of length-23 designed using
Program 10_8
0 0.2 0.4 0.6 0.8 1
-80
-60
-40
-20
0
&/%
Gain,
dB
fD i f Li Ph
7/24/2019 D- Filter Bank
40/40
Copyright 2001, S. K. Mitra40
Design of Linear-PhaseDesign of Linear-Phase
LLthth-Band Filters-Band Filters
The filter coefficients are given by
As expected,h[n] = 0for
;0]10[]10[;002315.0]11[]11[ $$##$$# hhhh
;0]8[]8[;005412.0]9[]9[ $$#$$# hhhh
;0]6[]6[;001586.0]7[]7[ $$##$$# hhhh
;0]4[]4[;003584.0]5[]5[ $$#$$# hhhh
;0]2[]2[;089258.0]3[]3[ $$##$$# hhhh;5.0]0[;3122379.0]1[]1[ $$$# hhh
10,8,6,4,2 :::::$n