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7/30/2019 Digital Filter Banks
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1Copyright 2001, S. K. Mitra
Digital 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 shown
below
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2Copyright 2001, S. K. Mitra
Digital Filter Banks
The subfilters in the analysis filter
bank 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
)(zHk
][nvk
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3Copyright 2001, S. K. Mitra
Digital Filter Banks
AnL-band synthesis filter bankis shown
below
It performs the dual operation to that of the
analysis filter bank
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4Copyright 2001, S. K. Mitra
Digital Filter Banks
The subfilters in the synthesis filter
bank are known assynthesis filters
The synthesis filter bank is used to combine
a set ofsubband signals (typically
belonging to contiguous frequency bands)
into one signaly[n] at its output
)(zFk
][nvk^
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5Copyright 2001, S. K. Mitra
Uniform Digital Filter Banks
A simple technique to design a class of
filter 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
][0 nh
)(0 zH
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6Copyright 2001, S. K. Mitra
Uniform Digital Filter Banks
Assume that has its passband edge
and stopband edge aroundp/M,whereM
is some arbitrary integer, as indicated below
)(0 zH
wp0 2p
pw sw
M
p
pw
sw
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7/30/2019 Digital Filter Banks
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8Copyright 2001, S. K. Mitra
Uniform Digital Filter Banks
i.e.,
The corresponding frequency response is
given by
Thus, the frequency response of is
obtained by shifting the response of
to the right by an amount2pk/M
),()( 0kMk zWHzH 10 Mk
),()( )/2(0Mkjj
k eHeHpww 10 Mk
)(zHk)(0 zH
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9Copyright 2001, S. K. Mitra
Uniform Digital Filter Banks
The responses of , , . . . ,
are shown below
)(zHk )(zHk )(zHk
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10Copyright 2001, S. K. Mitra
Uniform Digital Filter Banks
Note:The impulse responses are, in
general complex, and hence does
not necessarily exhibit symmetry withrespect tow = 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|)(| wjk eH
)(0 zH
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11 Copyright 2001, S. K. Mitra
Uniform 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
),()( 0kMk zWHzH 10 Mk
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12 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
Polyphase Implementation
Let the prototype lowpass transfer function
be represented in itsM-band polyphase
form:
where is the -th polyphase
component of :
1
00M MzEzzH
)()(
)(zH0
,][][)(
0 00 nn
nn znMhznezE
)(zE
10
M
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13 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
Substitutingzwith in the expression
for we arrive at theM-band polyphase
decomposition of :
In deriving the last expression we have used
the identity
)(zH0
kMzW
1
0M kM
MMk
Mk WzEWzzH )()(
)(zHk
101
0
MkzEWzM Mk
M
,)(
1kMMW
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14 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
The equation on the previous slide can be
written in matrix form as
].... )([)( kMMk
Mk
Mk WWWzH12
1
)(
)(
)(
)(
)(
MM
M
M
M
M
zEz
zEz
zEz
zE
11
22
110
10 Mk
..
.
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15 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
AllMequations on the previous slide can
be combined into one matrix equation as
In the aboveDis the DFT matrixMD
1
M
)(
)(
)(
)(
)(
M
M
M
M
M
M
zEz
zEz
zEz
zE
1
1
22
11
0
...
21121
1242
121
1
2
1
0
1
1
1
1111
)()()(
)(
)(
)(
)(
)()(
MM
MM
MM
MMMM
MMMM
M WWW
WWW
WWW
zH
zH
zHzH
.
.
.
.
.
.
.
.
.
.
.
.
...
...
...
...
.
.
.
.
.
.
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16 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
An efficient implementation of theM-band
uniform analysis filter bank, more
commonly known as the uniform DFT
analysis filter bank, is then as shown below
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17 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
The computational complexity of anM-band
uniform DFT filter bank is much smaller than
that of a direct implementation as shownbelow
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18 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
For example, anM-band uniform DFT
analysis filter bank based on anN-tap
prototype lowpass filter requires a total ofmultipliers
On the other hand, a direct implementation
requiresNMmultipliers
NMM 22log
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19 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
Following a similar development, we can
derive the structure for a uniform DFT
synthesis filter bankas shown below
Type I uniform DFT Type II uniform DFTsynthesis filter bank synthesis filter bank
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20 Copyright 2001, S. K. Mitra
Uniform DFT Filter Banks
Now can be expressed in terms of
The above equation can be used to
determine the polyphase components of an
IIR transfer function
)(
)(
)(
)(
)(
MM
M
M
M
M
zEz
zEzzEz
zE
11
22
1
10
M
1
)(
)(
)(
)(
zH
zH zH
zH
M 1
21
0
D.
.
.
.
.
.
)( Mi zE
)(zH0
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21 Copyright 2001, S. K. Mitra
Nyquist Filters
Under certain conditions, a lowpass filtercan be designed to have a number of zero-valued coefficients
When used as interpolation filters thesefilters 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 otherlowpass filters of same order
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22 Copyright 2001, S. K. Mitra
Lth-Band Filters
These filters, called theNyquist filtersorLth-band filters, are often used in single-rate and multi-rate signal processing
Consider the factor-of-L interpolator shownbelow
The input-output relation of the interpolatorin thez-domain is given by
L][nx ][ny)(zH][nxu
)()()( LzXzHzY
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23 Copyright 2001, S. K. Mitra
Lth-Band Filters
IfH(z) is realized in theL-band polyphase
form, then we have
Assume that thek-th polyphase component
ofH(z) is a constant, i.e., :
10 )()( Li Lii zEzzH
)(zEk
)(...)()()( 1)1(
11
0L
kkLL zEzzEzzEzH
)(...)( 1)1(
1)1( L
LLL
kk zEzzEz
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24 Copyright 2001, S. K. Mitra
Lth-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
1
0
)()()()(L
LLLk zXzEzzXzzY
k
][][ nxkLny
)1( L
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25 Copyright 2001, S. K. Mitra
Lth-Band Filters
A filter with the above property is called aNyquist filteror anLth-band filter
Its impulse response has many zero-valued
samples, making it computationallyattractive
For example, the impulse response of an
Lth-bandfilter fork= 0 satisfies thefollowing condition
][Lnh
otherwise,0
0, n
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26 Copyright 2001, S. K. Mitra
Lth-Band Filters
Figure below shows a typical impulse
response of athird-band filter(L = 3)
Lth-band filterscan be either FIR or IIR
filters
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27 Copyright 2001, S. K. Mitra
Lth-Band Filters
If the 0-thpolyphase component ofH(z) is aconstant, i.e., then it can be shown
that
(assuming = 1/L) Since the frequency response of is
the shifted version of ,
the sum of all of theseLuniformly shiftedversions of add up to a constant
)(0 zE
10 1)(
Lk kL LzWH
)( kLzWH
)( )/2( LkjeH pw )( wjeH
)( wjeH
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28 Copyright 2001, S. K. Mitra
Half-Band Filters
AnLth-bandfilter forL = 2 is called ahalf-
band filter
The transfer function of a half-band filter isthus given by
with its impulse response satisfying
)()( 211 zEzzH
]2[ nh
otherwise,00, n
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29 Copyright 2001, S. K. Mitra
Half-Band Filters
The condition
reduces to
(assuming = 0.5)
IfH(z) has real coefficients, then
Hence
)()( 211 zEzzH
1)()( zHzH
)()( )( wpw jj eHeH
1)()( )( wpw jj eHeH
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30 Copyright 2001, S. K. Mitra
Half-Band Filters
and add up
to 1 for allq
Or, in other words, exhibits asymmetry with respect to the half-band
frequencyp/2, hence the name half-band
filter
)( )2/( qpjeH )( )2/( qpjeH
)( wjeH
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31 Copyright 2001, S. K. Mitra
Half-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 aresymmetric with respect top/2, i.e.,
sp
pww sp
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32 Copyright 2001, S. K. Mitra
Half-Band Filters
Attractive property:About 50% of the
coefficients ofh[n] are zero
This reduces the number of multiplicationsrequired in its implementation significantly
For example, ifN= 101,an arbitrary Type 1
FIR transfer function requires about 50multipliers, whereas, aType 1half-band
filter requires only about 25 multipliers
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33 Copyright 2001, S. K. Mitra
Half-Band Filters
An FIR half-band filter can be designed
with 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 1||
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34
Half-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 form 2R+1 = 4K+3 [unlessH(z) is a
constant]
]2[ nhotherwise,0
0, n