multirate signal procesing -filterbanks
Post on 04-Jun-2018
228 Views
Preview:
TRANSCRIPT
-
8/13/2019 Multirate Signal Procesing -FilterBanks
1/21
-
8/13/2019 Multirate Signal Procesing -FilterBanks
2/21
Filter Banks (FBs)
Collection of filters, with common i/p or o/
Analysis FB: Splits a signalx(n)intoMscalled subband signals (Fig of AFB and F
Synthesis FB: CombineMsubband signsingle signal x(n)(Fig of SFB)
Freq. response could be marginally overlnon-overlapping or very much overlappin
-
8/13/2019 Multirate Signal Procesing -FilterBanks
3/21
DFT Relations
Based on DFT Relations for understandin
DFT Relations:X(k) =
M1m=0 x(m)e
2mk
M
x(m) = 1MM1
k=0 X(k)e2mk
M
X[k] =M1m=0 x(m)W
km
W =e2M
Wkm is aM MDFT matrixIn futureW refers toWkm
W is complex conjugate ofW
In case of DFT matrix,W
T
=W = W
W is transpose-conjugate ofW
W1 =W/M
x(m) = 1MM1
k=0 X(k)Wkm
-
8/13/2019 Multirate Signal Procesing -FilterBanks
4/21
DFT Filter Bank
To provide FB interpretation consider the
below
-
8/13/2019 Multirate Signal Procesing -FilterBanks
5/21
DFT FB (contd.)
We are interested in splitting i/p signalx(
parallel signalsxk(n)where,k= 0, 1, . . . (
Consider a case where fromx(n)we gensequencessi(n)by passingx(n)throughso thatsi(n) =x(n i)
From definition ofW = xk(n) =
M1i=0
Each o/p is connected to all i/ps via DFT
Above relation is same as IDFT except fo
Z.T. ofxk(n)is given byXk(z) =M1
i=0 SiSubstituting forSi(z), we have
Xk(z) =M1
i=0 ziX(z)Wki =
M1i=0 X(
-
8/13/2019 Multirate Signal Procesing -FilterBanks
6/21
DFT FB (contd.)
Xk(z) =Hk(z)X(z), whereHk(z)is define
H0(z) = 1 + z1 +z2 + . . .+z(M1)
System is equivalent to analysis bank witfiltersHk(z)
H0(z) = |H0(ej
)| = |sin(M /2)/sin(/H1(e
j) =H0(ejej2/M) =H0(e
j(2
In general,Hk(z)has response
Hk(ej) =H0(e
j(w (2k/M))), which is
version ofH0(ej
)Thus we haveManalysis filters which arshifted versions ofH0(z)
-
8/13/2019 Multirate Signal Procesing -FilterBanks
7/21
DFT FB. (contd.)
For synthesis purpose consider block con
matrix followed by delay elements
-
8/13/2019 Multirate Signal Procesing -FilterBanks
8/21
DFT FB. Contd.
I/p to this block arexk(n)from analysis b
Each o/p element is givenyl(n) =M1
k=0
Final o/p of synthesis bank
X(z) =M1
l=0 Yl(z)z((M1)l)
X(z) =
M1l=0 (
M1k=0 Xk(z)Wkl)z((M1)M1
l=0 (M1
k=0 Xk(z)(z((M1)l)Wkl))
X(z) =M1
l=0 Xk(z)Fk(z)and
Fk(z) =F0(z((M1)l)Wkl),
F0(z) =z(M1) + z(M2) +. . .+ z2 + z
-
8/13/2019 Multirate Signal Procesing -FilterBanks
9/21
DFT Fb. (contd.)
ConsiderH0(z)andF0(z)for illustration
s0(n)undergoes no delay inH0(z)and he
be delayed byz(M1) inF0(z)
s1(n)undergoesz1 delay inH0(z)and h
be delayed byz(M2) inF0(z)
sM1(n)undergoesz(M1) delay inH0(z
needs to be delayed by1inF0(z)
By this process, in effective all elements asame amount and hence possible to rege
combining them
Thus ifFk(z) = 1/Hk(z), then X(z) =X(z
-
8/13/2019 Multirate Signal Procesing -FilterBanks
10/21
Computational Comple
Let each filter be of order N = N comp
Assumingx(n)to be real, each filter requmultiplications
There areMfilters and accordingly2M N
Can we reduce number of multiplications
ByPolyphase Structuresfor realization
-
8/13/2019 Multirate Signal Procesing -FilterBanks
11/21
Computational Complexity of Uniform DFT Filter Bank:
Let each filter have Ncomplex coefficients for FIR
implementations (except H0(Z)).
Assuming x[n]to be real, each filter requires2N realmultiplications for a total of2MN real multiplicationsper sample of x[n].
So an inefficient realization.
Polyphase realization can significantly reduce the com-
putational complexity.
Polyphase Realization of FIR Filters:
Reference: Bellanger et.al., Digital filtering by polyphase
network: Application to sample-rate alteration and filter banks,
IEEE Trans. ASSP, vol. 34,no. 2,PP. 109-114, April 1976.
An important advancement in multi rate DSP is the
invention of polyphase representation.
Provides computationally efficient implementations of
decimation/interpolation filters, as well as filter banks.
For a filter of length M, the polyphase structureconsists of IFIR filters in parallel, where M is selected
to be integer multiple of I, i.e, Imust divide M.
10-1
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
12/21
-
8/13/2019 Multirate Signal Procesing -FilterBanks
13/21
Z1
x(n) y(n)
E (Z )
E (Z )
0
1
2
2
H(Z) = (h(0) + h(3).Z3) + (h(1).Z1 +h(4).z4) + (h(2).Z2 + h(5).z5)
Define:E0(Z) =h(0) + h(3).Z1
E1(Z) =h(1) + h(4).Z1
E2(Z) =h(2) + h(5).Z1
H(Z) =E0(Z3) + z1.E1(Z
3) + z2.E2(Z3)
10-3
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
14/21
Z1
Z1
x(n) y(n)H (Z )
H (Z )
H (Z )
3
3
3
0
1
2
In general, for a FIR filter of length M, a polyphase
representation withIparallel filters is given by:
H(Z) =E0(ZI) + Z1E1(Z
I) + Z2E2(ZI) + +
Z(I1)E(I1)(ZI)
H(Z) =(I1)
i=0ZiEi(Z
I)
Where,
E0(Z) =
h(0)+h(I)Z1
+h(2I)Z2
+ +h((k1)I)Z(k1)
E1(Z) =h(1) + h(I+ 1)Z1 + h(2I+ 1)Z2 +
+ h((k 1)I+ 1)Z(k1)
10-4
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
15/21
.
.
.
EI1(Z) =h(I 1) + h((I 1) + I)Z1 + h(2I+
(I 1))Z2 + + h((k 1)I+ (I 1))Z(k1)
Where, k=
M
I is an integer.
In general,
Ei(Z) =(k1)j=0
h(i +jI).zj 0 i (I 1)
Polyphase Realization of Uniform DFT Filter Bank
Polyphase realization can significantly reduce the
computational complexity of the uniform filter bank.
Decomposing H0(Z)in a polyphase shown as:
H0(Z) =
(M1)i=0
ziEi(zM) (1)
where, Ei(zM)is ith polyphase filter.
Ei(Z)is formed by picking every Mth coefficient of
H0(Z)starting from ith element.
10-5
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
16/21
Ei(Z) =h0[i] + h0[i + M].z1 + h0[i +
2M].z2 + .........+ h0[i + (M 1)M]z(M1)
Ei(Z) =M1
m=0
h0[i + mM]zm
(2)
kth filter in the uniform DFT filter bank is related to the
prototype filter by the relation
Hk(Z) =H0(z.ej2kM ) (3)
From equations(3)and(1)
Hk(Z) =H0
(z.ej2kM )
=(M1)i=0
(z.ej2kM )iEi((z.e
j2kM )M)
Hk(Z) =(M1)i=0
(zi.ej2kiM )Ei((z
M.ej2k))
10-6
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
17/21
Hk(Z) =
(M1)i=0
(zi.ej2kiM )Ei(z
M) (4)
H0(Z) =(M1)i=0
zi.Ei(zM)
H0(Z) =E0(zM) + z1.E1(z
M) +z2.E2(zM) + + zM1.EM1(zM)
H1(Z) =(M1)
i=0
zi.ej2kM .Ei(z
M)
H1(Z) =E0(zM) + z1.e
j2M .E1(z
M) + +
z(M1).ej2(M1)
M .E(M1)(zM)
Hk(Z) =D.E(zM)
10-7
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
18/21
where,
Hz(Z) =
H0(Z)H1(Z)
.
.
.
H(M1)(Z)
.
E(ZM
) =
E0(ZM)
z1.E1(ZM)
...
z(M1).EM1(ZM)
.
D=
1 1 1 1
1 e
j2M
e
j2.2M
e
j2(MM
.
.
.
1 ej2(M1)
M ej2(M1).2
M ej2(M
M
IDFT:
x[n] = 1M.(M1)k=0
X[k].ej2knM 0 n M1
10-8
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
19/21
D is essentially an inverse DFT matrix except for1M
.
If a polyphase structure is implemented using
E0(Z), E1(Z), , EM1(Z), then we must
perform an inverse DFT of the outputs in order to getthe signals Hk(Z)for k= 0, 1, , (M 1).
IDFT will perform 1M.D. Therefore, original signal must
be pre-multiplied by M.
Z 1
Z 1
E (ZM )
E (ZM )
E (ZM )
E (ZM )
M
M
M
M
x(n)0
1
2
M1
IDFT
X [0]
X [1]
X [2]
X [M1]
M
IDFT is simply a set of multipliers, and so the
10-9
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
20/21
down-samplers can be moved to precede the IDFT.
Furthermore, the down-sampling can be moved before
the filter using the identity
E(zM)( M) = ( M).E(Z)
This results in an efficient realization of uniform DFT
filter bank.
Z 1
Z 1
M
M
M
M
x(n) X [0]
X [1]
X [2]
X [M1]
M
IDFT
0
1E (Z
2
M1E (Z
E (Z )
)
E (Z)
)
10-10
w.jntuworld.com
-
8/13/2019 Multirate Signal Procesing -FilterBanks
21/21
Computational Complexity Of Polyphase Implementation
Of Uniform Dft Filter Bank:
Let H(Z)is an N-point filter.
Each polyphase filter therefore has NM
points.
For Mbranches, there is a total of Ncomplex
multipliers required for the filters M. NM
=N
IDFT: 2M. log2M real multipliers.
Therefore, Total multipliers =2N+ 2M. log2M.
Further there is a down sampler before every filter.
Therefore, number of real multipliers=
2 NM
+ 2. log2M
Comparison: let N=160 and M=16
Direct: 2MN=2*16*160=5120multipliers/sample
Polyphase : 2 NM
+ 2. log2M
= 216016 + 2. log216
= 20 + 2. log22
4
= 20 + 2.4= 23 multipliers/sample
Saving: 512023 = 230 times.
10-11
w.jntuworld.com
top related