Adsp 09 Msp Filterbanks Ec623 Adsp

8/10/2019 Adsp 09 Msp Filterbanks Ec623 Adsp

1/21

Multirate Signal ProcesFilter Banks

S. R. M. Prasanna

Dept of ECE,

IIT Guwahati,

[email protected]
http://prosper.sourceforge.net/http://prosper.sourceforge.net/


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


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,WT =W = W

W is transpose-conjugate ofW

W1 =W/M

x(m) = 1MM1

k=0 X(k)Wkm


4/21


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(


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)


7/21

DFT FB. (contd.)

For synthesis purpose consider block con

matrix followed by delay elements


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) =M1

l=0 (M1

k=0 Xk(z)Wkl)z((M1)M1l=0 (

M1k=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


9/21

DFT Fb. (contd.)

ConsiderH0(z)andF0(z)for illustration

s0(n)undergoes no delay in H0(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


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


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


12/21


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


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


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


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


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


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


19/21

Dis 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


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


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

Adsp 09 Msp Filterbanks Ec623 Adsp

Documents