Ployphase Decomposition

7/27/2019 Ployphase Decomposition

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1Copyright 2001, S. K. Mitra

Polyphase Decomposition

The Decomposition

Consider an arbitrary sequence {x[n]} with

az-transformX(z) given by

We can rewriteX(z) as

where

n

nznxzX][)(

1

0M

k

M

k

kzXzzX )()(

n

n

n

n

kk zkMnxznxzX ][][)(

10 Mk


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The subsequences are called the

polyphase components of the parent

sequence {x[n]} The functions , given by the

z-transforms of , are called the

polyphase componentsofX(z)

]}[{ nxk

]}[{ nxk

)(zXk


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The relation between the subsequences

and the original sequence {x[n]} are given

by

In matrix form we can write

]}[{ nxk

10 MkkMnxnxk ],[][

)(

)(

)(

....)( )(

M

M

M

M

M

zX

zX

zX

zzzX

1

10

111 ....


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4 Copyright 2001, S. K. Mitra


A multirate structural interpretation of the

polyphase decomposition is given below


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Thepolyphase decomposition of an FIR

transfer function can be carried out by

inspection

For example, consider a length-9 FIR

transfer function:

8

0n

nznhzH ][)(


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Its 4-branch polyphase decomposition is

given by

where

)()()()()( 4

3

34

2

24

1

14

0zEzzEzzEzzEzH

210 840

zhzhhzE ][][][)(

11 51

zhhzE ][][)(1

2 62 zhhzE ][][)(

13 73

zhhzE ][][)(


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Thepolyphase decomposition of an IIR

transfer functionH(z) =P(z)/D(z) is not that

straight forward

One way to arrive at anM-branch polyphase

decomposition ofH(z) is to express it in the

form by multiplyingP(z) and

D(z) with an appropriately chosenpolynomial and then apply anM-branch

polyphase decomposition to

)('/)(' MzDzP

)(' zP


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Polyphase Decomposition Example -Consider

To obtain a 2-band polyphase decomposition werewriteH(z) as

Therefore,

where

1

1

31

21

z

zzH )(

2

1

2

2

2

21

11

11

91

5

91

61

91

651

)31)(31(

)31)(21()(

z

z

z

z

z

zz

zz

zzzH

)()()( 2112

0 zEzzEzH

11

1

91

51

91

610

zz

zzEzE )(,)(


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Note:The above approach increases the

overall order and complexity ofH(z)

However, when used in certain multirate

structures, the approach may result in a

more computationally efficient structure

An alternative more attractive approach isdiscussed in the following example


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Example -Consider the transfer function of

a 5-th order Butterworth lowpass filter with

a 3-dB cutoff frequency at 0.5p:

It is easy to show thatH(z) can be expressedas

1 5

2 4

0.0527864 (1 )

1 0.633436854 0.0557281( )

z

z zH z

2

2

2

2

5278601

5278601

10557301

1055730

2

1

z

z

z

zzzH

.

.

.

.)(


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ThereforeH(z) can be expressed as

where

1

1

5278601

527860

2

11

z

zzE

.

.)(

1

1

10557301

1055730

2

10

z

zzE

.

.)(

)()()( 2112

0 zEzzEzH


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Note: In the above polyphase decomposition,

branch transfer functions arestable

allpass functions

Moreover, the decomposition has not

increased the order of the overall transfer

functionH(z)

)(zEi


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FIR Filter Structures Based on

Polyphase Decomposition We shall demonstrate later that a parallel

realization of an FIR transfer functionH(z)

based on the polyphase decomposition canoften result in computationally efficient

multirate structures

Consider theM-branch Type I polyphasedecomposition ofH(z):

)()(1

0M

k

M

k

k zEzzH


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Polyphase Decomposition A direct realization ofH(z)based on the

Type I polyphase decomposition is shown

below


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Polyphase Decomposition The transpose of the Type I polyphase FIR

filter structure is indicated below


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Polyphase Decomposition An alternative representation of the

transpose structure shown on the previous

slide is obtained using the notation

Substituting the above notation in the Type

I polyphase decomposition we arrive at theType II polyphase decomposition:

)()(1

0)1( MM M zRzzH

10),()( 1 MzEzRM

MM


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Polyphase Decomposition A direct realization ofH(z)based on the

Type II polyphase decomposition is shown

below


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Computationally Efficient

Decimators Consider first the single-stage factor-of-M

decimator structure shown below

We realize the lowpass filterH(z) using the

Type I polyphase structure as shown on the

next slide

M][nx )(zH ][nyv[n]


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Computationally EfficientDecimators

Using the cascade equivalence #1 we arriveat the computationally efficient decimatorstructure shown below on the right

Decimator structure based on Type I polyphase decomposition

y[n] y[n]x[n]x[n]

][nv


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Decimators To illustrate the computational efficiency of

the modified decimator structure, assume

H(z) to be a length-Nstructure and the inputsampling period to beT= 1

Now the decimator outputy[n] in the

original structure is obtained by down-sampling the filter outputv[n]by a factor of

M


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Decimators It is thus necessary to computev[n] at

Computational requirements are thereforeN

multiplications and additions per

output sample being computed

However, asnincreases, stored signals in

the delay registers change

...,2,,0,,2,... MMMMn

)1( N


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Decimators Hence, all computations need to be

completed in one sampling period, and for

the following sampling periods thearithmetic units remain idle

The modified decimator structure also

requiresNmultiplications andadditions per output sample being computed

)1( N

)1( M


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Decimators and Interpolators However, here the arithmetic units are

operative at all instants of the output

sampling period which isMtimes that ofthe input sampling period

Similar savings are also obtained in the case

of the interpolator structure developed usingthe polyphase decomposition


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Computationally EfficientInterpolators

Figures below show the computationally

efficient interpolator structures

Interpolator based onType I polyphase decomposition

Interpolator based onType II polyphase decomposition


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Decimators and Interpolators More efficient interpolator and decimator

structures can be realized by exploiting the

symmetry of filter coefficients in the case oflinear-phase filtersH(z)

Consider for example the realization of a

factor-of-3(M= 3) decimator using alength-12 Type 1linear-phase FIR lowpass

filter


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Decimators and Interpolators The corresponding transfer function is

A conventional polyphase decomposition of

H(z) yields the following subfilters:

54321 ]5[]4[]3[]2[]1[]0[)( zhzhzhzhzhhzH

11109876 ]0[]1[]2[]3[]4[]5[ zhzhzhzhzhzh

3210 ]2[]5[]3[]0[)( zhzhzhhzE

3211 ]1[]4[]4[]1[)(

zhzhzhhzE321

2 ]0[]3[]5[]2[)( zhzhzhhzE


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Decimators and Interpolators Note that still has a symmetric

impulse response, whereas is the

mirror image of These relations can be made use of in

developing a computationally efficient

realization using only 6 multipliers and 11two-input adders as shown on the next slide

)(1 zE

)(0 zE

)(2 zE


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Decimators and Interpolators Factor-of-3decimator with a linear-phase

decimation filter


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29 Copyright 2001 S K Mitra

A Useful Identity The cascade multirate structure shown

below appears in a number of applications

Equivalent time-invariant digital filter

obtained by expressingH(z) in itsL-term

Type I polyphase form

is shown below

L LH(z)x[n] y[n]

)(10L

kL

kk zEz

x[n] y[n])(0 zE

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Ployphase Decomposition

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