Polyphase Decomposition

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PolyphasePolyphase DecompositionDecompositionThe DecompositionThe Decomposition

• Consider an arbitrary sequence{ x[n]} with a z-transformX(z) given by

• We can rewriteX(z) as

where

∑∞−∞=

−= nnznxzX ][)(

∑ −=

−= 10

Mk

Mk

k zXzzX )()(

∑∑ ∞−∞=

−∞−∞=

− +== nn

nn

kk zkMnxznxzX ][][)(

10 −≤≤ Mk

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PolyphasePolyphase DecompositionDecomposition

• The subsequences are called the polyphasepolyphase componentscomponents of the parent sequence{ x[n]}

• The functions , given by the z-transforms of , are called thepolyphasepolyphase componentscomponents of X(z)

]}[{ nxk

]}[{ nxk

)(zX k

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PolyphasePolyphase DecompositionDecomposition• The relation between the subsequences

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

• In matrix form we can write

]}[{ nxk

10 −≤≤+= MkkMnxnxk ],[][

[ ]

=

−

−−−

)(

)(

)(

....)( )(

MM

M

M

M

zX

zX

zX

zzzX

1

1

0

111 ....

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• A multirate structural interpretation of the polyphase decomposition is given below

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• The polyphasepolyphase decomposition of an FIR decomposition of an FIR transfer functiontransfer function can be carried out by inspection

• For example, consider a length-9 FIR transfer function:

∑=

−=8

0n

nznhzH ][)(

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PolyphasePolyphase DecompositionDecomposition• Its 4-branch polyphase decomposition is

given by

where

)()()()()( 43

342

241

140 zEzzEzzEzzEzH −−− +++=

210 840 −− ++= zhzhhzE ][][][)(

11 51 −+= zhhzE ][][)(

12 62 −+= zhhzE ][][)(

13 73 −+= zhhzE ][][)(

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PolyphasePolyphase DecompositionDecomposition• The polyphasepolyphase decomposition of an IIR decomposition of an IIR

transfer functiontransfer function H(z) = P(z)/D(z) is not that straight forward

• One way to arrive at an M-branch polyphasedecomposition of H(z) is to express it in the form by multiplying P(z) and D(z) with an appropriately chosen polynomial and then apply an M-branch polyphase decomposition to

)('/)(' MzDzP

)(' zP

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PolyphasePolyphase DecompositionDecomposition•• ExampleExample - Consider

• To obtain a 2-band polyphase decomposition we rewriteH(z) as

• Therefore,

where

1

1

31

21−

−

+−=

z

zzH )(

2

1

2

2

2

21

11

11

91

5

91

61

91651

)31)(31(

)31)(21()( −

−

−

−

−

−−

−−

−−

−−

−+

−+−

−+−− +===

z

z

z

z

z

zz

zz

zzzH

)()()( 21

120 zEzzEzH −+=

11

1

91

5191

610 −−

−

−−

−+ ==

zz

z zEzE )(,)(

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• Note:The above approach increases the overall order and complexity ofH(z)

• However, when used in certain multiratestructures, the approach may result in a more computationally efficient structure

• An alternative more attractive approach is discussed in the following example

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PolyphasePolyphase DecompositionDecomposition•• ExampleExample - Consider the transfer function of

a 5-th order Butterworth lowpass filter witha 3-dB cutoff frequency at0.5π:

• It is easy to show thatH(z) can be expressed as

1 5

2 4

0.0527864 (1 )

1 0.633436854 0.0557281( ) z

z zH z

−

− −+

+ +=

+

= −

−

−

−

++−

++

2

2

2

2

5278601

5278601

10557301

105573021

z

z

z

z zzH.

.

.

.)(

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• Therefore H(z) can be expressed as

where

= −

−

++

1

1

5278601

52786021

1 z

zzE.

.)(

= −

−

++

1

1

10557301

105573021

0 z

zzE.

.)(

)()()( 21

120 zEzzEzH −+=

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• Note: In the above polyphase decomposition, branch transfer functions are stable stable allpassallpass functionsfunctions

• Moreover, the decomposition has not increased the order of the overall transfer function H(z)

)(zEi

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FIR Filter Structures Based on FIR Filter Structures Based on PolyphasePolyphase DecompositionDecomposition

• We shall demonstrate later that a parallel realization of an FIR transfer functionH(z) based on the polyphase decomposition can often result in computationally efficient multirate structures

• Consider theM-branch Type I Type I polyphasepolyphasedecompositiondecomposition of H(z):

)()( 10

Mk

Mk

k zEzzH ∑ −=

−=

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• A direct realization ofH(z) based on the Type I Type I polyphasepolyphase decompositiondecomposition is shown below

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• The transposeof the Type I polyphase FIR filter structure is indicated below

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• 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 Type II polyphasepolyphase decompositiondecomposition:

)()( 10

)1( MM M zRzzH ll

l∑ −=

−−−=

10),()( 1 −≤≤= −− MzEzR MM

Mlll

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• A direct realization ofH(z) based on the Type II Type II polyphasepolyphase decompositiondecomposition is shown below

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Computationally Efficient Computationally Efficient DecimatorsDecimators

• Consider first the single-stage factor-of-Mdecimator 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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• Using the cascade equivalence #1cascade equivalence #1 we arrive at the computationally efficient decimator structure 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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• To illustrate the computational efficiency of the modified decimator structure, assumeH(z) to be a length-N structure and the input sampling 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 ofM

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• It is thus necessary to computev[n] at

• Computational requirements are thereforeNmultiplications and additions per output sample being computed

• However, asn increases, stored signals in the delay registers change

...,2,,0,,2,... MMMMn −−=

)1( −N

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• Hence, all computations need to be completed in one sampling period, and for the following sampling periods the arithmetic units remain idle

• The modified decimator structure also requiresN multiplications and additions per output sample being computed

)1( −N

)1( −M

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Computationally Efficient Computationally Efficient Decimators and InterpolatorsDecimators and Interpolators

• However, here the arithmetic units are operative at all instants of the output sampling period which isM times that of the input sampling period

• Similar savings are also obtained in the case of the interpolator structure developed using the polyphase decomposition

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Computationally Efficient Computationally Efficient InterpolatorsInterpolators

• Figures below show the computationally efficient interpolator structures

Interpolator based on Type I polyphase decomposition

Interpolator based on Type II polyphase decomposition

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• More efficient interpolator and decimator structures can be realized by exploiting the symmetry of filter coefficients in the case of linear-phase filtersH(z)

• Consider for example the realization of afactor-of-3(M = 3) decimator using alength-12 Type 1linear-phase FIR lowpassfilter

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• The corresponding transfer function is

• A conventional polyphase decomposition ofH(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[)( −−− +++= zhzhzhhzE

3212 ]0[]3[]5[]2[)( −−− +++= zhzhzhhzE

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• 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 only6 multipliers and11 two-input adders as shown on the next slide

)(1 zE)(0 zE

)(2 zE

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• Factor-of-3decimator with a linear-phase decimation filter

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A Useful IdentityA Useful Identity• The cascade multirate structure shown

below appears in a number of applications

• Equivalent time-invariant digital filter obtained by expressingH(z) in its L-term Type Ipolyphase form is shown below

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

)(10

Lk

Lk

k zEz∑ −=

−

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

Polyphase Decomposition

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