polyphase decomposition - catholic university of...

Copyright © 2001, S. K. Mitra1

Polyphase Polyphase DecompositionDecompositionThe Decomposition• Consider an arbitrary sequence {x[n]} with

a z-transform X(z) given by

• We can rewrite X(z) as

where

∑∞−∞=

−= nnznxzX ][)(

∑ −=

−= 10

Mk

Mk

k zXzzX )()(

∑∑ ∞−∞=

−∞−∞=

− +== nn

nn

kk zkMnxznxzX ][][)(

10 −≤≤ Mk


PolyphasePolyphase Decomposition Decomposition

• The subsequences are called thepolyphase components of the parentsequence {x[n]}

• The functions , given by thez-transforms of , are called thepolyphase components of X(z)

]}[{ nxk

]}[{ nxk

)(zXk


PolyphasePolyphase Decomposition Decomposition• The relation between the subsequences

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

• In matrix form we can write

]}[{ nxk

10 −≤≤+= MkkMnxnxk ],[][

[ ]

=

−

−−−

)(

)(

)(

....)( )(

MM

M

M

M

zX

zXzX

zzzX

1

1

0111 ....



• A multirate structural interpretation of thepolyphase decomposition is given below



• The polyphase decomposition of an FIRtransfer function can be carried out byinspection

• For example, consider a length-9 FIRtransfer function:

∑=

−=8

0n

nznhzH ][)(


PolyphasePolyphase Decomposition Decomposition• Its 4-branch polyphase decomposition is

given by

where)()()()()( 4

334

224

114

0 zEzzEzzEzzEzH −−− +++=

210 840 −− ++= zhzhhzE ][][][)(

11 51 −+= zhhzE ][][)(

12 62 −+= zhhzE ][][)(

13 73 −+= zhhzE ][][)(


PolyphasePolyphase Decomposition Decomposition• The polyphase decomposition of an IIR

transfer function H(z) = P(z)/D(z) is not thatstraight forward

• One way to arrive at an M-branch polyphasedecomposition of H(z) is to express it in theform by multiplying P(z) andD(z) with an appropriately chosenpolynomial and then apply an M-branchpolyphase decomposition to

)('/)(' MzDzP

)(' zP


PolyphasePolyphase Decomposition Decomposition• Example - Consider

• To obtain a 2-band polyphase decompositionwe rewrite H(z) as

• Therefore,

where

1

1

3121

−

−

+−=

zzzH )(

2

1

2

2

2

21

11

11

915

9161

91651

)31)(31()31)(21()( −

−

−

−

−

−−

−−

−−

−−

−+

−+−

−+−− +===

zz

zz

zzz

zzzzzH

)()()( 21

120 zEzzEzH −+=

11

1

915

19161

0 −−

−

−

−

−+ ==

zzz zEzE )(,)(



• Note: The above approach increases theoverall order and complexity of H(z)

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

• An alternative more attractive approach isdiscussed in the following example


PolyphasePolyphase Decomposition Decomposition• Example - Consider the transfer function of

a 5-th order Butterworth lowpass filter witha 3-dB cutoff frequency at 0.5π:

• It is easy to show that H(z) can be expressedas

22

51

0557281063343685401105278640

−−

−

++

+=zz

zzH..

)(.)(

+

= −

−

−

−

+

+−+

+2

2

2

2

52786015278601

105573011055730

21

zz

zz zzH

.

.

.

.)(



• Therefore H(z) can be expressed as

where

= −

−

+

+1

1

5278601527860

21

1 zzzE

.

.)(

= −

−

+

+1

1

105573011055730

21

0 zzzE

.

.)(

)()()( 21

120 zEzzEzH −+=



• Note: In the above polyphase decomposition,branch transfer functions are stableallpass functions

• Moreover, the decomposition has notincreased the order of the overall transferfunction H(z)

)(zEi


FIR Filter Structures Based onFIR Filter Structures Based onPolyphasePolyphase Decomposition Decomposition

• We shall demonstrate later that a parallelrealization of an FIR transfer function H(z)based on the polyphase decomposition canoften result in computationally efficientmultirate structures

• Consider the M-branch Type I polyphasedecomposition of H(z):

)()( 10

Mk

Mk

k zEzzH ∑ −=

−=



• A direct realization of H(z) based on theType I polyphase decomposition is shownbelow



• The transpose of the Type I polyphase FIRfilter structure is indicated below



• An alternative representation of thetranspose structure shown on the previousslide is obtained using the notation

• Substituting the above notation in the TypeI polyphase decomposition we arrive at theType II polyphase decomposition:

)()( 10

)1( MM M zRzzH lll∑ −

=−−−=

10),()( 1 −≤≤= −− MzEzR MM

M lll



• A direct realization of H(z) based on theType II polyphase decomposition is shownbelow


Computationally EfficientComputationally EfficientDecimatorsDecimators

• Consider first the single-stage factor-of-Mdecimator structure shown below

• We realize the lowpass filter H(z) using theType I polyphase structure as shown on thenext slide

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



• 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



• To illustrate the computational efficiency ofthe modified decimator structure, assumeH(z) to be a length-N structure and the inputsampling period to be T = 1

• Now the decimator output y[n] in theoriginal structure is obtained by down-sampling the filter output v[n] by a factor ofM



• It is thus necessary to compute v[n] at

• Computational requirements are therefore Nmultiplications and additions peroutput sample being computed

• However, as n increases, stored signals inthe delay registers change

...,2,,0,,2,... MMMMn −−=

)1( −N



• Hence, all computations need to becompleted in one sampling period, and forthe following sampling periods thearithmetic units remain idle

• The modified decimator structure alsorequires N multiplications andadditions per output sample being computed

)1( −N

)1( −M


Computationally EfficientComputationally EfficientDecimators and InterpolatorsDecimators and Interpolators

• However, here the arithmetic units areoperative at all instants of the outputsampling period which is M times that ofthe input sampling period

• Similar savings are also obtained in the caseof the interpolator structure developed usingthe polyphase decomposition


Computationally EfficientComputationally EfficientInterpolatorsInterpolators

• Figures below show the computationallyefficient interpolator structures

Interpolator based on Type I polyphase decomposition

Interpolator based on Type II polyphase decomposition



• More efficient interpolator and decimatorstructures can be realized by exploiting thesymmetry of filter coefficients in the case oflinear-phase filters H(z)

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



• The corresponding transfer function is

• A conventional polyphase decomposition ofH(z) yields the following subfilters:

54321 ]5[]4[]3[]2[]1[]0[)( −−−−− +++++= zhzhzhzhzhhzH11109876 ]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



• Note that still has a symmetricimpulse response, whereas is themirror image of

• These relations can be made use of indeveloping a computationally efficientrealization using only 6 multipliers and 11two-input adders as shown on the next slide

)(1 zE)(0 zE

)(2 zE



• Factor-of-3 decimator with a linear-phasedecimation filter


A Useful IdentityA Useful Identity• The cascade multirate structure shown

below appears in a number of applications

• Equivalent time-invariant digital filterobtained by expressing H(z) in its L-termType I polyphase formis shown below

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

)(10

Lk

Lk

k zEz∑ −=

−

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


Arbitrary-Rate Sampling RateArbitrary-Rate Sampling RateConverterConverter

• The estimation of a discrete-time signalvalue at an arbitrary time instant between aconsecutive pair of known samples can besolved by using some type of interpolation

• In this approach an approximatingcontinuous-time signal is formed from a setof known consecutive samples of the givendiscrete-time signal



• The value of the approximating continuous-time signal is then evaluated at the desiredtime instant

• This interpolation process can be directlyimplemented by designing a digitalinterpolation filter


Ideal Sampling RateIdeal Sampling RateConverterConverter

• In principle, a sampling rate conversion byan arbitrary conversion factor can beimplemented as follows

• The input digital signal is passed through anideal analog reconstruction lowpass filterwhose output is resampled at the desiredoutput rate as indicated below



• Let the impulse response of the analoglowpass filter is denoted by

• Then the output of the filter is given by

• If the analog filter is chosen to bandlimit itsoutput to the frequency range ,its output can then be resampled at therate

)(tga

2/'Tg FF <

∑∞−∞= −= l ll )(][)( Ttgxtx aa

^

)(txa^

'TF



• Since the impulse response of an ideallowpass analog filter is of infinite durationand the samples have to becomputed at each sampling instant,implementation of the ideal bandlimitedinterpolation algorithm in exact form is notpractical

• Thus, an approximation is employed inpractice

)(tga

)'( TnTga l−



• Problem statement: Given inputsignal samples, x[k], k = , obtainedby sampling an analog signal at

, determine the sample value at time instant

where• Figure on the next slide illustrates the

interpolation process by an arbitrary factor

112 ++ NN21 NN ,...,−

)(txainkTt += 0

][)( αykTtx ina =+0 inkTtt += 0'

ktt =

21 NN ≤≤− α



• We describe next a commonly employedinterpolation algorithm based on a finiteweighted sum of input samples


LagrangeLagrange Interpolation InterpolationAlgorithmAlgorithm

• Here, a polynomial approximation to is defined as

where are the Lagrange polynomialsgiven by

)(txa

)(txa^

][)()( knxtPtxN

Nkka += ∑

−=

2

1

^

)(tPk

212

1

NkNtttttP

N

N k

kk ≤≤−

−−= ∏

−=,)(

l lk≠l



• Example - Design a fractional-rateinterpolator with an interpolation factor of3/2 using a 3rd-order polynomialapproximation with and

• The output y[n] of the interpolator is thuscomputed using

21 =N 12 =N

]1[)(]2[)(][ 12 −α+−α= −− nxPnxPny]1[)(][)( 10 +α+α+ nxPnxP



• Here, the Lagrange polynomials are givenby

)()( 361

6)1()1(

2 α+α−==α−

−αα+α−P

)2()( 2321

2)1()2(

1 α−α+α==α −αα+α−P

)22()( 2321

2)1)(1)(2(

0 −α−α+α−==α−

−α+α+αP

)3()( 2361

6)1)(2(

1 α+α+α==α−

α+α+αP



• Figure below shows the locations of thesamples of the input and the output for aninterpolator with a conversion factor of 3/2

• Locations of the output samples y[0], y[1],and y[2] in the input sample domain aremarked with an arrow

Input sample indexOutput sample index

↓ ↓ ↓



• From the figure on the previous slide it canbe seen that the value of α for computationof y[n], to be labeled , is 0

• Substituting this value of α in theexpressions for the Lagrange polynomialcoefficients derived earlier we get

,,

0α

0)( 02 =α−P 0)( 01 =α−P1)( 00 =αP 0)( 01 =αP



• The value of α for computation of y[n+1],to be labeled , is 2/3

• Substituting this value of α in theexpressions for the Lagrange polynomialcoefficients we get

,,

1α

0617.0)( 12 =α−P 2963.0)( 11 −=α−P7407.0)( 10 =αP 4938.0)( 11 =αP



• The value of α for computation of y[n+2],to be labeled , is 4/3

• Substituting this value of α in theexpressions for the Lagrange polynomialcoefficients we get

,,

2α

1728.0)( 22 −=α−P 7407.0)( 21 =α−P2963.1)( 20 −=αP 7284.1)( 21 =αP



• Substituting the values of the Lagrangepolynomial coefficients in the interpolatoroutput equation for n, n+1, and n+2, andcombining the three equations into a matrixform we arrive at

αααααααααααα

=++

−−−−−−

)()()()()()()()()()()()(

]2[]1[

][

212021221110111201000102

PPPPPPPPPPPP

nynyny

+

−−

]1[][]1[]2[

nxnx

nxnx



• The input-output relation of theinterpolation filter can be compactly writtenas

where H is the block coefficient matrix

+

−−

=++

]1[][]1[]2[

]2[]1[

][

nxnx

nxnx

nynyny

H



• For the factor-of-3/2 interpolator, we have

• It should be evident from an examination of

that the filter coefficients to computey[n+3], y[n+4], and y[n+5] are again givenby the same block matrix H

H

−−−=

7284.12963.17407.01728.04938.07407.02963.00617.00100

Output sample indexInput sample index

↓ ↓ ↓



• The desired interpolation filter is atime-varying filter

• A realization of the interpolator is givenbelow



• Note: In practice, the overall system delaywill be 3 sample periods

• Hence, the output sample y[n] actually willappear at the time index n+3

• A realization of the factor-of-3 interpolatorin the form of a time-varying filter is shownon the next slide



• The coefficients of the 5-th order time-varying FIR filter have a period of 3 and areassigned the values indicated below



• Substituting the expressions for the Lagrangepolynomials in the output equation we arriveat

])1[][]1[]2[(][61

21

21

613 ++−−+−−α= nxnxnxnxny

])1[][]1[(21

212 ++−−α+ nxnxnx

])1[][]1[]2[(31

21

61 +++−−−α+ nxnxnxnx][nx+



• A digital filter realization of the equation onthe previous slide leads to the Farrowstructure shown below

• In the above structurezzzzH 6

1211

212

61

0 )( +−+−= −−

zzzH 211

21

1 1)( +−= −

zzzzH 31

2112

61

2 )( ++−= −−



• In the Farrow structure only the value of a ischanged periodically with the remainingdigital filter structure kept unchanged

• Figures on the next slide show the input andthe output of the above interpolator for asinusoidal input of frequency of 0.05 Hzsampled at a 1-Hz rate



0 10 20 30 40 50

-0.1

-0.05

0

0.05

0.1

Time index n, sampling interval = 2/3 sec.

Am

plitu

de

Error Sequence

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

Time index n, sampling interval = 1 sec.

Am

plitu

de

Input Sinusoidal Sequence

0 10 20 30 40 50-1

-0.5

0

0.5

1

Time index n, sampling interval = 2/3 sec.

Am

plitu

de

Interpolator Output



Practical Considerations• A direct design of a fractional-rate sampling

rate converter in most applications is notpractical

• This is due to two main reasons:– length of the time-varying filter needed is

usually very large– real-time computation of the corresponding

filter coefficients is nearly impossible



• As a result, the fractional-rate sampling rateconverter is almost realized in a hybrid formas indicated below for the case of aninterpolator

• The digital sampling rate converter can beimplemented in a multistage form to reducethe computational complexity

polyphase decomposition - catholic university of...

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