14 Interpolation Decimation

1 Professor A G Constantinides

Interpolation & Decimation Interpolation & Decimation

• Sampling period T , at the output

• Interpolation by m:• Let the OUTPUT be [i.e. Samples

exist at all instants nT] • then INPUT is [i.e. Samples exist

at instants mT]

INPUT OUTPUT

Tjez

)(zY

)( mzX


Interpolation & Decimation Interpolation & Decimation • Let Digital Filter transfer function be

then• Hence is of the form i.e. its

impulse response exists at the instants mT. • Write

(.)H(.)).()( HzXzY m

(.)H )(zH

)1(21 ).1(...).2()1(.)0()( mzmhzhhzhzH)12()1( ).12(...).1().( mmm zmhzmhzmh

)13()12(2 )13(...).12().2( mmm zmhzmhzmh


Interpolation & Decimation Interpolation & Decimation • Or• Where

• So that

...)()()()( 32

21

1 mmm zHzzHzzHzH

...).2().()0()( 21 mmm zmhzmhhzH

...).12().1()1()( 22 mmm zmhzmhhzH

...).22().2()2()( 23 mmm zmhzmhhzH

)().()().()( 21

1mmmm zXzHzzXzHzY

...)().(32 mm zXzHz


Interpolation & Decimation Interpolation & Decimation • Hence the structure may be realised as

+

)(1mzH

)(2mzH

)(3mzH

OUTPUTINPUT

Samples across here are phasedby T secs. i.e. they do not interact in the adder.

Can be replaced by a commutator switch.


Interpolation & Decimation Interpolation & Decimation

• Hence

INPUTCommutator

OUTPUT

)(1mzH

)(2mzH

)(3mzH


Interpolation & Decimation Interpolation & Decimation • Decimation by m: • Let Input be (i.e. Samples exist at

all instants nT) • Let Output be (i.e. Samples exist at

instants mT)• With digital filter transfer function

we have

)(zX

)( mzY

)(zH

)().()( zHzXzY m


Interpolation & Decimation Interpolation & Decimation • Set

• And

• Where in both expressions the subsequences are constructed as earlier. Then

...)()()()( 32

21

1 mmm zHzzHzzHzH

)(.... )1( mm

m zHz

...)()()()( 32

21

1 mmm zXzzXzzXzX)(... )1( m

mm zXz

)(...)()()( )1(2

11

mm

mmmm zHzzHzzHzY

)(...)()( )1(2

11

mm

mmm zXzzXzzX


Interpolation & Decimation Interpolation & Decimation • Any products that have powers of less

than m do not contribute to , as this is required to be a function of .

• Therefore we retain the products

1z)( mzYmz

)()( 11mm zXzH )()( 2

mmm

m zXzHz

)...()( 31mm

mm zXzHz

)()(... 2m

mmm zXzHz


Interpolation & Decimation Interpolation & Decimation • The structure realising this is

+INPUT OUTPUT

Commutator

)(1mzH

)( mm zH

)(1m

m zH

)(2mzH


Interpolation & Decimation Interpolation & Decimation • For FIR filters why Downsample and then

Upsample?

#MULT/ACC N f s.LENGTH N

LOW PASSf s f s

LENGTH N

LOW PASS

LENGTH N

LOW PASSDOWNSAMPLE M:1 UPSAMPLE 1:M

f s f sMfs

#MULT/ACC N fM

s.#MULT/ACC

N fM

s.

TOTAL #MULT/ACC 2. .N fM

s


Interpolation & Decimation Interpolation & Decimation • A very useful FIR transfer function special

case is for : N odd, symmetric• with additional constraints on to be

zero at the points shown in the figure.

)(nh )(nh


Interpolation & Decimation Interpolation & Decimation • For the impulse response shown

• The amplitude response is then given

• In general

97531 ).5().4().3().2().1()0()( zhzhzhzhzhhzH9753 ).5().4().3().2().1( zhzhzhzhzh

)3cos(2).2()cos(2).1()0()( ThThhA

)5cos(2).3( Th

odd )cos(.

212)0()(

rTrrhhA


Interpolation & Decimation Interpolation & Decimation • Now consider• Then

21 ,T 21

odd 1 )cos(.

212)0()(

rTrrhhA

odd 12 cos.

212)0()(

rT

TrrhhA

odd 1 )cos(.

21)0(

rTrrhh


Interpolation & Decimation Interpolation & Decimation • Hence• Also

• Or• For a normalised response

)0(2)()( 21 hAA

odd .

2.cos.

212)0(

2 rT

Trrhh

TA

)0(h

T

AAA2

2)()( 21

1)0(A

T

A


Interpolation & Decimation Interpolation & Decimation • Thus• The shifted response

is useful

11)0(2 h 21)0( h

21)()(~ AA


Design of Decimator and Design of Decimator and InterpolatorInterpolator

• Example Develop the specs suitable for the design of a decimator to reduce the sampling rate of a signal from 12 kHz to 400 Hz

• The desired down-sampling factor is therefore M = 30 as shown below


Multistage Design of Multistage Design of Decimator and InterpolatorDecimator and Interpolator

• Specifications for the decimation filter H(z) are assumed to be as follows:

, , ,

Hz180pF Hz200sF0020.p 0010.s


Polyphase DecompositionPolyphase DecompositionThe 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


Polyphase DecompositionPolyphase Decomposition

• 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 components of X(z)

]}[{ nxk

]}[{ nxk)(zX k


Polyphase DecompositionPolyphase Decomposition• 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

zXzX

zzzX

1

1

0111 ....



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



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

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

8

0n

nznhzH ][)(


Polyphase DecompositionPolyphase Decomposition• 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 ][][)(


Polyphase DecompositionPolyphase Decomposition• The polyphase decomposition of an IIR

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

• One way to arrive at an M-branch polyphase decomposition 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


Polyphase DecompositionPolyphase Decomposition• Example - Consider

• To obtain a 2-band polyphase decomposition we 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 )(,)(



• The above approach increases the overall order and complexity of H(z)

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

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


Polyphase DecompositionPolyphase Decomposition• Example - Consider the transfer function of

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

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

40557281.02633436854.01

5)11(0527864.0)(

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



• In the above polyphase decomposition, branch transfer functions are stable allpass functions (proposed by Constantinides)

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

)(zEi


FIR Filter Structures Based on FIR Filter Structures Based on Polyphase DecompositionPolyphase Decomposition

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

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

)()( 10

Mk

Mk

k zEzzH



• A direct realization of H(z) based on the Type I polyphase decomposition is shown below



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

14 Interpolation Decimation

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