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
2 Professor A G Constantinides
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
3 Professor A G Constantinides
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
4 Professor A G Constantinides
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.
5 Professor A G Constantinides
Interpolation & Decimation Interpolation & Decimation
• Hence
INPUTCommutator
OUTPUT
)(1mzH
)(2mzH
)(3mzH
6 Professor A G Constantinides
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
7 Professor A G Constantinides
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
8 Professor A G Constantinides
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
9 Professor A G Constantinides
Interpolation & Decimation Interpolation & Decimation • The structure realising this is
+INPUT OUTPUT
Commutator
)(1mzH
)( mm zH
)(1m
m zH
)(2mzH
10 Professor A G Constantinides
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
11 Professor A G Constantinides
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
12 Professor A G Constantinides
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
13 Professor A G Constantinides
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
14 Professor A G Constantinides
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
15 Professor A G Constantinides
Interpolation & Decimation Interpolation & Decimation • Thus• The shifted response
is useful
11)0(2 h 21)0( h
21)()(~ AA
16 Professor A G Constantinides
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
17 Professor A G Constantinides
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
18 Professor A G Constantinides
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
19 Professor A G Constantinides
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
20 Professor A G Constantinides
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 ....
21 Professor A G Constantinides
Polyphase DecompositionPolyphase Decomposition
• A multirate structural interpretation of the polyphase decomposition is given below
22 Professor A G Constantinides
Polyphase DecompositionPolyphase Decomposition
• 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 ][)(
23 Professor A G Constantinides
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 ][][)(
24 Professor A G Constantinides
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
25 Professor A G Constantinides
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 )(,)(
26 Professor A G Constantinides
Polyphase DecompositionPolyphase Decomposition
• 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
27 Professor A G Constantinides
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
..
..)(
28 Professor A G Constantinides
Polyphase DecompositionPolyphase Decomposition
• Therefore H(z) can be expressed as
where
1
1
5278601527860
21
1 zzzE
..)(
1
1
105573011055730
21
0 zzzE
..)(
)()()( 21
120 zEzzEzH
29 Professor A G Constantinides
Polyphase DecompositionPolyphase Decomposition
• 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
30 Professor A G Constantinides
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
31 Professor A G Constantinides
FIR Filter Structures Based on FIR Filter Structures Based on Polyphase DecompositionPolyphase Decomposition
• A direct realization of H(z) based on the Type I polyphase decomposition is shown below
32 Professor A G Constantinides
FIR Filter Structures Based on FIR Filter Structures Based on Polyphase DecompositionPolyphase Decomposition
• The transpose of the Type I polyphase FIR filter structure is indicated below