1 interpolation decimation
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Professor A G
Constantinides© 1
AGC
DSP
AGC
DSP
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
Professor A G
Constantinides© 2
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DSP
AGC
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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
Professor A G
Constantinides© 3
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AGC
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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
Professor A G
Constantinides© 4
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AGC
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Interpolation & Decimation Interpolation & Decimation Hence the structure may be realised as
Samples across here are phased
by T secs. i.e. they do not interact in the adder.
Can be replaced by a commutator switch.
)(1mzH
)(2mzH
)(3mzH
INPUT+ OUTPUT
Professor A G
Constantinides© 5
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DSP
AGC
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Interpolation & Decimation Interpolation & Decimation
HenceINPUT
Commutator
OUTPUT
)(1mzH
)(2mzH
)(3mzH
Professor A G
Constantinides© 6
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AGC
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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
Professor A G
Constantinides© 7
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AGC
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Interpolation & Decimation Interpolation & Decimation Set
And
Then
...)()()()( 32
21
1 mmm zHzzHzzHzH
)(.... )1( mm
m zHz
...)()()()( 32
21
1 mmm zXzzXzzXzX
)(... )1( mm
m zXz
)(...)()()( )1(2
11
mm
mmmm zHzzHzzHzY
)(...)()( )1(2
11
mm
mmm zXzzXzzX
Professor A G
Constantinides© 8
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AGC
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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)( mzY
mz
)()( 11mm zXzH )()( 2
mmm
m zXzHz )...()( 31
mmm
m zXzHz
)()(... 2m
mmm zXzHz
Professor A G
Constantinides© 9
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AGC
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Interpolation & Decimation Interpolation & Decimation The structure realising this is
+INPUT OUTPUT
Commutator
)(1mzH
)( mm zH
)(1m
m zH
)(2mzH
Professor A G
Constantinides© 10
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DSP
AGC
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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 PASS
DOWNSAMPLE M:1 UPSAMPLE 1:M
f s f sMfs
#MULT/ACC N f
Ms.
#MULT/ACC N f
Ms.
TOTAL #MULT/ACC 2. .N f
Ms
Professor A G
Constantinides© 11
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DSP
AGC
DSP
Interpolation & Decimation Interpolation & Decimation
A very useful FIR transfer function special case is for : N odd, symmetricwith additional constraints on to be zero at alternate sampling points other than at zero time index.
)(nh )(nh
Professor A G
Constantinides© 12
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AGC
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Interpolation & Decimation Interpolation & Decimation For this impulse response
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(.
21
2)0()(r
Trr
hhA
Professor A G
Constantinides© 13
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AGC
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Interpolation & Decimation Interpolation & Decimation Now consider Then
21 ,T 21
odd 1 )cos(.
21
2)0()(r
Trr
hhA
odd 12 cos.
21
2)0()(r
TT
rr
hhA
odd 1 )cos(.
21
)0(r
Trr
hh
Professor A G
Constantinides© 14
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AGC
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Interpolation & Decimation Interpolation & Decimation Hence Also
Or
For a normalised response
)0(2)()( 21 hAA
odd .
2.cos.
21
2)0(2 r
TT
rr
hhT
A
)0(h
T
AAA2
2)()( 21
1)0(A
T
A
Professor A G
Constantinides© 15
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AGC
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Interpolation & Decimation Interpolation & Decimation Thus
The shifted response
is useful
11)0(2 h
21
)0( h
21
)()(~ AA
Professor A G
Constantinides© 16
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DSP
AGC
DSP
Design of Decimator and Design of Decimator and InterpolatorInterpolator
Example Develop the specifications 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
Professor A G
Constantinides© 17
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AGC
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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
Professor A G
Constantinides© 18
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AGC
DSP
Polyphase DecompositionPolyphase DecompositionThe Decomposition Consider an arbitrary sequence {x[n]}
with a z-transform X(z) given by
Write X(z) as
nnznxzX ][)(
10
Mk
Mk
k zXzzX )()(
nn
nn
kk zkMnxznxzX ][][)(
10 Mk
Professor A G
Constantinides© 19
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AGC
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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
Professor A G
Constantinides© 20
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AGC
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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
zX
zX
zzzX
1
1
0
111 ....
Professor A G
Constantinides© 21
AGC
DSP
AGC
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Polyphase DecompositionPolyphase Decomposition A multirate structural interpretation of
the polyphase decomposition is given below
Professor A G
Constantinides© 22
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DSP
AGC
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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 ][)(
Professor A G
Constantinides© 23
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DSP
AGC
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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 ][][)(
Professor A G
Constantinides© 24
AGC
DSP
AGC
DSP
Polyphase DecompositionPolyphase Decomposition The polyphase decomposition of an IIR
transfer function H(z) = P(z)/D(z) is not that straightforward
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
Professor A G
Constantinides© 25
AGC
DSP
AGC
DSP
Polyphase DecompositionPolyphase Decomposition Example - Consider
To obtain a 2-band polyphase decomposition we rewrite H(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
)()()( 21
120 zEzzEzH
11
1
91
5191
610
zz
z zEzE )(,)(
Professor A G
Constantinides© 26
AGC
DSP
AGC
DSP
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
Professor A G
Constantinides© 27
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DSP
AGC
DSP
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:
We can show that H(z) can be expressed as
40557281.02633436854.01
5)11(0527864.0)(
zz
zzH
2
2
2
2
5278601
5278601
10557301
105573021
z
z
z
z zzH.
.
.
.)(
Professor A G
Constantinides© 28
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DSP
AGC
DSP
Polyphase DecompositionPolyphase Decomposition 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
Professor A G
Constantinides© 29
AGC
DSP
AGC
DSP
Polyphase DecompositionPolyphase Decomposition
In the above polyphase decomposition, branch transfer functions are stable allpass functions (proposed by Constantinides)
(More on this later) Moreover, the decomposition has not
increased the order of the overall transfer function H(z)
)(zEi
Professor A G
Constantinides© 30
AGC
DSP
AGC
DSP
FIR Filter Structures Based FIR Filter Structures Based on Polyphase Decompositionon Polyphase Decomposition
We demonstrate 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): )()( 1
0M
kMk
k zEzzH
Professor A G
Constantinides© 31
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AGC
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FIR Filter Structures Based FIR Filter Structures Based on Polyphase Decompositionon Polyphase Decomposition
A direct realization of H(z) based on the Type I polyphase decomposition is shown below
Professor A G
Constantinides© 32
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AGC
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FIR Filter Structures Based FIR Filter Structures Based on Polyphase Decompositionon Polyphase Decomposition
The transpose of the Type I polyphase FIR filter structure is indicated below
Professor A G
Constantinides© 33
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AGC
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IIR Filters Structures Based IIR Filters Structures Based on Polyphase Formon Polyphase Form
Consider the two-phase system below (proposed by Constantinides)
(You’ll see these structures again later on when we are dealing with Wavelets)
)(1 zH
)(2 zH
Professor A G
Constantinides© 34
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AGC
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IIR Filters Structures Based IIR Filters Structures Based on Polyphase Formon Polyphase Form
Where
Then the overall transfer function (+ sign) is
)(11 )( jezH
)(22 )( jezH
)(2)(121 )()()( jj eezHzHzG
)2/)]()(cos([)( 212/)](2)(1[ jezG
Professor A G
Constantinides© 35
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DSP
AGC
DSP
IIR Filters Structures Based IIR Filters Structures Based on Polyphase Formon Polyphase Form
Efficient computation is achieved when
These forms correspond to the half-band filters mentioned earlier
The concept can be generalised in a polyphased form
Reconstruction?
)()( 211 zFzH
)()( 22
12 zFzzH
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