Transcript DSP & Digital Filters

Interpolation & Decimation
jT
z

e
• Sampling period T ,
INPUT
1
at the output
OUTPUT
• Interpolation by m:
• Let the OUTPUT be Y (z ) [i.e. Samples
exist at all instants nT]
m
• then INPUT is X ( z ) [i.e. Samples exist
at instants mT]
Professor A G Constantinides
Interpolation & Decimation
• Let Digital Filter transfer function be H (.)
m
then Y ( z )  X ( z ).H (.)
• Hence H (.) is of the form H (z ) i.e. its
impulse response exists at the instants mT.
• Write
1
H ( z )  h(0)  z .h(1)  h(2).z
 h(m).z
 h(2m).z
2
m
2 m
 h(m  1).z
( m 1)
 h(2m  1).z
2
 ...  h(m  1).z
 ...  h(2m  1).z
( 2 m 1)
( m 1)
( 2 m 1)
 ...  h(3m  1) z
( 3m 1)
Professor A G Constantinides
Interpolation & Decimation
m
1
m
2
m
H
(
z
)

H
(
z
)

z
H
(
z
)

z
H
(
z
)  ...
• Or
1
2
3
• Where
H1 ( z m )  h(0)  h(m).z m  h(2m).z 2 m  ...
m
m
2 m
H 2 ( z )  h(1)  h(m  1).z  h(2m  1).z
 ...
H 3 ( z )  h(2)  h(m  2).z
m
m
 h(2m  2).z
2 m
 ...
• So that
1
Y ( z )  H1 ( z ). X ( z )  z H 2 ( z ). X ( z ) 
m
m
m
m
 z 2 H 3 ( z m ). X ( z m )  ...
3
Professor A G Constantinides
Interpolation & Decimation
• Hence the structure may be realised as
H1 ( z m )
INPUT
H2 (zm )
+
OUTPUT
H3 (z m )
Samples across here are
phased
by T secs. i.e. they do not
interact in the adder.
Can be replaced by a
commutator switch.
4
Professor A G Constantinides
Interpolation & Decimation
• Hence
H1 ( z m )
m
INPUT
H2 (z )
H3 (z m )
5
Commutator
OUTPUT
Professor A G Constantinides
Interpolation & Decimation
• Decimation by m:
• Let Input be X (z ) (i.e. Samples exist at
all instants nT)
m
Y
(
z
) (i.e. Samples exist at
• Let Output be
instants mT)
• With digital filter transfer function H (z )
we have
Y ( z )  X ( z ).H ( z )
m
6
Professor A G Constantinides
Interpolation & Decimation
• Set
1
2
H ( z )  H1 ( z )  z H 2 ( z )  z H 3 ( z )  ...
m
 ...  z
( m 1)
m
m
m
.H m ( z )
• And X ( z )  X 1 ( z m )  z 1 X 2 ( z m )  z 2 X 3 ( z m )  ...
 ...  z ( m1) X m ( z m )
• Where in both expressions the subsequences
are constructed as earlier. Then


Y ( z m )  H1 ( z m )  z 1H 2 ( z m )  ...  z ( m1) H m ( z m ) 
7
X ( z
1
m
1
)  z X 2 ( z )  ...  z
m
( m 1)
m
X (z )

mG Constantinides
Professor A
Interpolation & Decimation
• Any products that have powers of z 1 less
m
than m do not contribute to Y ( z ) , as this
is required to be a function of z m .
• Therefore we retain the products
H1 ( z ) X 1 ( z )  z  m H m ( z m ) X 2 ( z m )
m
z
m
m
m
m
H m1 ( z ) X 3 ( z )...
...  z m H 2 ( z m ) X m ( z m )
8
Professor A G Constantinides
Interpolation & Decimation
• The structure realising this is
Commutator
H1 ( z m )
Hm (zm )
INPUT
H m1 ( z m )
+
OUTPUT
H2 (zm )
9
Professor A G Constantinides
Interpolation & Decimation
• For FIR filters why Downsample and then
Upsample?
fs
LOW PASS
fs
LENGTH N
#MULT/ACC  N . f s
fs
DOWNSAMPLE M:1
LOW PASS
LENGTH N
N. fs
#MULT/ACC  M
TOTAL #MULT/ACC
10
UPSAMPLE 1:M
LOW PASS
fs
M
fs
LENGTH N
N. fs
#MULT/ACC  M

2. N . f s
M
Professor A G Constantinides
Interpolation & Decimation
• A very useful FIR transfer function special
case is for : N odd, h(n) symmetric
• with additional constraints on h(n) to be
zero at the points shown in the figure.
11
Professor A G Constantinides
Interpolation & Decimation
• For the impulse response shown
1
3
5
7
H ( z )  h(0)  h(1).z  h(2).z  h(3).z  h(4).z  h(5).z
 h(1).z  h(2).z 3  h(3).z 5  h(4).z 7  h(5).z 9
9
• The amplitude response is then given
A( )  h(0)  h(1).2 cos(T )  h(2).2 cos(3T )
 h(3).2 cos(5T )
• In general
r  1

A( )  h(0)  2  h
. cos( rT )
r odd
12
 2 
Professor A G Constantinides
Interpolation & Decimation
• Now consider 1, 2
• Then
1  2 

T
r  1

A(1 )  h(0)  2  h
. cos( rT )
r odd  2 
r  1
 



A(2 )  h(0)  2  h
. cos r   1 T 
r odd  2 
 
 T
r  1

 h(0)   h
. cos( r1T )
r odd  2 
13
Professor A G Constantinides
Interpolation & Decimation
• Hence A(1 )  A(2 )  2h(0)
 
r  1   


• Also
A   h(0)  2  h
. cos r. .T 
 2T 
r odd
 2 
 2T

 h(0)
 

A(1 )  A(2 )  2 A 
 2T 
• Or
• For a normalised response


A   
A(0)  1  
T 
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Professor A G Constantinides
Interpolation & Decimation
• Thus 2h(0)  1      1
• The shifted response
1
~
A( )  A( ) 
2
is useful
15
1
h(0) 
2
Professor A G Constantinides
Design of Decimator and
Interpolator
• 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
16
Professor A G Constantinides
Multistage Design of
Decimator and Interpolator
• Specifications for the decimation filter H(z)
are assumed to be as follows:
Fp  180 Hz , Fs  200 Hz ,
 p  0.002 ,  s  0.001
17
Professor A G Constantinides
Polyphase Decomposition
The Decomposition
• Consider an arbitrary sequence {x[n]} with
a z-transform X(z) given by

X ( z )  n x[n]z n
• We can rewrite X(z) as
M 1  k
M
X ( z )  k 0 z X k ( z )
where


n
n
X k ( z )  n xk [n] z  n x[Mn  k ] z
18
0  k  M 1
Professor A G Constantinides
Polyphase Decomposition
• The subsequences {xk [n]} are called the
polyphase components of the parent
sequence {x[n]}
• The functions X k (z ), given by the
z-transforms of {xk [n]}, are called the
polyphase components of X(z)
19
Professor A G Constantinides
Polyphase Decomposition
• The relation between the subsequences {xk [n]}
and the original sequence {x[n]} are given
by
xk [n]  x[Mn  k ], 0  k  M  1
• In matrix form we can write
X ( z )  1
20
z
1
.... z ( M 1)
 X0(zM ) 


M
 X1.( z ) 
..


.
M 
X
(
z
)
Professor
M
1
A G Constantinides

Polyphase Decomposition
• A multirate structural interpretation of the
polyphase decomposition is given below
21
Professor A G Constantinides
Polyphase Decomposition
• The polyphase decomposition of an FIR
transfer function can be carried out by
inspection
• For example, consider a length-9 FIR
transfer function:
H ( z) 
8
 h[n] z
n
n 0
22
Professor A G Constantinides
Polyphase Decomposition
• Its 4-branch polyphase decomposition is
given by
4
1
4
2
4
3
4
H ( z )  E0 ( z )  z E1( z )  z E2 ( z )  z E3 ( z )
where
1
2
E0 ( z )  h[0]  h[4]z  h[8]z
E1( z )  h[1]  h[5]z 1
1
23
E2 ( z )  h[2]  h[6]z
1
E3 ( z )  h[3]  h[7]z
Professor A G Constantinides
Polyphase 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
M
P
'
(
z
)
/
D
'
(
z
) by multiplying P(z) and
form
D(z) with an appropriately chosen
polynomial and then apply an M-branch
polyphase decomposition to P '( z )
24
Professor A G Constantinides
Polyphase Decomposition
1 2 z 1
H ( z) 
13 z 1
• Example - Consider
• To obtain a 2-band polyphase decomposition we
rewrite H(z) as
(12 z 1 )(13 z 1 )
H ( z) 
(13 z 1 )(13 z 1 )
15 z 1 6 z 2

19 z 2
16 z 2

19 z 2
5 z 1

19 z 2
• Therefore,
1
where H ( z )  E0 ( z )  z E1( z )
2
25
16 z 1
E0 ( z ) 
,
1
19 z
E1( z ) 
2
5
19 z 1
Professor A G Constantinides
Polyphase 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
26
Professor A G Constantinides
Polyphase Decomposition
• Example - Consider the transfer function of
a 5-th order Butterworth lowpass filter with
a 3-dB cutoff frequency at 0.5:
H ( z)
27
0.0527864 (1 z 1 )5

1 0.633436854z  2  0.0557281z  4
• It is easy to show that H(z) can be expressed
as
2
2
1 0.52786  z 
1  0.105573  z 
H ( z )  
z 


2
 2 
2  1 0.105573 z 
 1 0.52786 z 
Professor A G Constantinides
Polyphase Decomposition
• Therefore H(z) can be expressed as
2
1
2
H ( z )  E0 ( z )  z E1( z )
where
1
1  0.105573  z 
E0 ( z )  
2  1 0.105573 z 1 
1
1  0.52786  z 
E1( z )  
2  1 0.52786 z 1 
28
Professor A G Constantinides
Polyphase Decomposition
• In the above polyphase decomposition,
branch transfer functions Ei (z ) are stable
allpass functions (proposed by
Constantinides)
• Moreover, the decomposition has not
increased the order of the overall transfer
function H(z)
29
Professor A G Constantinides
FIR Filter Structures Based on
Polyphase 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):
30
H ( z)  
M 1 k
M)
z
E
(
z
k
k 0
Professor A G Constantinides
FIR Filter Structures Based on
Polyphase Decomposition
• A direct realization of H(z) based on the
Type I polyphase decomposition is shown
below
31
Professor A G Constantinides
FIR Filter Structures Based on
Polyphase Decomposition
• The transpose of the Type I polyphase FIR
filter structure is indicated below
32
Professor A G Constantinides