4. Multirate Systems and their Applications
Inefficient Implementation of Downsampling
x[n ]
H z 
y[m]
N
We compute here … and throw away most of them here!!!!
N
Inefficient Implementation of Upsampling
x[n ]
N
H z 
y[m]
Most terms here are zero … and waste time to process them here!!!!
N
H z 
$$
$$
y[m]
Recall the Noble Identities
x[n ]
 
H z
N
y[m]
N

x[n ]
N
H z 
y[m]
Same!!!
x[n ]
N
 
H zN
y[m]

x[n ]
H z 
y[m]
N
Example…
h[n ]

h[n]  0 for n odd
H ( z )   h[n]  h[0]  h[2]z 2  h[4]z 4  ...  H ( z 2 )
n
x[n ]
 
H z
2
2
y[m]
 
H z 2  2.0  1.5 z 2  1.5 z 4  1.2 z 6
x[n ]
2
H z 
y[m]
H z   2.0  1.5z 1  1.5 z 2  1.2 z 3
… continued
 
x[n ]
H z2
2
y[m]
h[n ]
h[m ]
m
n
x[n ]
2
H z 
y[m]
Example…
Same h[n ]
x[n ]
2
 
y[m]
H z2
 
H z 2  2.0  1.5 z 2  1.5 z 4  1.2 z 6
x[n ]
H z 
2
H z   2.0  1.5z 1  1.5 z 2  1.2 z 3
y[m]
… continued
x[n ]
 
y[m]
H z2
2
h[m ]
h[n ]
m
n
x[n ]
H z 
2
y[m]
General Filter: Polyphase Decomposition
In general: Polyphase Decomposition.
Take N=2:
2
4


H z  h[0]  h[2]z  h[4]z  ...
1

2
5

 z h[1]  h[3]z  h[5]z  ...
 
 
H  z   H 0 z  z H1 z
2
with
1
2
H 0 ( z 2 )   h[2m]z 2 m
m
H1 ( z 2 )   h[2m  1]z 2 m
m
Polyphase
Components
Downsampling using Polyphase Decomposition
x[n ]
H z 
y[m]
2
2
x[n ]
H1 z 
z 1
2
H 0 z 
y[m]
POLYPHASE
 
NOBLE
IDENTITY
H1 z 2
x[n ]
z 1
 
H0 z2
y[m]
2
Serial to Parallel (Buffer)
x[n ]
2
x1[m]  x[2m  1]
2
x0 [m]  x[2m]
z 1
This is a Serial to Parallel (a Buffer):
x[n ]
S/P
x1[m]
0 1 3 5
x0 [m]
0 2
0001234567
2
n0
m0
4
6
Upsampling using Polyphase Decomposition
x[n ]
2
H z 
y[m]
NOBLE IDENTITY
H 0 z 
x[n ]
H1 z 
POLYPHASE
x[n ]
2
 
H0 z2
 
H1 z 2
y[m]
z 1
y[m]
2
z 1
2
NOBLE
IDENTITY
Parallel to Serial (Unbuffer or Interlacer)
y0 [n]
y[ m ]
2
z 1
y1[n]
2
This is a Parallel to Serial (an
Unbuffer):
5
y0 [ n ]
0 2 4 6
y1[n]
0 1
n0
3
P/S
2
y[ m ]
0001234567
m0
General Polyphase Decomposition
Given any integer N:
H z  

n
h
[
n
]
z

n  
 



 Nk 
  z   h[kN  ]z 
k 0
  



N 1
 
 
Hk zN
 
H ( z )  H 0 z N  z 1 H1 z N  ...  z  ( N 1) H N 1 z N
Example: take N=3
H ( z )  2.1  .5z 1  .3z 2  4.2 z 3  0.6 z 4  1.4 z 5  0.2 z 6  0.7 z 7

H 0 ( z 3 )  2.1  4.2 z 3  0.2 z 6
H1 ( z 3 )  .5  0.6 z 3  0.7 z 6
H 2 ( z 3 )  .3  1.4 z 3
Apply to Downsampling…
H z 
x[n ]
y[m]
N
POLYPHASE
H N 1  z N 
z 1
H2  zN 
z 1
H1  z N 
x[n ]
z 1
H0  z N 
y[m]
N
… apply Noble Identity
N
H N 1  z 
N
H2  z 
N
H1  z 
z 1
z 1
x[n ]
z 1
N
H0  z 
y[m]
Serial to Parallel (Buffer)
N
z 1

N
x1[m]  x[mN  1]
N
x0 [m]  x[mN ]
z 1
x[n ]
xN 1[m]  x[(m  1) N  1]
Serial to Parallel (Buffer):
x[n ]
1
2
3
4
5
S/P
6
N
n  N
n0
xN 1[m]
1
3
5
2
4
6
x0 [m]
m0
Same for Upsampling…
N
x[n ]
H z 
y[m]
POLYPHASE
x[n ]
N
 
H0 z N
 
y[m]
z 1
H1 z N

 
H N 1 z N
z 1
… apply Noble Identity
x[n ]
H z 
N
y[m]
NOBLE IDENTITY
x[n ]
H 0 z 
H1 z 
z 1
N

H N 1 z 
y[m]
N
z 1
N
Parallel to Serial (Unbuffer or Interlacer)
N
y0 [n]
N
y N 1[n ]
y[ m ]
z 1
This is a Parallel to Serial (an
Unbuffer):
1
3
5
2
4
6
y0 [ n ]
y N 1[n ]
n0
n 1
y[ m ]
P/S
N
1
m0
2
3
mN
4
5
6
Processing Data by Blocks
• In most efficient implementations we process data by blocks, rather than one sample
at a time.
• Real Time simulation and design software such as Simulink are designed to take
advantages of block processing for two purposes:
• efficient computations, thus faster simulations;
• efficient design.
“Sample Based” and “Frame Based” Signals
N
Sample Based:
[M  N ]

M
time
They are MN distinct signals arranged in a matrix
Particular Case:
[ M  1]
x[n ]
x1[n]
is like M distinct signals
xM [n]

“Sample Based” and “Frame Based” Signals
M
Frame Based:
x1[n]
[M  N ]

xN [ n]
They are N distinct signals, each one represented
as a sequence of frames of length M
Particular Case:
[ M  1]
x[n ]
one signal as a sequence of
M
frames of length M
Convert “to Sample” and “to Frame”
Example:
5

y0= 
y1= 
y2=y0= 
5


Frame based
Sample based
Frame based
Serial to Parallel in Simulink
Serial to Parallel is implemented by the “Buffer” Simulink block in
Signal Processing Blcokset > Signal Management > Buffers:
x[n ]


S/P
N
xN 1[m]
x0 [m]


[ N 1]
[11]
Sample based
Frame based
Buffer
Example of Downsampling
DF FIR
x1[m]
[2x1]
[2x1]
1
x[n]
H1(z)
1
x0[m]
Add
Buffer
y[m]
DF FIR
H0(z)
x1[m ]
2
H1 z 
2
H 0 z 
z 1
x[n]
x0 [ m ]
y[m]
Parallel to Serial in Simulink
Parallel to Serial is implemented by the “UnBuffer” Simulink block in
Signal Processing Blcokset > Signal Management > Buffers:
y0 [ n ]


y N 1[n ]
y[ m ]
P/S
N
[ N 1]


[11]
Frame based
Sample based
Unbuffer
Example of Upsampling
DF FIR
y0[n]
1
x[n]
H0(z)
2
2
DF FIR
y1[n]
To
Frame
[2x1]
[2x1]
1
y[m]
Frame Conversion
Unbuffer
H1(z)
x[n]
H 0 z 
y0 [ n ]
y[m]
2
z 1
H1 z 
y1[n ]
2