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Class Report
馬睿峰 :
Multirate Signal Processing
Introduction
Multirate Signal Processing
• Sampling rate conversion: The process of converting a
signal from a given rate to a different rate
• Multirate Systems: Systems handling more than one
sampling rate
• Objective: Minimize the complexity of the signal
processing under Nyquist Sampling criterion
• Key Operations: Sampling rate conversion for
communications
-- Decimation: Down sampling at receiver side
-- Interpolation: Up sampling at transmission side
• DSP Functions:
Decimation Filtering
Interpolation Filtering
Why and Where to Use Multirate
• Multirate systems can reduce required system
computation, bandwidth, or storage requirements
• Multirate systems are used for converting data
between A/D and D/A devices that run at different
sampling rates
• Common multirate applications:
–
–
–
–
–
–
–
Sampling systems
Audio/speech encoders
Image/video encoders
Music synthesizers
CAT scan, image systems
Echo cancellers
Modems, data multiplexers
Sampling Rate Conversion
• It is often advantages to digitally alter the
sampling rate during processing, after the
initial A/D conversion has been finished:
– High sampling rates (perhaps many times the
Nyquist rates) can be used to help remove noise
and quantization error
– Low sampling rates (near the Nyquist rate) are
best for computation speed and memory
requirements
Sampling – Time Domain (Multiplication)
 Analog to digital conversion process:
s(t)
xc(t)
xc(t)
s(t)
x[n] = xc(nt)
x[n]
Sampling – Frequency Domain (Convolution)

If a continuous signal xc(t) is band limited with |Xc(j)| = 0,
for ||  2Fc, then xc(t) can be uniquely reconstructed without
error, if Fsam  2Fc, where Fsam= 1/T is the sampling frequency.
Xc(j)
-2Fc 0 2Fc

S(j)
-2F
-F

F=2F
0
Xc(j)  S(j)
3
2
F

2
0
F

2
F
3
2

F
Aliasing (Under Sampling)
 Fourier transform of continuous-time signal
-/T
/T
 Fourier transform of sampled signal
-2
-
High Frequency noise

2
Multirate Antialiasing Filter
Analog
LPF
A/D
3
Digital
LPF
4
Sampling and Reconstruction
Sampling
f [n]  f c (nT )
Reconstruction
f c [ n] 


n 
 t  nT 
f [n]sin c 

 T 
sin c( x) 
sin( x)
x

Nyquist sampled
Nyquist Condition:
Under sampled
Fc ( )   f c (t )e jt dt

f c [n]  f c (t )iff Fc ( )  0|  |

T
Multirate signal processing - Decimation
The M-fold decimator :
The M-fold decimator which takes an input sequence
x(n) and produces the output sequence
x[n]
y[n] = x[Mn]
where M is an integer.
w[n]
h[n]
F
y[m]
M
F’
F
Only those samples of x[n] which occur at time equal to
multiples of M are retained by the decimator .
Over-sampled signal
Decimated signal
3 to 1 decimation
time
0
time
0
Decimation
• The decimator is also called as downsampler ,
subsampler, sampling rate compressor or merely
a compressor
• Aliasing can be avoided if x[n] is a lowpass signal
bandlimited to the region |  | < /M
• In most applications ,the
decimator is preceeded by
a lowpass filter called the
decimation filter . The
filter ensures that signal
being decimated is
bandlimited
Decimation by Integer Factor
x[n]
h[n]
w[n]
F
y[m]
M
F’
F
Time-domain input-output relation:
y[m]  w[ Mm] 

 h[k ]x[Mm  k ]
T ' Fs
M  '
T Fs
k 
Frequency-domain input-output relation:
1
Y ( z) 
M
Y (e
j '
1
)
M
M 1
 H (e
 j 2l / M
z1/ M ) X (e i 2l / M z1/ M )
l 0
M 1
 H (e
j ( ' 2l ) / M
) X (e j ( ' 2l ) / M )
l 0
Idea frequency response of decimation filter:


C , |  |
j
H (e ) 


 0,
M
otherwise
Decimation by Integer Factor
x[n]
w[n]
h[n]
F
y[m]
M
F’
F
|X(ej)|
F

0
2
|H(ej)|
F
0
/M
|W(ej)|
Y (e
0

j '
1
)
M
/M
M 1
 H (e
2
j ( ' 2l ) / M
) X (e j ( '2l ) / M )
l 0
F

2
|Y(ej’)|
F’=F/M
0

2
4
6
Interpolation by Integer Factor
Definition: A up-sampler with an up-sampling factor
L. The aim is to get a new sequence with
with a higher sampling frequency without
changing the spectrum.
x[n]
F
L
y[m]
w[m]
h[m]
F’
F’
T Fs'
L 
T ' Fs
1 to 3 interpolation
time
        
0
time
0
Interpolation by Integer Factor
x[n]
y[m]
w[m]
L
h[m]
F’
F
F’ = L  F
Time-domain input-output relation:
y[ m] 

 h[k ]w[m  k ]
k 
 m
 x[ ],
w[m]   L

 0,
m  0,  L, 2 L,...
otherwise
Frequency-domain input-output relation:
j '
j '
Y (e )  H (e ) X (e
j ' L
)
Idea frequency response of Interpolation filter:


C ,
|  |
j '
H (e ) 


 0,
L
otherwise
Interpolation by Integer Factor
x[n]
L
y[m]
w[m]
h[m]
F’ = L  F
F’
F
|X(ej)|
0

2
|W(ej’)|
F’
0
/L

2
Y (e j ' )  H (e j ' ) X (e j 'L )
|Y(ej’)|
F’
0
/L

2
Conversion by Rational Factor M/L
Interpolation by L
x[n]
L
h1(k)
F
Decimation by M
s[k]
h2(k)
M
F”=LF
x[n]
F
L
h(k)
F”=LF
F”
M
y(m)
F’ = (L/M)F
y[m]
F’ = (L/M)F
 

C , |  "| min | , |
H ( e j " )  
L M
 0,
otherwise
References of Multirate Signal Processing
• Ronald E. Crochiere and Lawrence R.
Rabiner, Multirate Digital Signal
Processing, Prentice Hall, 1983.
• P. P. Vaidyanathan, Multirate Systems and
Filter Bank, Prentice Hall, 1993.
• P. P. Vaidyanathan, Multirate Digital Filters,
Filter Banks, Polyphase Networks and
Applications: A Tutorial, IEEE Proceeding,
Vol.78, Jan. 1990.