Transcript Document

Changing the Sampling Rate
Quote of the Day
There is no branch of mathematics, however
abstract, which may not someday be applied to
the phenomena of the real world.
Nicolai Lobachevsky
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Changing the Sampling Rate
• A continuous-time signal can be represented by its samples as
xn  xc nT 
• We can use bandlimited interpolation to go back to the
continuous-time signal from its samples
• Some applications require us to change the sampling rate
– Or to obtain a new discrete-time representation of the same
continuous-time signal of the form
x' n  xc nT'
where T  T'
• The problem is to get x’[n] given x[n]
• One way of accomplishing this is to
– Reconstruct the continuous-time signal from x[n]
– Resample the continuous-time signal using new rate to get x’[n]
– This requires analog processing which is often undersired
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Sampling Rate Reduction by an Integer Factor: Downsampling
• We reduce the sampling rate of a sequence by “sampling” it
xd n  xnM  xc nMT
• This is accomplished with a sampling rate compressor
• We obtain xd[n] that is identical to what we would get by
reconstructing the signal and resampling it with T’=MT
• There will be no aliasing if



 N
T' MT
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Frequency Domain Representation of Downsampling
• Recall the DTFT of x[n]=xc(nT)
   2k  
1 
j
Xe 
Xc  j 
 

T k     T
T 
• The DTFT of the downsampled signal can similarly written as
 
 
Xd e
j
   2r  
  
1 
1 
2r  




Xc  j 
Xc  j

 
 


T' r     T'
T'   MT r     MT MT  
• Let’s represent the summation index as
r  i  kM
 
Xd e
j
• And finally
where
  
1 M1  1 
2k 2i  

    X c  j


 
M i  0  T r     MT
T
MT  
 
Xd e j
Copyright (C) 2005 Güner Arslan
-   k   and 0  i  M
  2 i 
1 M1  j M  M  
 X e

M i0 


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Frequency Domain Representation of Downsampling: No Aliasing
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Frequency Domain Representation of Downsampling w/ Prefilter
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Increasing the Sampling Rate by an Integer Factor: Upsampling
• We increase the sampling rate of a sequence interpolating it
xi n  xn / L  xc nT / L 
• This is accomplished with a sampling rate expander
• We obtain xi[n] that is identical to what we would get by
reconstructing the signal and resampling it with T’=T/L
• Upsampling consists of two steps
– Expanding

xn / L  n  0,L,2L,...
xe n  
  xk n  kL
else
k  
 0
– Interpolating
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351M Digital Signal Processing
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Frequency Domain Representation of Expander
• The DTFT of xe[n] can be written as
 
 

 
  jn
Xe e     xk n  kLe
  xk e jLk  X e jL
n   k  
k  

• The output of the expander is frequency-scaled
j

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351M Digital Signal Processing
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Frequency Domain Representation of Interpolator
• The DTFT of the desired interpolated signals is
• The extrapolator output is given as
• To get interpolated signal we apply the following LPF
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Interpolator in Time Domain
• xi[n] in a low-pass filtered version of x[n]
• The low-pass filter impulse response is
sinn / L 
n / L
• Hence the interpolated signal is written as
hi n 
sinn  kL / L 
xi n   xk 
n  kL / L
k  

• Note that
hi 0  1
hi n  0
n  L,2L,...
• Therefore the filter output can be written as
xi n  xn / L  xc nT / L   xc nT'
Copyright (C) 2005 Güner Arslan
for n  0,L,2L,...
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Changing the Sampling Rate by Non-Integer Factor
• Combine decimation and interpolation for non-integer factors
• The two low-pass filters can be combined into a single one
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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