digital signal processing sampling

1. Chap 4
Sampling of Continuous-
Time Signals

2. Changing The Sampling Rate
Using Discrete-Time
Processing
 Sampling Rate Reduction by an Integer Factor
 Increasing the Sampling Rate by an Integer
Factor
 Changing the Sampling Rate by a
Noninteger Factor
 Sampling Rate Reduction  Increase
period
 Decrease
frequency

4.  Sampling Rate Reduction by an Integer Factor
xd[n] = x[nM] = xc(nMT)
system called Sampling Rate Compressor or Compressor
operation called Downsampling
 xd[n] is an exact representation of xc(t) if /(MT) > N
 sampling rate can be reduced by a factor of M without aliasing
if the original sampling rate was at least M times the Nyquist
rate or if the bandwidth of the sequence is first reduced by a
factor of M by discrete-time filtering
 Fourier transform of discrete-time sampled sequence xd[n] is
where r = i+kM, -infinity < k < infinity and 0 < and = i < and = M-
1
 Lowpass filter+Compressor = Decimator
Sampling Rate Reduction by an Integer
Factor
MT MT
MT
X ( j  j
)
j

Xd (e )
 r

 c
1 
2r
M
X(e
)
j

Xd (e )

 j(/ M2i/
M)
i
0
1 M1

6. General system for sampling rate reduction
by integer factor M
Lowpass filter
Gain = 1
Cutoff = /M
M
x[n] x~[n] xd
~[n] = x~[nM]
Sampling
period T
Sampling
period T
Sampling
period T’ = MT

9. Sampling rate reduction by
2

10. Downsample with M=3

11. Downsampling with aliasing (a to c) and
with prefiltering to avoid aliasing (d
to f).

12. Increasing the Sampling Rate by an
Integer Factor
 Increasing the Sampling Rate by an Integer Factor
xi[n] = x[n/L] = xc(nT/L), n = 0, ±L, ± 2L, ...
and
 Xi(ej) can be obtained from Xe(ej) by correcting the amplitude
scale from 1/T to L/T and by removing all the frequency-scaled
images of Xc(j) except at integer multiples of 2
k 
 system called Sampling Rate Expander or
Expander
 operation called Upsampling or Interpolation

xe [n]   x[k][n 
kL]
j

j
L
 X (e )
(4.85)
 
 j
n

 j
Lk
k 
 
x[k]e


n  k 

Xe (e )     x[k]
[n  kL]e

xi [n]  
x[k]
k 
sin[(n  kL) / L]
(n  kL) / L

13. Increasing the Sampling Rate by an Integer
Factor

15. General system for sampling rate increase by
integer factor L
L
Lowpass filter
Gain = L
Cutoff = /L
x[n] xe[n] xi[n]
Sampling
period
T
Sampling
period T’ = T/L
Sampling
period T’ = T/L

16. Increasing sampling rate (Interpolation)
by 2