Transcript Document

Discrete-Time Processing of
Continuous-Time Signals
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Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Reconstruction of Bandlimited Signal From Samples
• Sampling can be viewed as modulating with impulse train
• If Sampling Theorem is satisfied
– The original continuous-time signal can be recovered
– By filtering sampled signal with an ideal low-pass filter (LPF)
• Impulse-train modulated signal
x s t  

 xnt  nT 
n  
• Pass through LPF with impulse response hr(t) to reconstruct
xr t  

 xnh t  nT 
r
n  
x[n]
Copyright (C) 2005 Güner Arslan
Convert from
sequence to
impulse train
Ideal
reconstruction
filter
Hr(j)
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xr(t)
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Ideal Reconstruction Filter
• Ideal LPC with cut of frequency of c=/T or fc=2/T
sint / T 


hr t 
t / T
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351M Digital Signal Processing
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Reconstructed Signal
xr t  

 xn
n  
sint  nT  / T 
t  nT  / T
sinc function is 1 at t=0
sinc function is 0 at nT


Xr j  X e jT Hr j
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Discrete-Time Processing of Continuous-Time Signals
xc(t)
C/D
xn Discrete- yn
Time
System
D/C
yr(t)
• Overall system is equivalent to a continuous-time system
– Input and output is continuous-time
• The continuous-time system depends on
– Discrete-time system
– Sampling rate
• We’re interested in the equivalent frequency response
– First step is the relation between xc(t) and x[n]
– Next between y[n] and x[n]
– Finally between yr(t) and y[n]
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Effective Frequency Response
• Input continuous-time to discrete-time
 
xn  xc nT 
Xe
j
• Assume a discrete-time LTI system
   He Xe 
Ye
j
j
j
   2k  
Xc  j 
 

T 
k  
 T
 
1
 H e j
T
• Output discrete-time to continuous-time
sint  nT  / T 
yr t    yn
t  nT  / T
n  
   2k  
1 

X
j 
 

c

T k     T
T 


TY e jT
Yr j  
 0


  /T
otherwise
• Output frequency response


H e jT Xc j    / T
Yr j  
0
otherwise

• Effective Frequency Response
Yr j  Heff jXc j
Copyright (C) 2005 Güner Arslan

H e jT
Heff j  
 0
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
  /T
otherwise
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Example
• Ideal low-pass filter implemented as a discrete-time system
Continuous-time
input signal
Sampled continuoustime input signal
Apply discrete-time
LPF
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Example Continued
Signal after discretetime LPF is applied
Application of
reconstruction filter
Output continuoustime signal after
reconstruction
Copyright (C) 2005 Güner Arslan
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Impulse Invariance
• Given a continuous-time system Hc(j)
– how to choose discrete-time system response H(ej)
– so that effective response of discrete-time system Heff(j)=Hc(j)
• Answer:
• Condition:
 
H ej  Hc j / T 
Hc j  0

  /T
• Given these conditions the discrete-time impulse response can
be written in terms of continuous-time impulse response as
hn  Thc nT 
• Resulting system is the impulse-invariant version of the
continuous-time system
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Example: Impulse Invariance
• Ideal low-pass discrete-time filter by impulse invariance
1   c
Hc j  
else
0
• The impulse response of continuous-time system is
sinc t 
t
• Obtain discrete-time impulse response via impulse invariance
hc t  
hn  Thc nT   T
sincnT  sincn

nT
n
• The frequency response of the discrete-time system is
 
Hc e
Copyright (C) 2005 Güner Arslan
j
1
  c

0 c    
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