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Signals and SystemsLecture 23
•DT Processing of CT Signals
•Digital Differentiator and Half-Sample Delay
•DT Decimation and Interpolation
2
DT Processing of Band-limited CT Signals
Chapter 7 Sampling
Why do this?— Inexpensive, versatile, and higher noise margin.
How do we analyze this system?— We will need to do it in the frequency domain in both CT and DT— In order to avoid confusion about notations, specify
ω— CT frequency variable Ω — DT frequency variable (Ω =ωΤ)
Step 1: Find the relation between xc(t) and xd[n], or Xc(jω) and Xd(ejΩ)
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Time-Domain Interpretation of C/D Conversion
Chapter 7 Sampling
4
Frequency-Domain Interpretation
n
cn
cp nTtnTxnTttxtx )()()()()(
Chapter 7 Sampling
kscp kjX
TjX ))((
1)(
n
nTjc enTx )(
n
njd
jd enxeX ][)(
n
njc enTx )(
CT----periodic with period ωs=2π/T
DT----periodic with period 2π
)()(T
jXeX pj
d
Note: ωs<=>2π CT DT
(Ω = ωT)
5
Illustration of C/D Conversion in the Frequency-Domain
Chapter 7 Sampling
k
scp kjXT
jX ))((1
)(
n
nTjc enTx )(
n
njd
jd enxeX ][)(
n
njc enTx )(
)()(T
jXeX pj
d
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D/C Conversion: Reverse of the process of C/D conversion
Chapter 7 Sampling
Again, Ω = ωT
)()( Tjdp eYjY Reverse frequency scaling
otherwise
eTYjYsTj
dc
02
)()(
bandlimited
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Now the whole picture
Chapter 7 Sampling
•Overall system is time-varying if sampling theorem is not satisfied;•It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs xc(t), with ωM<2ωs/2•When the input xc(t) is band-limited (X(jω) = 0 at |ω|>ωM)sampling theorem is satisfied (ωs> 2ωM), then
)(*)()()()()( txthtyjHjHjY cccccc LTI
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Frequency-domain Illustration of DT Processing of CT Signals
Chapter 7 Sampling
Original signal
After sampling
After C/D conversion
)/()( TjXeX pj
d
k
scp kjXT
jX ))((1
)(
)( jX c
)/(1
TjXT c
#step 1-C/D:
-periodic
-periodic
- π < Ω < π if no aliasing
ω=Ω/T
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Frequency-domain Illustration of DT Processing of CT Signals
)()(1 jXeHT c
Tjd
)()(1 jXeHT p
Tjd
Chapter 7 Sampling
DT processing
After D/C conversion
Equivalent CT system
)()( Tjdp eYjY
0
)()(
Tjd
c
eTYjY
)()()( jd
jd
jd eXeHeY
)/()(1
TjXeHT c
jd
#step 2-Digital Filter:
Ω=ωT
- π < Ω < π
#step 3-D/C:
)()( jXeH cTj
d
ωs/2<ω<ωs/2
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Design Digital Filter
Chapter 7 Sampling
otherwise
jXeHjYs
cTj
dc
02
)()()(
otherwise
eHjHsTj
dc
02
)()(
In practice, first specify the desired Hc(jω), then design Hd(ejΩ).
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Example: Digital Differentiator
Chapter 7 Sampling
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Construction of Band-limited Digital Differentiator
c
cc
jjH
0
)(
periodic
TjHeH cj
d
)/()(
Chapter 7 Sampling
Desired:
Set ωs=2ωc=>cs
T
2
Assume ωM<ωc→Nyquist rate is met
Choice for Hd(ejΩ):
periodic
Tj /
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Band-limited Digital Differentiator (continued)
Chapter 7 Sampling
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Digital Differentiator in the Time-Domain
00
0)1(1
)cos(1
)]cos([11
)sin()(2
2)(
2
1][
00
02
n
nnT
dnn
nnT
dnjTjdeeHnh
n
oddHnjj
dd
d
periodic
TjeH j
d )(
Chapter 7 Sampling
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Example 7.2
t
Tttxc
)sin()( 2
)sin()cos()(
t
Tt
tT
Tttyc
][1
)(][ nT
nTxnx cd
)0(,][)1(
)(][][
kknT
nTyTnyTnhk
cdd
Chapter 7 Sampling
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Half-Sample Delay
)()( txty cc )()( jXejY cj
c
otherwise
ejH c
j
c0
)(
,)( Tjjd eeH
Chapter 7 Sampling
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Half-Sample Delay in the Frequency Domain
Chapter 7 Sampling
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Half-Sample Delay in the Time Domain
Chapter 7 Sampling
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Impulse-Train Sampling
k
p kNnkNxnpnxnx ][][][][][
Chapter 7 Sampling
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Illustration of Impulse-Train Sampling in Time
Chapter 7 Sampling
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Impulse-Train Sampling in Frequency Domain
deXePeX jjjp )()(
2
1)( )(
2
1
0
)( )(1
)(N
k
kjj seXN
eX
1
0
)(2
)(N
ks
j kN
eP
Chapter 7 Sampling
22
Illustration of Impulse-Train Sampling in Frequency
Chapter 7 Sampling
deXeP
eX
jj
jp
)()(2
1
)(
)(
2
1
0
)( )(1
)(N
k
kjj seXN
eX
1
0
)(2
)(N
ks
j kN
eP
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Recovery of a Discrete-Time Signal
Chapter 7 Sampling
n
nNnh
c
cc
sin
][
][][][ nhnxnx pr
k
r kNnhkNx ][][
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Discrete-Time Decimation
][][ nNxnx pb
Chapter 7 Sampling
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Discrete-Time Decimation in Frequency Domain
k
kjb
jb ekxeX ][)(
k
kjp ekNx ][
n
Nnjp enx ][
)()( Njp
jb eXeX
Chapter 7 Sampling
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Prepare of Downsampling
Chapter 7 Sampling
27
Upsampling
Chapter 7 Sampling
28
Sampling Rate Conversion
Chapter 7 Sampling
29
Summary
• DT Processing of CT Signals
• Digital Differentiator and Half-Sample Delay
• DT Decimation and interpolation
30
Problem Set
• P567 7.31
