Signals and Systems
Fall 2003
Lecture #14
23 October 2003
1,Review/Examples of Sampling/Aliasing
2,DT Processing of CT Signals
Sampling Review
Demo,Effect of aliasing on music.
Strobe Demo
> 0,strobed image moves forward,but at a slower pace
= 0,strobed image still
< 0,strobed image moves backward.
Applications of the strobe effect (aliasing can be useful sometimes):
— E.g.,Sampling oscilloscope
DT Processing of Band-Limited CT Signals
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
— DT frequency variable (? = ωΤ)
Step 1,Find the relation between x
c
(t) and x
d
[n],or X
c
(jω) and X
d
(e
j?
)
Time-Domain Interpretation of C/D Conversion
Note,Not full
analog/digital
(A/D) conversion
– not quantizing
the x[n] values
Frequency-Domain Interpretation of C/D Conversion
Note,ω
s
2π
CT DT
Illustration of C/D Conversion in the Frequency-Domain
)(eX
j?
d
)(eX
j?
d
1
ωT?=
2
ωT?=
D/C Conversion y
d
[n] → y
c
(t)
Reverse of the process of C/D conversion
Now the whole picture
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 x
c
(t),with
When the input x
c
(t) is band-limited (X(jω) = 0 at |ω| > ω
Μ
) and the
sampling theorem is satisfied (ω
s
> 2ω
M
),then
ω
M
<
ω
s
2
DT omege needs to changed
Frequency-Domain Illustration of DT Processing of CT Signals
Sampling
DT filter
Interpolate
(LPF)
equivalent
CT filter
CT freq → DT freq
DT freq → CT freq
Assuming No Aliasing
In practice,first specify the desired H
c
(jω),then design H
d
(e
j?
).
Example,Digital Differentiator
Applications,Edge Enhancement
Courtesy of Jason Oppenheim.
Used with permission.
Courtesy of Jason Oppenheim.
Used with permission.
Construction of Digital Differentiator
Bandlimited Differentiator
Band-Limited Digital Differentiator (continued)
CT DT