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Signals and SystemsSpring 2008
Lecture #14(4/3/2008)
Covers O & W pp. 527-5431. Review/Examples of
Sampling/Aliasing2. DT Processing of CT Signals
“Figures and images used in these lecture notes by
permission,copyright 1997 by Alan V. Oppenheim and Alan S.
Willsky”
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If X j!( ) = 0, ! >! M " x(t) is band - limited
and ! s =2#
T> 2!M
then, assuming we choose !M
Sampling Review
Aliasing –– if the sampling criterion is not satisfied, that is:
ωs < 2ωM.
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A Simple Example of aliasingx(t) = cos(!
0t)
After a LPF, getour originalsignal back. Noproblem.
Q: Whathappens to thesignals within theLPF when thesignal
frequencyωo increases?
In this case,ωM = ωo
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Demo:The effect of sampling on sinusoidal audio signals
cos(ωot)ωo is varied
The cut-off frequency ωc/2π of theLPF is kept at ~4 kHz ~
ωs/4π.
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Demo:The effect of sampling on music
The sampling frequency ωs/2π is varied and displayed onthe
frequency meter.
ωs/2π
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Demo: The effect of quantization on music.
In addition to sampling to convert CT signals to DTsignals, the
DT signals are further quantized in theiramplitudes. That is, x →
(10110101001010), convert x intoa binary number with a limited
precision and maximumvalue, so that they can be processed by
DSPs.
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Strobe Demo
Applications of the strobe effect (aliasing can be useful
sometimes):— Sampling oscilloscope (will be discussed in
recitation)
M
∆ = 0, strobed image still∆ > 0, strobed image moves forward,
but at a slower pace∆ < 0, strobed image moves backward.
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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 variableΩ — DT frequency variable (Ω = ωΤ)
Step 1: Find the relation between xc(t) and xd[n], or Xc(jω) and
Xd(ejΩ)
Hc(jω)
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Time-Domain Interpretation of C/D Conversion
DiscontinuousCT signal
DT Sequencexd[n] = xc(nT)
Both have a periodic spectrumperiodicityin ω?
periodicityin Ω?
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Illustration of C/D Conversion in the Frequency-Domain
Note: Ω = ωT2π ⇔ ωs
DT CT
ωs=
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Frequency-Domain Interpretationof C/D Conversion
x p(t) = xc(t) ! " t # nT( )n= #$
+$
% = xc(nT)" t # nT( )n =#$
+$
%
Note:xd[n] = xc(nT)
b b
Xp j!( ) =1
TXc j ! " k!s( )( )
n= "#
+#
$ = xc(nT)e" j!nT
n="#
+#
$ — CT (1)
— periodic with period !s =2%T( )
! Compare Eqs. (1) & (2), and remember " = #T
Xd ej"( ) = Xp j
"
T
$ %
& '
$ % (
& ' )
Xd ej!( ) = xd n[ ] e" j! n
n= "#
+#
$ = xc (nT)e" j!nn= "#
+#
$ — DT (2)
— periodic with period 2%( )
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D/C Conversion yd[n] → yc(t)Reverse of the process of C/D
conversion
Again, ! ="T
Yp j"( ) = Yd ej"T( ) — Reverses frequency scaling
Yc j!( ) =TYd e
j!T( )0
! <!s2
— band - limited
otherwise
" # $
% $
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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 xc(t), with
• When the input xc(t) is band-limited (X(jω) = 0 at |ω| >
ωΜ) and thesampling theorem is satisfied (ωs > 2ωM), then
!M<!
s
2
Yc ( j!) = Hc( j!) Xc( j! )" # $ yc( t) = hc (t) % xc (t)
LTI
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Frequency-domain Illustration of DT Processing of CT Signals
OriginalCT signal
aftersampling
after C/Dconversion
DTprocessing
after D/Cconversion
equivalentCT system
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Assuming No Aliasing (using an AAF)Step #1 CT ! DT: Xd e
j"( ) = Xp j" / T( ) — periodic
=1
TXc j" / T( ) , # $ < " < $ if no aliasing
In practice, first specifies the desired Hc(jω), then design
Hd(ejΩ)=Hc(jΩ/T).
Step # 2 DTsystem: Yd ej!( ) = Hd ej!( )Xd ej!( )
=1
THd e
j!( )Xc j! / T( ) , " #
Step# 3 D / C: Yc j!( ) =TYd e
j!T( ) = Hd ej!T( )Xc j!( )
0
,
,
"! s2
! s2
otherwise
# $ %
!
Hc j"( ) =Hd e
j"T( )0
,
,
" < "s2
otherwise
# $ %
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Example: Digital DifferentiatorApplications: Edge
Enhancement
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Hc(jω)
Construction of Band-limited Digital Differentiator
Desired: Hc j!( ) =j!
0
,
,
! !c
" # $
Choice for Hd(ejΩ): Hd ej!( ) =
Hc j! / T( ) , ! < "
periodic ! # "
$ % &
' &
→ Nyquist rate is metSet !s= 2!
c" T =
2#
!s
=#
!c
. Assume !M
c
=j ! / T( ) , ! < "
periodic ! # "
$ % &
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Band-limited Digital Differentiator (continued)
Desired CT system DT implementation
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Digital Differentiator in the Time-Domain
For: Hd (ej!) =
j(! / T ) |! | < "
periodic |! |# "
$ % &
hd[n] is the corresponding “Fourier series” of the triangle
wave
hd[n] =1
2!Hd2!" (e
j#)e
j#nd# =
Hd odd 2
2!j(# / T ) $ jsin(#n) d#
0
!
"
=
1
nT
(!1)n n " 0
0 n = 0
#
$ %
& %
= !1
"T
1
n!# cos(#n)[ ]
0
"!1
ncos(#n)d#
0
"
$% & '
( ) *
Next lecture covers O & Wpp. 582-599
