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Chengbin Ma UM-SJTU Joint Institute
Class#13
Chapter 4: Applications of Fourier Representations to Mixed Signal Classes
- Fourier transform representations of periodic signals (4.2)
- Convolution and multiplication with mixtures of periodic and non-periodic
signals (4.3)
Midterm#2 is postponed, now Apr. 3rd, next Friday.
FS FT FS (FT is from FS, but a generalization of FS)
Slide 1
Chengbin Ma UM-SJTU Joint Institute
Review of Previous Lecture
Relations between time domain and frequency domain.
Multiplication property: windowing
Parseval relationships: conservation of energy, connect
the time-domain and frequency-domain responses
Time-bandwidth product: relationship between the speed
of response (time-domain) and bandwidth (frequency-
domain)
Definitions of Td and Bw; Proof of time-bandwidth
product
Duality: time and frequency axes are interchangeable
Inverse Fourier Transform: use partial-fraction expansion
(three special cases)
Slide 2
Chengbin Ma UM-SJTU Joint Institute
This Lecture
Expand FT to analyze periodic signals
Discuss this expansion in the cases of
convolution and multiplication.
i.e., transform from X[k]k to X[jw]w
Slide 3
Chengbin Ma UM-SJTU Joint Institute
Class#13
Chapter 4: Applications of Fourier Representations to Mixed Signal Classes
- Fourier transform representations of periodic signals (4.2)
- Convolution and multiplication with mixtures of periodic and non-periodic
signals (4.3)
- Sampling (4.5)
Slide 4
Chengbin Ma UM-SJTU Joint Institute
Mixing of Signals (1)
Periodic and nonperiodic signals (e.g., x(t)
could be periodic, but h(t) is usually a non-
periodic one)
Slide 5
Chengbin Ma UM-SJTU Joint Institute
Mixing of Signals (2)
Continuous- and discrete-time signals
Slide 6
This lecture will focus on the
pervious case, mixture of periodic
and non-periodic signals.
Chengbin Ma UM-SJTU Joint Institute
Expansion of FT to Include FS
FT can be expanded to analyze periodic
signals.
Purpose: analyze the mixtures of periodic and
non-periodic signals.
Slide 7
dtetxjX tjww )()(
Ttjkdtetx
TkX
0
0)(1
][w
Fourier Series Fourier Transform
Chengbin Ma UM-SJTU Joint Institute
FS representation of periodic signal x(t) is
Slide 8
Complex Sinusoidal
k
tjkekXtx 0][)(
w
X[k] versus k
to
X(jw) versus w
Chengbin Ma UM-SJTU Joint Institute
Problem: FT of
Inverse Fourier transform of
Slide 9
FT representation of the complex sinusoids
tjke 0
w
02 ww k
ww
w dejXtx tj)(2
1)( 020 ww
w FT
tje
Refer to slide 22, class#10.
Chengbin Ma UM-SJTU Joint Institute Slide 10
Frequency Shift
k
FT
k
tjk
kkXjX
ekXtx
0][2)(
][)( 0
www
w
020 www
FTtj
e
X[k] versus k X[jw] versus w
Chengbin Ma UM-SJTU Joint Institute Slide 11
Example (1)
Strength of 2X[k] spaced by the fundamental
frequency w0.
k
kkXjX 0][2)( www
Ttjkdtetx
TkX
0
0)(1
][w
Chengbin Ma UM-SJTU Joint Institute
Example (2)
Example 4.2, p344: FT of a unit impulse train
P 4.1, p344: FT of square wave (period T=4)
FS coefficients (Ex. 3.13, p221):
FT?
Slide 12
k
kkX
2/sin][
Ttjkdtetx
TkX
0
0)(1
][w
k
kkXjX 0][2)( www
Strength of 2X[k] spaced by the fundamental frequency w0.
Chengbin Ma UM-SJTU Joint Institute
Class#13
Chapter 4: Applications of Fourier Representations to Mixed Signal Classes
- Fourier transform representations of periodic signals (4.2)
- Convolution and multiplication with mixtures of periodic and non-
periodic signals (4.3)
Slide 13
Chengbin Ma UM-SJTU Joint Institute Slide 14
Convolution with Mixed Signals
FT ( ) ( )* ( ) ( ) ( ) ( )
FS ( ) ( ) ( ) [ ] [ ] [ ]
where ( ) ( ) ( ) ( )
(periodic convolution)
T
y t h t x t Y j H j X j
y t h t x t Y k TH k X k
h t x t h x t d
w w w
=
Chengbin Ma UM-SJTU Joint Institute Slide 15
Derivation (x(t) is periodic)
k
FT
kkXjXtx 0][2)()( www
k
FT
jHkkXjYthtxty wwww 0][2)()(*)()(
k
FT
jkHkkXjYthtxty 00][2)()(*)()( wwww
)()()()(*)()( www jHjXjYthtxtyFT
MichaelRectangle
Chengbin Ma UM-SJTU Joint Institute
Ex 4.4, p349: An LTI system with impulse
response h(t)=(1/(t))sin(t) and the periodic
square wave. Find its output.
Slide 16
Example
k
tjkekXtx 0][)(
w
020 www
FTtj
e
Low-pass filter
k
kkX
2/sin][
Chengbin Ma UM-SJTU Joint Institute Slide 17
Multiplication with Mixed Signals
ww
w jXjGjYtxtgtyFT
*2
1)()()()(
k
kkXjX 0][2)( www
0][*)()()()( wwww kkXjGjYtxtgtyk
FT
0][)()()()( www kjGkXjYtxtgtyk
FT
(x(t) is periodic)
Chengbin Ma UM-SJTU Joint Institute
Ex. 4.5, p352: Consider a system with output
y(t)=g(t)x(t). Let x(t) be the square wave and
g(t)=cos(t/2), sketch Y(jw) .
Slide 18
Example (1.1)
0][)()()()( www kjGkXjYtxtgtyk
FT
24
,2/sin
][
0
w
T
k
kkX
020 www
FTtj
e
2][)(
ww kjGkXjY
k
G(jw)
-1/2 1/2
Chengbin Ma UM-SJTU Joint Institute
Example (1.2)
Slide 19
2/12 :Hint
)2/2/1()2/2/1()2/sin(
)(
/k
kkk
kjY
k
ww
w
Magnitude of G(jw) is
scaled by X[k], and
G(jw) is continuously
shifted by kw0 =k/2.
For example, when
k=0, its magnitude is
2/)2/sin(
lim0
k
k
k
0][)()()()( www kjGkXjYtxtgtyk
FT
Chengbin Ma UM-SJTU Joint Institute
Homework
Problem 4.18(a)(b)(e)
Problem 4.20(a)(c)
Due: 02:00PM of next Thursday
Slide 20