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