1

1

Covers O & W pp. 385-427• DTFT Properties and Examples• Duality• Magnitude/Phase of Fourier

transforms

2

Convolution Property ExampleExample 5.13(O & W p385)

!

h[n] =" nu[n] , x[n] = # n

u[n] , |" |, |# |<1

!

"

y[n] = h[n]# x[n]$ % & Y (ej') =

1

1() e( j'*

1

1( + e( j'

– ratio of polynomials in , = e( j'

!

H(ej") =

1

1#$ e# j", X(e

j") =

1

1# % e# j"

!

" #$ : Y (ej%) =PFE A

1&$ e& j%+

B

1& " e& j%

A, B – determined by partial fraction expansion

!

y[n] = A" nu[n] + B# n

u[n] .

!

" =# : Y (ej$) =

1

(1%# e% j$ )2

y[n] = ?

!

ny[n ]" jdY (e

j#)

d#$ % $ $ $ $ $ y[n] = (n +1)& nu[n]

EECE233 Signals and Systems Lecture Note #11 Prof. Joon Ho Cho (Slides thanks to Qing Hu*, D. Boning, D. Freeman, T. Weiss, J. White, and A. Willsky)

* These lecture notes are based on the lecture notes used in 6.003 taught by Prof. Qing Hu at M.I.T. These notes are adopted for EECE233 under the permission of Prof. Qing Hu.

2

3

DT LTI System Described by LCCDE’s

— Rationalfunction of e-jω,use PFE to geth[n].

!

aky[n " k] =k= 0

N

# bkx[n " k]k= 0

M

#

!

From time - shifting property : x[n " k]# $ % e" jk&

X(ej&)

!

"

FT on both sides : ake# jk$

Y (ej$) =

k= 0

N

% bke# jk$

X(ej$)

k= 0

M

%

!

"

Devide both sides by ake# jk$

k= 0

N

% : Y (e j$) =

bke# jk$

k= 0

M

%

ake# jk$

k= 0

N

%

&

'

( ( ( (

)

*

+ + + +

H (e j$ )

1 2 4 3 4

X(ej$)

4

Example: First-order recursive system

!

y[n] "#y[n "1] = x[n] , |# |<1

!

with the condition of initial rest " causal

!

b

h[n] =" nu[n] – infinite duration

!

FT on both sides : (1"#e" j$ )Y (e j$) = X(e

j$)

!

H(ej") =

Y (ej")

X(ej")

=1

1#$e# j"

3

5

DTFT Multiplication Property

!

y[n] = x1[n] " x

2[n]# $ % Y (e

j&) =

1

2'X1(e

j()X

2(e

j(&)( ))d(

2'*

=1

2'X1(e

j&)+ X

2(e

j&) – periodic convolution

!

=1

2"X1(e

j#)X

2(e

j($%# ))d#

2"& QED

!

Proof : Y (ej") = x

1[n] # x

2[n]e

$ j" n

n=$%

%

&

!

Use synthesis Eq. for x1[n] : =

1

2"X1(e

j#)e

j# nd#

2"$

%

& '

(

) *

x1[n]

1 2 4 4 4 4 3 4 4 4 4

x2[n]e

+ j, n

n=+-

-

.

!

Exchange order of " and # : =1

2$X1(e

j%) x

2[n]e

& j('&% )n

n=&(

(

"

X 2 (ej (' &% )

)

1 2 4 4 3 4 4

)

*

+ +

,

+ +

-

.

+ +

/

+ +

d%2$#

6

Calculating Periodic Convolutions

Suppose we integrate from -π to π:

ˆ X 1

ej!( ) =

X1

ej!( ) , ! " #

0 , otherwise

$ % &

!

Y (ej") =

1

2#X1(e

j$)X

2(e

j("%$ ))d$

%#

#

&

!

can be written as : =1

2"ˆ X

1(e

j#)X

2(e

j($%# ))d#

%&

&

'

Changes a periodic convolution to a regular convolution.

where

4

7

y n[ ] =sin ! n 4( )

! n

"

# $

%

& '

2

= x1 n[ ] ( x2 n[ ] , x1 n[ ] = x2 n[ ] =sin ! n 4( )

! n

Y(e j) ) =1

2!X1(e

j) )* X2 (ej) )

Example:

8

Duality in Fourier AnalysisFourier Transform is highly symmetric

CTFT: Both time and frequency are continuous and in general aperiodic

x(t) =1

2!X j"( )e j"td"

#$

+$

%

X j"( ) = x(t)e# j" t

dt#$

+$

%&

Suppose f(*) and g(*) are two functions such that they are a Fourier-transform pair (Eg. f is a square and g is a sinc),

x1(t) = f(t) ↔ X1(jω) = g(ω).

Then for another time-domain function such that,

x2(t) = g(t)

then without doing any math, we should know that

X2(jω) = 2πf (-ω)

Same operation

5

9

Example of CTFT dualitySquare pulse in either time or frequency domain

10

Duality between CTFS and DTFT

x(t) = akejk! 0t

k ="#

+#

$ = x(t + T ) !o =2%

T

ak =1

Tx(t)e

" jk! 0tdtT&

x n[ ] =1

2!X(e j" )e j"nd"

2!#

X(e j" ) = x n[ ]e$ j"n

n= $%

+%

& = X(ej " + 2!( ))

CTFS

DTFT

Periodic in time ! " # discrete in frequency

Discrete in time ! " # periodic in frequency

6

11

Illustration: Periodic in time ! " # discrete in frequency

⇓

It is clear that x(t) = ˜ x ( t)! "(t # nT) — sampling functionn= #$

$

%

Any periodic signalx(t) is aconvolution of x(t)in one period with

!

"(t # nT)n=#$

$

%!

"(t # nT)n=#$

$

%

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Illustration (continued):

Also discrete

From the symmetry of Fourier Transform, the reverse is also true:

Discrete in time DTFT

! " # # # periodic in frequency

Discrete

X( j! )1 2 3 =˜ X ( j! ) "

2#

T$(! %

k2#

T)

k= %&

&

'1 2 4 4 4 3 4 4 4

— also a sampling function

7

13

DTFS

x n[ ] = akejk!

0n

k=<N>

" = x n + N[ ] !o =2#

N

ak =1

Nx n[ ]e$ jk!0 n

k =<N >

" = ak+N

%

Discrete & periodic in time ! " # periodic & discrete in frequency

Suppose f[*] and g[*] are two functions such that they are a FT pair,

x1[n] = f [n] ↔ ak = g[k].

Then for another time-domain function such that,

bk ↔ x2[n] = g[n]

then without doing any math, we should know that

bk = N f [-k]

Samemathematicaloperation

14

Magnitude and Phase of FT, and Parseval Relation

CT:

Parseval Relation:

DT:

Parseval Relation:

x(t) =1

2!X j"( )ej"td"

#$

+$

%

X j"( ) = X j"( )e j&X j"( )

x(t)2dt

#$

+$

% =1

2!X j"( )

2

1 2 4 3 4 d"

#$

+$

%

x n[ ] =1

2!X e

j"( )e j"nd"2!#

X ej"( ) = X e

j"( )ej$X e j"( )

x n[ ]2

=n =%&

+&

'1

2!X e j"( )

2

d"2!#

Energy distribution in ω

8

15

Effects of Phase• Not on signal energy distribution as a function of

frequency

• Can have dramatic effect on signal shape/character

— Constructive/Destructive interference

• Is that important?

— Depends on the signal and the context, not as important as the amplitude in hearing, but very important for image processing.

16

Demo: 1) Effect of phase on Fourier Series

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17

Demo: 2) Effect of phase on image processing

Original image of a boat

Amplitude of FT

Phase of FT

18

Demo:2) Effect of phase on image processing (cont.)

Constructed with the amplitude ofFT of the boat, but with auniform phase.

Constructed with the phase ofFT of the boat, but with auniform amplitude.

10

19

Demo:2) Effect of phase on image processing (cont.)

This image was constructed with the phase of FT of the boat, but withamplitude from a very different image –– you wouldn’t have guess it.

20

Demo:2) Effect of phase on image processing (cont.)

The amplitude information for the previous image is from our oldfriend! Cannot make a monkey out of a boat!Next lecture covers O & W pp. 427-472.