Signals and Systems

Continuous-Time Fourier

Transform

Chang-Su Kim

continuous time discrete time

periodic (series) CTFS DTFS

aperiodic (transform) CTFT DTFT

Lowpass Filtering – Blurring or Smoothing

Original Strong LPF

e.g. 21x21 moving

average filter

Less strong LPF

e.g. 11x11 moving

average filter

average

Highpass Filtering – Edge Extraction

CTFT Formula and Its Derivation

Bridge Between Fourier Series and Transform

Consider the periodic signal x(t)

Its Fourier coefficients are

0

0

)sin(

,2

10

1

k

k

k

Tk

T

T

ak

x(t)

t

-T1 0 T1 T-T

Sketch ak on the k-axis

The sketch is obtained by sampling the sinc function.

For each value of k, the signal x(t) has a periodic component with weight ak. So, the above sketch shows the frequency content of the signal x(t).

Bridge Between Fourier Series and Transform

ak

k-2 -1 0 1 2

2T1/T

The same sketch ak on the -axis:

On the -axis, the distance between two consecutive

ak’s is 0=2/T, which is the fundamental frequency.

ak

-20 -0 0 0 20

2T1/T

Bridge Between Fourier Series and Transform

The same sketch Tak on the -axis:

The distance between two adjacent ak’s is 0=2/T.

As T, 00. The distance between two consecutive ak’s becomes zero

The sketch of ak becomes continuous

The continuous curve X(jw) is called as Fourier Transform

Tak

-20 -0 0 0 20

2T1

Bridge Between Fourier Series and Transform

X(jw)

On the other hand, as T, the signal x(t) becomes an aperiodic signal

Fourier Transform can represent an aperiodic signalin frequency domain

x(t)

t-T1 0 T1

Bridge Between Fourier Series and Transform

How can we use this formula for a non-

periodic (aperiodic) function x(t)?

. . .

0 T 2T

)(~ tx

0

x(t)

( ) lim ( )T

x t x t

From CTFS to CTFT: Formal Derivation

( ) lim ( )T

x t x t

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Given the relationships

derive the following CTFT formula

From CTFS to CTFT: Formal Derivation

CTFT Formula – Fourier Transform Pair

( ) ( ) j tX j x t e dt

1( ) ( )

2

j tx t X j e d

Forward Transform

Inverse Transform

X(jw) represents the strength of frequency

component at w in x(t)

Time Domain vs. Frequency Domain

Fourier analysis (series or transform) is a tool

to determine the frequency contents of a given

signal

Conversion from time domain to frequency domain.

It is always possible to move back from

frequency domain to time domain

( ) ( ) j tX j x t e dt

1( ) ( )

2

j tx t X j e d

Some Examples

1)()( )()( F

dtetjXttx tj

t

1

0

Ex 1) Impulse function → constant function

Ex 2) Rectangular pulse → sinc function

1

1

1 11

2sin( ) 2 sinc( )

sin( )where sinc( )

Tj t

T

T Tj e dt T

tt

t

-T1 T1

1

2T1

1T

1T

t

Some Examples

Note the inverse relationship between time and frequency domains

More Examples

Unified Framework for CTFS and CTFT:

Periodic Signals Can Also Be Represented as Fourier Transform

Fourier Transform for Periodic Signals

Consider the inverse Fourier transform of

So, we can deduce that

k

k kaj )(2)( 0

0 Fourier

0( ) ( ) 2 ( )jk t

k k

k k

x t a e j a k

. . .

0 02 03

32 a

. . .

ja

jattx SF

2

1 ,

2

1)sin()( 11

..

0

Fourier Transform for Periodic Signals

Ex 1) sin function

0

0

)( j

/j

-/j

2

1)cos()( 11

..

0 aattx SF

0 0

)( j

Ex 2) cos function

0/ 2

. .

/ 2

1 1( ) ( ) ( )

2 2( ) ( )

Tjk tF S

kT

k

x t t kT a t e dtT T

kX j

T T

Fourier Transform for Periodic Signals

Ex 3) Fourier transform of impulse trains

1

T 2T

…

2/T

2/T 4/Tt

... …… F

)( jX)(tx

Properties of CTFT

Properties of CTFT

1. Linearity

2. Time shifting

3. Conjugation and conjugate symmetry

4. Differentiation and integration

( ) ( ) ( ) ( )Fa x t b y t aX j bY j

0

0( ) ( )j tFx t t e X j

* *

*

( ) ( )

( ) ( ) [ ( ) real]

Fx t X j

X j X j x t

( )( )

1( ) ( ) (0) ( )

F

tF

dx tj j

dt

x d X j Xj

Properties of CTFT

5. Time and frequency scaling

6. Parseval’s relation

7. Duality

)()(

)(||

1)(

jtx

a

j

aatx

F

F

djdttx 22 |)(|

2

1|)(|

)(2)( )()( FF gjtGjGtg

Convolution Property of CTFT

( ) ( ) ( ) ( ) ( ) ( )Fy t h t x t Y j H j X j

Two approaches for proof and understanding

1. LTI interpretation

Note that the frequency response H(jw) is just the

CTFT of the impulse response h(t).

2. Direct equation manipulation

Lowpass Filter

0

0 0

0

0,

( ) ( ),

0,

Y j X j

Convolution Property of CTFT

H(j)

-0 0

1X(j) Y(j)

Highpass Filter:

0

0 0

0

( ),

( ) 0,

( ),

X j

Y j

X j

Convolution Property of CTFT

H(j)

-0 0

1X(j) Y(j)

Bandpass Filter:

1 2

1 2

( ),

( ) ( ),

0, otherwise

X j

Y j X j

Convolution Property of CTFT

H(j)

-2 -1 1 2

1X(j) Y(j)

Examples

CTFT Table

CTFT Table

CTFT Table

Multiplication Property of CTFT

1( ) ( ) ( ) ( ) ( ) ( ( ))

2

Fr t s t p t R j S j P j d

This is a dual of the convolution property

- 00

A/2

G(j)

modulation

FT of p(t)=cos0t

- 00

P(j)

FT of r(t)

A

R(j)

g(t)p(t) Q(j) F

A/2

-200 20

A/4

Original signal is recovered after a low-pass filter

g(t) = r(t)p(t)

demodulation

Multiplication Property of CTFT

Idea of AM (amplitude modulation)

Multiplication Property of CTFT

A communication system

modulation

A

B

N

A B N

demodulation+ bandpass filtering

Causal LTI Systems Described

by Differential Equations

Linear Constant-Coefficient Differential

Equations

The DE describes the relation between the input x(t)

and the output y(t) implicitly

In this course, we are interested in DEs that

describe causal LTI systems

Therefore, we assume the initial rest condition

which also implies

M

kk

k

k

N

kk

k

kdt

txdb

dt

tyda

00

)()(

0 0If ( ) 0 for , then ( ) 0 forx t t t y t t t

1

0 00 1

( ) ( )( ) 0

N

N

dy t d y ty t

dt dt

Frequency Response

What is the frequency response H(jw) of the

following system?

It is given by

M

kk

k

k

N

kk

k

kdt

txdb

dt

tyda

00

)()(

0

0

( )( )

( )

M k

kk

N k

kk

b jH j

a j

Example

2

2

( ) ( ) ( )4 3 ( ) 2 ( ),

( ) ( ).t

d y t dy t dx ty t x t

dt dt dt

x t e u t

2

2

1 1 14 2 4

2

( ) ( ) ( )

2 1

( ) 4( ) 3 1

2

( 1) ( 3)

1 ( 1) 3

Y j H j X j

j

j j j

j

j j

j j j

31 1 1( ) ( )

4 2 4

t t ty t e te e u t

Q)

A)