Discrete-Time Fourier

Transform

Chang-Su Kim

continuous time discrete time

periodic (series) CTFS DTFS

aperiodic (transform) CTFT DTFT

DTFT Formula and Its Derivation

DTFT Formula

DTFT

cf) CTFT

Note that in DT case, 𝑋(𝑒𝑗𝜔) is periodic with period 2𝜋 and the inverse transform is defined as a integral over one period

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Examples

11

(a) [ 1]2

(b) [ 1] [ 1]

n

u n

n n

2 , 0(a) ( )

2 , 0

jwj w

X ej w

Find the Fourier transforms of

Find the inverse Fourier transform of

Derivation of DTFT from DTFS

x[n]

N-N-N1 N2

][~ nx

0

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

As , [ ] [ ]

and the DTFS formula becomes the desired DTFT formula

N x n x n

DTFT of Periodic Functions

0

0[ ] ( ) 2 ( )jk n F j

k k

k N k

x n a e Χ e a k

Periodic functions can also be represented as

Fourier Transforms

Examples

DTFT of periodic functions

0

(a) cosine function

[ ] cos

(b) periodic impulse train

[ ] [ ] k

x n w n

y n n kN

Selected Properties of DTFT

Shift in Frequency

This property can be used to

convert a lowpass filter to a

highpass one, or vice versa

0 0( )

( )

[ ] ( )

( 1) [ ] ( )

jw n j w wF

Fn j w

e x n X e

x n X e

Differentiation in Frequency

( )[ ]

jwF dX enx n j

dw

Parseval’s Relation

22

2

1[ ] ( )

2

jw

n

x n X e dw

Time Expansion

otherwise , 0

of multipleinteger an is if , ]/[][)(

knknxnx k

. . .

][nx

1 2 3

][)3( nx

3 6 94 5

For a natural number k, we define

)(][ )(jkF

k enx

Convolution

[ ] [ ] [ ] ( ) ( ) ( )F jw jw jwy n x n h n Y e X e H e

Multiplication

( )

1 2 1 22

1[ ] [ ] [ ] ( ) ( ) ( )

2

F j j jy n x n x n Y e X e X e d

Multiplication in time domain corresponds to the

periodic convolution in frequency domain

Summary of Fourier Series and

Transform Expressions

All the Four Formulas

CT DT

Periodic(series)

CTFS DTFS

Aperiodic(transform)

CTFT DTFT

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Time Frequency

aperiodic continuousperiodic discretediscrete periodiccontinuous aperiodic

[ ], ( )x n x t

[ ], ( )x n x t

[ ]x n

( )x t

( ), ( )jX e X j

( ), ( )jX e X j

( )jX e

( )X j

Properties

CT DT

Periodic(series)

CTFS DTFS

Aperiodic(transform)

CTFT DTFT

0

0

0

( )

1(

) )

)

( 2 (

jk t

k

k

jk t

k

k

k

T

x t a e

a x t e d

a k

t

X

T

j

2

2

( )

(

0

)

[ ]

1[

( 2 (

]

) )

N

N

jk n

k

k N

jk n

k

k

N

k

n

j

x n a e

a x n eN

Χ e a k

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

sampling of continuous

functions => discrete

Dualities

CT DT

Periodic(series)

CTFS DTFS

Aperiodic(transform)

CTFT DTFT

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Dualities

CT DT

Periodic(series)

CTFS DTFS

Aperiodic(transform)

CTFT DTFT

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Dualities

CT DT

Periodic(series)

CTFS DTFS

Aperiodic(transform)

CTFT DTFT

0

0

( ) ,

1( )

jk t

k

k

jk t

k

T

x t a e

a x t e dtT

2

2

( )

( )

[ ]

1[ ]

N

N

jk n

k

k N

jk n

k

n N

x n a e

a x n eN

2

1[ ] ( )

2

( ) [ ]

j j n

j j n

n

x n X e e d

X e x n e

1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Causal LTI Systems Described

by Difference Equations

Linear Constant-Coefficient Difference

Equations

The DE describes the relation between the input x[n]

and the output y[n] implicitly

In this course, we are interested in DEs that

describe causal LTI systems

Therefore, we assume the initial rest condition

0 0

[ ] [ ]N M

k k

k k

a y n k b x n k

0 0If [ ] 0 for , then [ ] 0 forx n n n y n n n

Frequency Response

What is the frequency response H(ejw) of the

following system?

It is given by

0

0

( )

M jkw

kjw k

N jkw

kk

b eH e

a e

0 0

[ ] [ ]N M

k k

k k

a y n k b x n k

Example

3 1[ ] [ 1] [ 2] 2 [ ],

4 8

1[ ] [ ]. What is [ ]?

4

n

y n y n y n x n

x n u n y n

Q)

Analogy between Differential Equations

and Difference Equations

Many properties, learned in differential equations,

can be applied to solve interesting problems

described by difference equations

Fibonacci sequence

a0 = a1 = 1

an = an-1 + an-2 (n≥2)

Golden ratio = 1.61803

Golden Ratio

Golden rectangle is said to be the most visually pleasing geometric

form

1

1.618

Golden Ratio

Golden rectangle is said to be the most visually

pleasing geometric form

1

1.618