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Discrete-Time Fourier
Transform
Chang-Su Kim
continuous time discrete time
periodic (series) CTFS DTFS
aperiodic (transform) CTFT DTFT
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
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
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
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
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
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
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