Discrete Time Fourier Transform

Discrete Time Fourier Transform
• Why Fourier Transform
• General Properties & Symmetry relations
• Formula and steps
• magnitude and phase spectra
• Convergence Condition
• mean-square convergence
• Gibbs phenomenon
• Direct Delta
• Energy Density Spectrum

Image Transforms
• Many times, image processing tasks can be
best performed in a domain other than the
spatial domain.
• Key steps
(1) Transform the image
(2) Carry the task(s) in the transformed domain.
(3) Apply inverse transform to return to the
spatial domain.

Why is FT Useful?
• Easier to remove undesirable frequencies in
the frequency domain.
• Faster to perform certain operations in the
frequency domain than in the spatial domain.
– i.e., using the Fast Fourier Transform (FFT)

Frequency Filtering: Main Steps
1. Take the FT of f(x):
2. Remove undesired frequencies:
3. Convert back to a signal:
x

Example: Removing undesirable frequencies
remove high
frequencies
filtered
signal
frequencies
noisy signal

FT - Definitions
• F(u) is a complex function:
• Magnitude of FT (or spectrum):
• Phase of FT:
• Magnitude-Phase representation:
• Power of f(x): P(u)=|F(u)|2=

Discrete-Time Fourier Transform
• Definition - The Discrete-Time Fourier
Transform (DTFT) of a sequence
x[n] is given by
• In general, is a complex function of
the real variable w and can be written as
)
( w
j
e
X
)
( w
j
e
X






n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(
)
(
)
(
)
( im
re
w
w
w j
j
j
e
X
j
e
X
e
X 


• and are, respectively, the
real and imaginary parts of , and are
real functions of w
• can alternately be expressed as
where
)
( w
j
e
X
)
( ω
re
j
e
X )
( ω
im
j
e
X
)
( w
j
e
X
)
(
)
(
)
( w

w
w  j
j
j e
e
X
e
X
)}
(
arg{
)
( w

w
 j
e
X

• is called the magnitude function
• is called the phase function
• Both quantities are again real functions of w
• In many applications, the DTFT is called the
Fourier spectrum
• Likewise, and are called the
magnitude and phase spectra
)
( w
j
e
X
)
(w

)
( w
j
e
X )
(w


• Inverse Discrete-Time Fourier Transform:
 w





w
w
d
e
e
X
n
x n
j
j
)
(
2
1
]
[

Symmetry relations of the DTFT of a
complex sequence

Symmetry relations of the DTFT
of a real sequence
x[n]: A real sequence

DTFT of unit Impulse Sequence
• Example - The DTFT of the unit sample
sequence d[n] is given by
1
]
0
[
]
[
)
( 
d

d

w
 w




n
j
n
e
n






n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(

DTFT of Causal Sequence
• Example - Consider the causal sequence


 



otherwise
0
0
1
]
[
,
1
],
[
]
[
n
n
n
n
x n










n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(


 




w




w

w
0
]
[
)
(
n
n
j
n
n
n
j
n
j
e
e
n
e
X
w





w


 
 j
e
n
n
j
e
1
1
0
)
(

DTFT of Causal Sequence
• The magnitude and phase of the DTFT
are shown below
)
5
.
0
1
/(
1
)
( w

w

 j
j
e
e
X
-3 -2 -1 0 1 2 3
0.5
1
1.5
2
w/
Magnitude
-3 -2 -1 0 1 2 3
-0.4
-0.2
0
0.2
0.4
0.6
w/
Phase
in
radians

Convergence Condition
• - An infinite series of the form
may or may not converge
• Consider the following approximation





w

w
n
n
j
j
e
n
x
e
X ]
[
)
(




w

w
K
K
n
n
j
j
K e
n
x
e
X ]
[
)
(

• Then for uniform convergence of ,
• If x[n] is an absolutely summable sequence, i.e., if
for all values of w
• Thus, the absolute summability of x[n] is a sufficient
condition for the existence of the DTFT
)
( w
j
e
X
0
)
(
)
(
lim 
 w
w


j
K
j
K
e
X
e
X






n
n
x ]
[












w

w
n
n
n
j
j
n
x
e
n
x
e
X ]
[
]
[
)
(

• Example - The sequence for
is absolutely summable as
and therefore its DTFT converges to
uniformly
]
[
]
[ n
n
x n



1







 

 





 1
1
]
[
0
n
n
n
n
n
)
( w
j
e
X
)
1
/(
1 w


 j
e

• Since
an absolutely summable sequence has always
a finite energy
• However, a finite-energy sequence is not
necessarily absolutely summable
,
]
[
]
[
2
2














 n
n
n
x
n
x

• - The sequence
has a finite energy equal to
• However, x[n] is not absolutely summable since the
summation
does not converge.
Example




]
[n
x 0
0
1
1


n
n
n
,
,
/
6
1 2
1
2


 








n
x
n
E







1
1
1
1
n
n n
n

Mean Square Convergence
• To represent a finite energy sequence that is not
absolutely summable by a DTFT, it is necessary to
consider a mean-square convergence of
where
)
( w
j
e
X
0
)
(
)
(
lim
2

w
 



w
w


d
e
X
e
X j
K
j
K




w

w
K
K
n
n
j
j
K e
n
x
e
X ]
[
)
(

Mean Square Convergence
• Here, the total energy of the error
must approach zero at each value of w as K
goes to
• In such a case, the absolute value of the error
may not go to zero as K goes to and the
DTFT is no longer bounded


)
(
)
( w
w
 j
K
j
e
X
e
X
)
(
)
( w
w
 j
K
j
e
X
e
X

Example
• - Consider the DTFT
shown below





w

w
w

w


w
c
c
j
LP e
H
,
0
0
,
1
)
(
)
( w
j
LP e
H
c
w 
0
1

 c
w

w

Example
• The inverse DTFT of is given by
• The energy of is given by
• is a finite-energy sequence, but
it is not absolutely summable
)
( w
j
LP e
H
]
[n
hLP
,
sin
2
1
n
n
jn
e
jn
e c
n
j
n
j c
c

w












w

w
w



w
w

w
d
e
n
h
c
c
n
j
LP
2
1
]
[




 n

w /
c
]
[n
hLP

Example
• As a result
does not uniformly converge to for all
values of w, but converges to in the
mean-square sense
)
( w
j
LP e
H
)
( w
j
LP e
H
 
w




w



w

K
K
n
n
j
c
K
K
n
n
j
LP e
n
n
e
n
h
sin
]
[

Example
• The mean-square convergence property of the
sequence can be further illustrated by
examining the plot of the function
for various values of K as shown next
 
w



w

w
K
K
n
n
j
c
j
K
LP e
n
n
e
H
sin
)
(
,
]
[n
hLP

Example
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w/
Amplitude
N = 20
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w/
Amplitude
N = 30
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w/
Amplitude
N = 10
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w/
Amplitude
N = 40

Example
• As can be seen from these plots,
independent of the value of K there are
ripples in the plot of around
both sides of the point
• The number of ripples increases as K
increases with the height of the largest
ripple remaining the same for all values of K
)
(
,
w
j
K
LP e
H
c
w

w

Gibbs Phenomenon
• As K goes to infinity, the condition
holds indicating the convergence of
to
• The oscillatory behavior of
approximating in the mean-square
sense at a point of discontinuity is known as
the Gibbs phenomenon
)
(
,
w
j
K
LP e
H
)
( w
j
LP e
H
0
)
(
)
(
lim
2
, 
w
 



w
w


d
e
H
e
H j
K
LP
j
LP
K
)
(
,
w
j
K
LP e
H
)
( w
j
LP e
H

Dirac delta function d(w)
• The DTFT can also be defined for a certain
class of sequences which are neither
absolutely summable nor square summable
• Examples of such sequences are the unit step
sequence [n], the sinusoidal sequence
and the exponential sequence
• For this type of sequences, a DTFT
representation is possible using the Dirac delta
function d(w)
)
cos( 

w n
o
n
A

Dirac delta function d(w)
• A Dirac delta function d(w) is a function of
w with infinite height, zero width, and unit
area
• It is the limiting form of a unit area pulse
function as  goes to zero, satisfying
)
(w

p











 w
w
d
w
w d
d
p )
(
)
(
lim
0
w
2


2

0

1
)
(w

p

Example
• Example - Consider the complex exponential
sequence
• Its DTFT is given by
where is an impulse function of w and
n
j o
e
n
x w

]
[
 

w

w
d




w
k
o
j
k
e
X )
2
(
2
)
(
)
(w
d


w


 o

Example
• The function
is a periodic function of w with a period 2
and is called a periodic impulse train
• To verify that given above is indeed
the DTFT of we compute the
inverse DTFT of
 

w

w
d




w
k
o
j
k
e
X )
2
(
2
)
(
)
( w
j
e
X
n
j o
e
n
x w

]
[
)
( w
j
e
X

Example
• Thus
where we have used the sampling property of
the impulse function )
(w
d
  w


w

w
d








w
k
n
j
o d
e
k
n
x )
2
(
2
2
1
]
[
n
j
n
j
o
o
e
d
e w



w

w
 w

w
d
 )
(

Energy Density Spectrum
• The total energy of a finite-energy sequence
g[n] is given by
• From Parseval’s relation given above we
observe that





n
g n
g
2
]
[
E
 w







w



d
e
G
n
g j
n
g
2
2
2
1
)
(
]
[
E

• The quantity
is called the energy density spectrum
• Therefore, the area under this curve in the
range divided by 2 is the energy
of the sequence


w



2
)
(
)
( w

w j
gg e
G
S

Example
• - Compute the energy of the sequence
• Here
where






w
 n
n
n
n
h c
LP ,
sin
]
[
 w






w



d
e
H
n
h j
LP
n
LP
2
2
)
(
2
1
]
[





w

w
w

w


w
c
c
j
LP e
H
,
0
0
,
1
)
(

• Therefore
• Hence, is a finite-energy sequence



w

 w



w
w




c
n
LP
c
c
d
n
h
2
1
]
[
2
]
[n
hLP

Discrete Time Fourier Transform

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