The document discusses the discrete-time Fourier transform (DTFT) and its properties. It begins by introducing the discrete-time Fourier series for periodic signals, then defines the DTFT by applying a periodic extension to non-periodic signals. Key steps include deriving the DTFT and its inverse, and examining common examples like the unit pulse and exponentially decaying signal. The DTFT is shown to be periodic with examples of how filters are represented.

1. ECE 8443 – Pattern Recognition
EE 3512 – Signals: Continuous and Discrete
• Objectives:
Derivation of the DTFT
Transforms of Common Signals
Properties of the DTFT
Lowpass and Highpass Filters
• Resources:
Wiki: Discete-Time Fourier Transform
JOS: DTFT Derivation
MIT 6.003: Lecture 9
CNX: DTFT Properties
SKM: DTFT Properties
LECTURE 16: DISCRETE-TIME FOURIER TRANSFORM
Audio:
URL:

2. EE 3512: Lecture 16, Slide 1
Discrete-Time Fourier Series
• Assume x[n] is a discrete-time periodic signal. We want to represent it as a
weighted-sum of complex exponentials:
Note that the notation <N> refers to performing the summation over an N
samples which constitute exactly one period.
• We can derive an expression for the coefficients by using a property of
orthogonal functions (which applies to the complex exponential above):
• Multiplying both sides by and summing over N terms:
• Interchanging the order of summation on the right side:
• We can show that the second sum on the right equals N if k = r and 0 if k  r:


 




 otherwise
,
0
...
,
2
,
,
0
,
)
/
2
( N
N
k
N
e
N
n
n
N
jk 
   
 
 
 



 







N
n N
k
n
N
r
k
j
k
N
n N
k
n
N
jr
n
N
jk
k
N
n
n
N
jr
e
c
e
e
c
e
n
x )
/
2
)(
(
)
/
2
(
)
/
2
(
)
/
2
( 



  T
e
c
e
c
n
x
N
k
n
jk
k
N
k
n
T
jk
k /
2
, 0
)
/
2
( 0






 
 



n
N
jr
e )
/
2
( 

  

 








N
n
n
N
r
k
j
N
k
k
N
n
n
N
jr
e
c
e
n
x )
/
2
)(
(
)
/
2
( 

 




N
n
n
N
jk
k e
n
x
N
c )
/
2
(
1 

3. EE 3512: Lecture 16, Slide 2
Discrete-Time Fourier Series (Cont.)
• This provides a closed-form expression for the Discrete-Time Fourier Series:
• Note that the DT Fourier Series coefficients can be interpreted as in the CT
case – they can be plotted as a function of frequency (k0) and demonstrate
the a periodic signal has a line spectrum.
• As expected, this DT Fourier Series (DTFS) obeys the same properties we
have seen for the CTFS and CTFT.
• Next, we will apply the DTFS to nonperiodic signals to produce the Discrete-
Time Fourier Transform (DTFT).
 
    (analysis)
1
1
)
(synthesis
/
2
,
0
0
)
/
2
(
0
)
/
2
(



















N
n
n
jk
N
n
n
N
jk
k
N
k
n
jk
k
N
k
n
T
jk
k
e
n
x
N
e
n
x
N
c
T
e
c
e
c
n
x







4. EE 3512: Lecture 16, Slide 3
Periodic Extension
• Assume x[n] is an aperiodic,
finite duration signal.
• Define a periodic extension of x[n]:
• Note that:
• We can apply the complex Fourier series:
• Define: . Note this is periodic in  with period 2.
• This implies: . Note that these are evenly spaced samples of our
definition for .
 
2
2
],
[
~ N
n
N
n
x
n
x 


  ]
[
~
~ n
x
N
n
x 

  

 N
as
n
x
n
x ]
[
~
 
n
jk
n
n
jk
N
N
n
n
jk
N
N
n
n
n
jk
N
N
k
k
e
n
x
N
e
n
x
N
e
n
x
N
c
T
e
c
n
x
0
0
2
1
0
0
]
[
1
]
[
~
1
]
[
~
1
/
2
,
~
2
/
2
/
0
2
/
2
/



























  n
j
n
j
e
n
x
N
e
X 
 




 ]
[
1
 
0
1 
jk
n e
X
N
c 
 

j
e
X

5. EE 3512: Lecture 16, Slide 4
• Derive an inverse transform:
• As
• This results in our Discrete-Time Fourier Transform:
Notes:
• The DTFT and inverse DTFT are not symmetric. One is integration over a
finite interval (2π), and the other is summation over infinite terms.
• The signal, x[n] is aperiodic, and hence, the transform is a continuous
function of frequency. (Recall, periodic signals have a line spectrum.)
• The DTFT is periodic with period 2. Later we will exploit this property to
develop a faster way to compute this transform.
The Discrete-Time Fourier Transform
        0
0
0
0
0
0
0
2
1
)
2
(
2
1
)
1
(
~ 








 n
jk
N
N
k
jk
n
jk
N
N
k
jk
n
jk
N
N
k
jk
e
e
X
N
e
e
X
e
e
X
N
n
x 

















 











 


 d
N
n
x
n
x
N 0
0 ,
0
2
],
[
]
[
~
,
 
  equation)
(analysis
]
[
equation)
(synthesis
2
1
]
[
2








n
n
j
j
n
j
j
e
n
x
e
X
d
e
e
X
n
x








6. EE 3512: Lecture 16, Slide 5
Example: Unit Pulse and Unit Step
• Unit Pulse:
The spectrum is a constant (and periodic over the range [-,].
• Shifted Unit Pulse:
Time delay produces a phase shift linearly proportional to . Note that these
functions are also periodic over the range [-,].
• Unit Step:
Since this is not a time-limited function, it has
no DTFT in the ordinary sense. However, it can
be shown that the inverse of this function is
a unit step:
  1
]
[
]
[
]
[
]
[














n
n
j
n
n
j
j
e
n
e
n
x
e
X
n
n
x





 
    0
0
0
0
0
0
and
1
]
[
]
[
]
[
]
[
n
e
e
X
e
e
X
e
e
n
n
e
n
x
e
X
n
n
n
x
n
j
j
n
j
j
n
j
n
n
j
n
n
j
j





































]
[
]
[ n
u
n
x 
  )
(
1
1






  j
j
e
e
X

7. EE 3512: Lecture 16, Slide 6
Example: Exponential Decay
• Consider an exponentially decaying signal:
1
]
[
]
[ 
 a
n
u
a
n
x n
 





j
n
n
j
n
n
j
n
n
n
j
j
ae
ae
e
a
e
n
x
e
X















 


1
1
)
(
]
[
0
0
  2
2
2
2
2
2
2
2
)
cos(
2
1
1
Hence,
)
cos(
2
1
)
(
sin
)
(
cos
)
cos(
2
1
))
sin(
(
))
cos(
1
(
:
Note
a
a
e
X
a
a
a
a
a
a
a
j




















 
2
2
))
sin(
(
))
cos(
1
(
1
)
sin(
))
cos(
1
(
1
1
1






a
a
ja
a
ae
e
X j
j







 

8. EE 3512: Lecture 16, Slide 7
The Spectrum of an Exponentially Decaying Signal
 
 
a
a
a
a
a
a
e
X
a
a
a
a
a
a
e
X
j
j






















1
1
)
1
(
1
)
2
1
(
1
)
cos(
2
1
1
1
1
)
1
(
1
)
2
1
(
1
)
0
cos(
2
1
1
2
2
2
2
2
2
0





 Lowpass Filter:
Highpass Filter:

9. EE 3512: Lecture 16, Slide 8
Finite Impulse Response Lowpass Filter












1
1
1
1
,
0
,
1
,
0
]
[
N
n
N
n
N
N
n
n
h
  )
2
/
sin(
))
2
/
1
(
sin(
)
( 1
1
1
1
1




 


 
 




 N
e
e
e
X
N
N
n
n
j
N
N
n
n
j
j
• The frequency response of
a time-limited pulse is a
lowpass filter.
• We refer to this type of
filter as a finite impulse
response (FIR) filter.
• In the CT case, we obtained
a sinc function (sin(x)/x) for the frequency response. This is close to a sinc
function, and is periodic with period 2.

10. EE 3512: Lecture 16, Slide 9
Example: Ideal Lowpass Filter (Inverse)
 




c
c
n
n
d
e
n
h c
n
j







sin
)
1
(
2
1
]
[
The Ideal Lowpass
Filter Is Noncausal!

11. EE 3512: Lecture 16, Slide 10
Properties of the DTFT
• Periodicity:
• Linearity:
• Time Shifting:
• Frequency Shifting:
Example:
    CTFT)
the
from
(different
)
2
( 

 j
j
e
X
e
X 

   

 j
j
e
bY
e
aX
n
by
n
ax 

 ]
[
]
[
   

 j
n
j
e
X
e
n
n
x 0
0



   
)
( 0
0 

 

 j
n
j
e
X
n
x
e
  ]
[
)
1
(
]
[ n
x
n
x
e
n
y n
n
j


 
 Note the role
periodicity
plays in the
result.

12. EE 3512: Lecture 16, Slide 11
Properties of the DTFT (Cont.)
• Time Reversal:
• Conjugate Symmetry:
Also:
• Differentiation in
Frequency:
• Parseval’s Relation:
• Convolution:
 ]
[
]
[ 

j
e
X
d
d
j
n
nx 
   

 j
j
e
X
e
X
n
x *
real
]
[ 
 
   
 
   
  functions
odd
are
Im
and
functions
even
are
Re
and




j
j
j
j
e
X
e
X
e
X
e
X

 
  odd
and
imaginary
purely
is
odd,
and
real
is
]
[
even
and
real
is
even,
and
real
is
]
[


j
j
e
X
n
x
e
X
n
x
 

j
e
X
n
x 

 ]
[
  
 

d
e
X
n
x j
n
2
2
2
2
1
]
[ 
 



)
(
)
(
)
(
]
[
*
]
[
]
[ 

 j
j
j
e
X
e
H
e
Y
n
h
n
x
n
y 



13. EE 3512: Lecture 16, Slide 12
Properties of the DTFT (Cont.)
• Time-Scaling:
)
(
otherwise
0
k
of
multiple
a
is
n
if
]
/
[
]
[ 
jk
k e
X
k
n
x
n
x 





14. EE 3512: Lecture 16, Slide 13
Example: Convolution For A Sinewave

15. EE 3512: Lecture 16, Slide 14
Summary
• Introduced the Discrete-Time Fourier Transform (DTFT) that is the analog of
the Continuous-Time Fourier Transform.
• Worked several examples.
• Discussed properties: