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Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Q. 1) Compute the DFT of each the following finite length sequences considered to be of length N
(Where N is even).
a) [ ] [ ]x n n
b) 0 0[ ] [ ] 0 1x n n n n N
c) 1, 0 1
[ ]0, 0 1
n even n Nx n
n odd n N
d) 1, 0 ( / 2) 1
[ ]0, ( / 2) 1
n Nx n
N n N
e) , 0 1
[ ]0,
na n Nx n
otherwise
Solution: Q. 1a)
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0,1,2,........., 1N j kn
N
n
X k x n e for k N
Given: 1 0
( ) ( )0
nx n n
elsewhere
Therefore
1 2
0
( ) ( )
( ) 1 0,1,2,.........., 1
N j knN
n
X k n e
X k for k N
Thus we got the important DFT pair as given below
( )n 1 DFT
N-point
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Solution: Q. 1b)
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0,1,2,........., 1N j kn
N
n
X k x n e for k N
Given: 0
0
1( ) ( )
0
n nx n n n
elsewhere
Therefore
0
1 2
0
2
( ) ( )
( ) 0,1,2,.........., 1
N j knN
n
j knN
X k n e
X k e for k N
Thus we got the important DFT pair as given below
Solution: Q. 1c)
Given: 1, 0 1
[ ]0, 0 1
n even n Nx n
n odd n N
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0,1,2,........., 1N j kn
N
n
X k x n e for k N
( )n
02j knNe
DFT
N-point
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
In this question ( )x n is only present for the even values of ‘n’ and the sequence ( )x n is zero for odd values of
‘n’. Hence the length of the sequence is2
N. Thus the formula for computing the DFT of this sequence will be as
follows:
2
122
0
( ) (2 ) 0,1,2,.........., 1N
N
j kn
n
X k x n e for k N
As it is given that, the value of the sequence is 1 for even values of ‘n’ thus the above equation can be
written as
12 4
0
2
( )/
( ) 0,1,2,.........., 1
1( )
1
N
j knN
n
j k
j k N
X k e for k N
eX k
e
Thus, 2 2
0,( )
0
NN kX k
elsewhere
Solution: Q. 1d)
Given: 1, 0 ( / 2) 1
[ ]0, ( / 2) 1
n Nx n
N n N
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0,1,2,........., 1N j kn
N
n
X k x n e for k N
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Thus
12 2
0
( ) 0,1,2,........., 1
N
j knN
n
X k e for k N
(2 )/
1( ) 0,1,2,........., 1
1
j k
j k N
eX k for k N
e
2
(2 )/
0
2( )
1
0 , 0 ( 1)
N
j k N
k
X k k odde
k even k N
Solution: Q. 1e)
Given: , 0 1
[ ]0,
na n Nx n
otherwise
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0,1,2,........., 1N j kn
N
n
X k x n e for k N
Thus,
1 2
0
( ) 0,1,2,........., 1N j kn
n N
n
X k a e for k N
1 2
0
( ) 0,1,2,........., 1
nN j kN
n
X k ae for k N
2
(2 )/
1( ) 0,1,2,........., 1
1
N j k
j k N
a eX k for k N
ae
(2 )/
1( ) 0,1,2,........., 1
1
N
j k N
aX k for k N
ae
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Q. 2) Consider the complex sequence
0 , 0 1[ ]
0,
j ne n N
x notherwise
a) Find the Fourier transform X(ejw
) of x[n].
b) Find the N-point DFT X[k] of the finite-length sequence x[n].
c) Find the DFT of x[n] for the case 00
2 k
N
, where k0 is an integer.
Solution: Q. 2a)
The Fourier transform X(ejw
) of x[n] is given by the formula.
( ) ( )j j n
n
X e x n e
0
1
0
( )N
j nj j n
n
X e e e
0
1( )
0
( )N
j nj
n
X e e
0
0
( )
( )
1( )
1
j Nj
j
eX e
e
0
00
00
0
( )2
( ) ( )2 2
( )/2( )/2
( )/2
1
( )1
Nj
N Nj j
j
jj
j
e
e eX ee
ee
0 0
00
0 0
( ) ( )2 2( )
( )/22( )/2 ( )/2
( )
N Nj j
Nj
jj
j j
e eX e e e
e e
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
0 0
0
0 0
( ) ( )12 2( )
2
( )/2 ( )/2( )
N Nj jN
jj
j j
e eX e e
e e
0
1 0( )2
0
2sin ( )2
( )( )
2sin2
Nj
j
N
X e e
0
1 0( )2
0
sin ( )2
( )( )
sin2
Nj
j
N
X e e
Solution: Q. 2b)
Given: 0 , 0 1
[ ]0,
j ne n N
x notherwise
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0 1N j kn
N
n
X k x n e for k N
0
1 2
0
( ) 0 1N j kn
j n N
n
X k e e for k N
0
1 2
0
( ) 0 1
nN j kj N
n
X k e e for k N
0
21
0
( ) 0 1
nkN j
N
n
X k e for k N
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
0
0
2
2
1( )
1
kj N
N
kj
N
eX k
e
0
00
0 0
0
2
22 2
2 2
2 2
2 2
2
2
1
( )1
k Nj
Nk N k Nj jN N
j k j k
N N
j k
N
e
e eX k
e e
e
0 0
0 0
0 0
2 22 2 2 2
2 2
2 2
2 2
( )
k N k Nj j
k N j k N NjN N
j k j k
N N
e eX k e e
e e
0
2 1 0
2
0
22sin
2( )
1 22sin
2
k Nj
N
k N
NX k e
k
N
0
2 1 0
2
0
2sin
2( )
1 2sin
2
k Nj
N
k N
NX k e
k
N
Note that 2( ) ( )jk
N
X k X e
Solution: Q. 2c)
Here we need to compute the DFT of x[n] for the case 00
2 k
N
, where k0 is an integer. To find this we
need to replace 00
2 k
N
in the solution of Q. 2b)
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
1 2
0
( ) ( ) 0 1N j kn
N
n
X k x n e for k N
01 2 2
0
( ) 0 1N j k n j kn
N N
n
X k e e for k N
01 2 ( )
0
( ) 0 1N j k k n
N
n
X k e for k N
0
0
2 ( )
2 ( )
1( )
1
j k k
k kj
N
eX k
e
0
2 1( )
02
0
sin ( )( )
sin ( ) /
Nj k k
Nk k
X k ek k N
Q. 3) Find the N-point DFT of the following finite duration sequence of length L (N>=L)
, 0 1[ ]
0,
A n Lx n
otherwise
Solution: Q. 3)
Given: , 0 1
[ ]0,
A n Lx n
otherwise
We know that the N-point DFT of a Discrete Time Sequence ( )x n is given by the formula
1 2
0
( ) ( ) 0 1N j kn
N
n
X k x n e for k N
1 2
0
( ) 0 1L j kn
N
n
X k Ae for k N
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
1 2
0
( )L j kn
N
n
X k A e
2 /
2
1( )
1
j kL N
kj
N
eX k A
e
( 1)/ sin( / )( )
sin( / )
j L N kL NX k Ae
k N
Q. 4) Compute the DFT of the following sequence x(n)= [0, 1, 2, 1] and check the validity of your
answer by calculating its IDFT.
Try to solve this question by your own.
Q. 5) Let ( )jX e denote the Fourier transform of the sequence1
[ ] ( )2
n
x n u n
. Let [ ]y n denote
a finite duration sequence of length 10; i.e., [ ] 0, 0, [ ] 0, 10.y n n and y n n The 10-point
DFT of [ ]y n , denoted by [ ]Y k , corresponds to 10 equally spaced samples of ( )jX e i.e.,
2 /10[ ] ( )j kY k X e . Determine [ ]y n .
Solution: Q. 5)
We know that, Fourier transform of the sequence1
[ ] ( )2
n
x n u n
is given by
( ) ( )j j n
n
X e x n e
0
1( )
2
n
j j n
n
X e e
0
1( )
2
n
j j
n
X e e
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
1( )
11
2
j
j
X e
e
To find [ ]Y k , will take 10 equally spaced samples of ( )jX e
2 /10( ) ( ) , 0 9j
kY k X e k
Thus we have the 10-point DFT of [ ]y n
(2 /10)
1( ) , 0 9 (1)
11
2
j k
Y k k
e
But here we are asked to find [ ]y n . So we need to take IDFT of ( )Y k . But here I will use the formula of
DFT which will give us the relationship between [ ]y n and ( )Y k
210
9
0
( ) [ ] 0 9 (2)j kn
n
Y k y n e k
Recall that
point DFT
(2 / )
1(3)
1
NNn
j k N
aa
ae
So, by comparing equation number 1, 2 and 3, we may infer that
10
1
2[ ] , 0 9
11
2
n
y n n
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Solve the following questions
Q. 1) Determine the 8-point DFT of the sequence 1 0 3
[ ]0
nx n
otherwise
Q. 2) Determine the 8-point DFT of the sequence [ ] {1,2,3,4}x n
Q. 3) Determine the 6-point DFT of the sequence [ ] cos5
nx n
Q. 4) Determine the 4-point DFT of the sequence
1[ ] 2 ( ) 4 ( 1) 2 ( 2) 4 ( 3)x n n n n n
2[ ] ( ) 6 ( 1) 3 ( 3)x n n n n
Unsolved Problems from the book: “Digital signal processing: Principles, Algorithms, and
Applications” by John G. Proakis & D. G. Manolakis, Fourth Edition.
Chapter No. 7
Problem Nos.: 7.1, 7.5, 7.6, 7.12, 7.16, 7.17, 7.23
Department of Electronics and Telecommunication Engineering Yeshwantrao Chavan College of Engineering Nagpur
ET317: Digital signal Processing
Tutorial No: 2: Discrete Time Fourier Transform
©Course Instructors: Prof. V. R. Gupta, E.T. Department, YCCE, Nagpur.
Some Basic Formulas are required for solving these numerical which are as follows:
2
2
1 2
0
1
0
1 2
0
1
0
: ( ) ( ) 0 1
( ) ( ) 0 1; ,
1: ( ) ( ) 0 1
1( ) ( ) 0 1; ,
jN
jN
N j knN
n
Nkn
N N
n
N j knN
n
Nkn
N N
n
DFT X k x n e for k N
OR
X k x n W for k N where W e
IDFT x n X k e for n NN
OR
x n X k W for k N where W eN
2
1 2
1
1 2
1
1 1
11
Nn N N
N
N N for a
a a afor a
a
1
0
1
11
1
Nn N
N for a
a afor a
a
cos sinie i
2 cos(2 ) sin(2 ) cos(2 ) 1 for all the valuesof
sin(2 ) 0 for all the valuesof
j ke k i k k k
k k
1cos( ) sin( ) cos( )
1
sin( ) 0 for all the valuesof
j kfor k even
e k i k kfor k odd
k k