DFT - Discrete Fourier Transform and its Properties

DFT - Discrete Fourier Transform and its Properties

1. By
Ms.J.Shiny Christobel, AP/ECE
Sri Ramakrishna Institute of Technology, Coimbatore

2. 1.Linearity
  
 
   kbXkaXnbxnax
kXnx
kXnx
21
DFT
21
2
DFT
2
1
DFT
1



2.Duality  
  
N
DFT
DFT
kNxnX
kXnx


3.Circular Shift of a Sequence  
    
 mN/k2jDFT
N
DFT
ekX1-Nn0 mnx
kXnx




3. Periodicity
•X(K) is N-point DFT of a finite sequence x(n)
then x(n+N)=x(n) for all n
 DFT [x(k+N)]=x(k)
Time reversal of the sequence
•The time reversal of an N-point sequence x(n) is
attained by wrapping the sequence x(n) around the
circle in clockwise direction
Then X((-n))N=x(N-n)
 DFT [X(N-m)]=X(N-k)

4. Circular frequency shift
•If DFT[x(n)]=X(k)
 then DFT[ x(n)??????
��??????��
??????]=X((k-l))N
Complex conjugate
•If DFT[x(n)]=X(k)
 then DFT[ x*(n)]=X*(N-k)=X*((-k))N
Circular convolution
• x1(n)&x2(n) are finite duration sequence both of
length N
The DFTS X1(k),X2(k)
•Circular convolution of x1(n)&x2(n) represented as
 x3(n)=x1(n) x2(n)
Then DFT[x1(n) x2(n) ]=X1(k)X2(k)
N
N N

6. Relation ship between DTFT& DFT
•DTFT is a continuous periodic function of??????.
•DFT is obtained by sampling DTFT at a finite
number of equally spaced point over one period

7. Example1
1.Find the DFT of sequence ??????�=1,1,0,0
•Solution
•Let us assume N=L=4
•We have to find

•formula 1.

2.??????
−�??????
=??????�????????????−� ??????��??????
•Step1.Find x(0),x(1),x(2),x(3). Where k=0,1,2,..N-1
Find x(0)
x(0)= x(n)??????
−�2????????????.�/??????3
??????=0 = x(0)??????
−�2??????0.0/4
+ x(1)??????
−�2??????1.0/4
+
x(2)??????
−�2??????2.0/4
+ x(3)??????
−�2??????3.0/4
.
 = x(0)??????
0
+ x(1)??????
0
+ x(2)??????
0
+x(3)??????
0
.
X(0)=1+1+0+0=2 .
 ]n[xkX
DFT
 
 





1N
0n
knN/2j
e]n[xkX

8. Find x(1)
x(1)= ??????(�)
�
�=� ??????
−��??????�.�/�
= x(0)??????
−��??????�.�/�
+ x(1)??????
−��??????�.�/�
+
x(2)??????
−��??????�.�/�
+ x(3)??????
−��??????�.�/�
.
 =x(0).1+ x(1).??????�????????????/�-j??????��??????/�+ x(2).??????�????????????-
j??????��??????+ x(3).??????�??????�??????/�-j??????���??????/�.
 We know ??????�=�,�,�,�
 Therefore x(1) =1.1+1.(0-j)+0(??????�????????????-j??????��??????)+0.(
??????�??????�??????/�-j??????���??????/�).
 x(1) =1-j.
same way find x(2) =0, x(3) = 1+j
final answer assign X(K)= ??????�,??????�,??????�,??????(�)
X(K)= �,�−�,�,�+�

9. Example2
1.Find the IDFT of sequence ????????????=2,0,2,0.
Solution
Let us assume N= 4
We have to find x(n)
Formula
 1.X(n)=??????/?????? ??????(??????)
??????−�
??????=�??????
��??????�.�/??????
�.??????
−�??????
=??????�????????????−� ??????��??????
Step1.Find x(0),x(1),x(2),x(3). Where N=0,1,2,..N-1
Find X(0) N=4
X(n)= ??????/� ??????(??????)??????
��??????�.�/��
??????=�
 =??????/� ??????(??????)??????
�??????�.�/��
??????=�
for n=0
X(0)=
??????
�
[ ??????�??????
�??????
�.�
�+??????�??????
�??????
�.�
�+??????�??????
�??????
�.�
�+ ??????�??????
�??????
�.�
�]
 =
??????
�
[ �.??????
�
+�.??????
�
+�.??????
�
+ �.??????
�
]=
??????
�
[ �.�+�+�.�+ �]
 =4/4=1

 ]n[xkX
DFT


10. for n=1
X(1)=
??????
�
[ ??????�??????
�??????
�.�
�+??????�??????
�??????
�.�
�+??????�??????
�??????
�.�
�+ ??????�??????
�??????
�.�
� ] ??????�????????????/�-
j????????????�??????/�

=
??????
�
[ �.�+�.[??????�????????????/�+j????????????�??????/�]+
�.[??????�????????????+j????????????�??????]+ �.[??????�??????�??????/�+j????????????��??????/�]]
 =
??????
�
[ �+�+�−��]
=0
same way find x(2) =1, x(3) = 0
 final answer assign x(n)= ??????�,??????�,??????�,??????�
 x(n)= �,�,�,�