Discrete Fourier Transform
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This document summarizes key aspects of the discrete Fourier transform (DFT). It defines the DFT, provides the formula for calculating it, and explains that the DFT transforms a discrete-time signal from the time domain to the frequency domain. It also outlines several important properties of the DFT, including linearity, shift property, duality, symmetry, and circular convolution. Examples are provided to illustrate duality and symmetry. References for further information on the discrete Fourier transform are also included.
![Discrete Fourier Transform
• Definition - For a length-N sequence x[n],
defined for 0 ≤ n ≤ N −1 only N samples of its
DFT are required, which are obtained by
uniformly sampling X (e jω
) on the ω-axis
between 0 ≤ ω≤ 2π at ωk = 2πk/ N, 0 ≤ k ≤ N −1
• From the definition of the DFT we thus have
N−1
ω=2πk/ N
= ∑ x[n]e− j2πk/ N ,
k=0
X[k] = X (e jω
)
0 ≤ k ≤ N −1](https://image.slidesharecdn.com/140120109005072141005-160407134247/85/Discrete-Fourier-Transform-5-320.jpg)
![Discrete Fourier Transform
X[k] is also a length-N sequence in the
frequency domain
• The sequence X[k] is called the Discrete
Fourier Transform (DFT) of thesequence
x[n]
• Using the notation WN = e− j2π/ N
the
DFT is usually expressed as:
N−1
n=0
X[k] = ∑ x[n]W kn
, 0 ≤ k ≤ N −1N](https://image.slidesharecdn.com/140120109005072141005-160407134247/85/Discrete-Fourier-Transform-6-320.jpg)
![Discrete Fourier Transform
• To verify the above expression we multiply
N
and sum the result from n = 0 to n = N −1
both sides of the above equation by W ln
1
∑ , 0 ≤ n ≤ N −1X[k]Wx[n]=
• The Inverse Discrete Fourier Transform
(IDFT) is given by
N−1
N k=0
−kn
N](https://image.slidesharecdn.com/140120109005072141005-160407134247/85/Discrete-Fourier-Transform-7-320.jpg)
![Discrete Fourier Transform
resulting in
∑ ( ∑
N −1 1 N−1
n=0 k=0
−kn
N−1
∑
n=0
WN
N
l nl n
X[k]WNx[n]WN =
=
1
∑ ∑
N−1N−1
n=0 k=0N
X[k]WN
−(k−l)n
=
1
∑ ∑
N−1N−1
k=0 n=0N
X[k]WN
−(k−l)n
)](https://image.slidesharecdn.com/140120109005072141005-160407134247/85/Discrete-Fourier-Transform-8-320.jpg)
![Discrete Fourier Transform
=
• Making use of the identity
N−1
n=0
∑ WN
−(k−l )n
0, otherwise
N, for k − l = rN, r an integer
we observe that the RHS of the last
equation is equal to X[l]
• Hence
Nx[n]W ln = X[l]
N−1
∑
n=0
{](https://image.slidesharecdn.com/140120109005072141005-160407134247/85/Discrete-Fourier-Transform-9-320.jpg)
Discrete Fourier Transform
- 1. Gandhinagar Institute Of Technology Subject – Signals and Systems ( 2141005) Branch – Electrical Topic – Discrete Fourier Transform
- 2. Name Enrollment No. Abhishek Chokshi 140120109005 Soham Davra 140120109007 Keval Darji 140120109006 Guided By – Prof. Hardik Sir
- 3. finite-duration Discrete Fourier Transform DFT is used for analyzing discrete-time signals in the frequency domain Let be a finite-duration sequence of length outside . The DFT pair of such that is: and
- 4. time domain frequency domain ... ...... ... discrete and finite discrete and finite
- 5. Discrete Fourier Transform • Definition - For a length-N sequence x[n], defined for 0 ≤ n ≤ N −1 only N samples of its DFT are required, which are obtained by uniformly sampling X (e jω ) on the ω-axis between 0 ≤ ω≤ 2π at ωk = 2πk/ N, 0 ≤ k ≤ N −1 • From the definition of the DFT we thus have N−1 ω=2πk/ N = ∑ x[n]e− j2πk/ N , k=0 X[k] = X (e jω ) 0 ≤ k ≤ N −1
- 6. Discrete Fourier Transform X[k] is also a length-N sequence in the frequency domain • The sequence X[k] is called the Discrete Fourier Transform (DFT) of thesequence x[n] • Using the notation WN = e− j2π/ N the DFT is usually expressed as: N−1 n=0 X[k] = ∑ x[n]W kn , 0 ≤ k ≤ N −1N
- 7. Discrete Fourier Transform • To verify the above expression we multiply N and sum the result from n = 0 to n = N −1 both sides of the above equation by W ln 1 ∑ , 0 ≤ n ≤ N −1X[k]Wx[n]= • The Inverse Discrete Fourier Transform (IDFT) is given by N−1 N k=0 −kn N
- 8. Discrete Fourier Transform resulting in ∑ ( ∑ N −1 1 N−1 n=0 k=0 −kn N−1 ∑ n=0 WN N l nl n X[k]WNx[n]WN = = 1 ∑ ∑ N−1N−1 n=0 k=0N X[k]WN −(k−l)n = 1 ∑ ∑ N−1N−1 k=0 n=0N X[k]WN −(k−l)n )
- 9. Discrete Fourier Transform = • Making use of the identity N−1 n=0 ∑ WN −(k−l )n 0, otherwise N, for k − l = rN, r an integer we observe that the RHS of the last equation is equal to X[l] • Hence Nx[n]W ln = X[l] N−1 ∑ n=0 {
- 10. Discrete Fourier Transform-DFT 1, 0 1 ( ) is a square-wave sequence ( ) 0, otherwise N N n N R n R n ， we use (( )) to denote (n modulo N)Nn (0)x (1)x (2)x (3)x (4)x (6)x (7)x (8)x (11)x (10)x (9)x (5)x 12N 12 20 8x x 12 1 11x x
- 11. Properties of DFT Since DFT pair is equal to DFS pair within , their properties will be identical if we take care of the values of and when the indices are outside the interval 1. Linearity Let and be two DFT pairs with the same duration of . We have: Note that if and are of different lengths, we can properly append zero(s) to the shorter sequence to make them with the same duration.
- 12. 2. Shift of Sequence If , then sure that the resultant time , we need shift, Note that in order to make index is within the interval of which is defined as where the integer is chosen such that
- 14. 3. Duality If , then 4. Symmetry If , then and
- 15. Example: Duality
- 16. be two DFT pairs with the 5. Circular Convolution Let and same duration of . We have where is the circular convolution operator.
- 18. References •Techmax and Technical •Wikipedia •Youtube Channel https://www.youtube.com/results?sea rch_query=properties+of+discrete+fouri er+transform