Gandhinagar Institute Of Technology

Subject – Signals and Systems ( 2141005)Branch – Electrical Topic – Discrete Fourier Transform

Name Enrollment No. Abhishek Chokshi 140120109005 Soham Davra 140120109007Keval Darji 140120109006

Guided By – Prof. Hardik Sir

finite-durationDiscrete Fourier TransformDFT is used for analyzing discrete-time signals in the frequency domain

Let be a finite-duration sequenceof length outside . The DFT

pair of

such that is:

and

time domain frequency domain

... ...... ...

discrete and finite discrete and finite

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 = 2π k / N, 0 ≤ k ≤ N

−1

• From the definition of the DFT we thus have

N −1

=ω 2π k / N

= ∑ x[n]e− j 2π k /

N ,k =0

X [k ] = X (e jω)

0 ≤ k ≤ N −1

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 the sequence x[n]

• Using the notation WN = e− j 2π / N

the DFT is usually expressed as:N −1

n=0X [k ] = ∑ x[n]W kn, 0 ≤ k ≤ N −1N

Discrete Fourier Transform

• To verify the above expression we multiplyN

and sum the result from n = 0 to n = N −1

both sides of the above equation by W l n

1 ∑ , 0 ≤ n ≤ N −1X

[k ]Wx[n] =

• The Inverse Discrete Fourier Transform(IDFT) is given by

N −1

N k =0

−knN

Discrete Fourier Transform

resulting in

∑ ( ∑N −1 1 N −1

n=0 k =0

−knN −1∑n=0

WNNl nl n X

[k ]WN

x[n]WN

=

= 1 ∑ ∑

N −1N −1n=0 k =0

NX [k ]WN

−(k − l)n

= 1 ∑ ∑

N −1N −1k =0 n=0

NX [k ]WN

−(k − l)n

)

Discrete Fourier Transform

=

• Making use of the identityN −1

n=0∑ WN

−(k −l )n0,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 l n = X [ l]

N −1∑n=0

{

Discrete Fourier Transform-DFT

1, 0 1( ) is a square-wave sequence ( )

0, otherwiseN N

n NR 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

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.

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

3. Duality

If , then

4. Symmetry

If , then

and

Example: Duality

be two DFTpairs

withthe

5. Circular Convolution

Let andsame duration of . We have

where is the circular convolution operator.

Symmetry Properties

References• Techmax and Technical • Wikipedia• Youtube Channel https://

www.youtube.com/results?search_query=properties+of+discrete+fourier+transform

THANK-YOU