**Theory:**

FFT is a most efficient algorithm used in speech processing, frequency estimation and communication. FFT utilizes full advantage of two symmetry properties in order to provide efficient algorithm for DFT. while considering a divide and conquer approach, a computationally efficient algorithm can be developed. This methods depends on the decomposition of an N-point DFT into successively smaller size then DFTs. While in the N-point sequence, the term N can be expressed as N = r_{1}r_{2}r_{3}, …, r_{m}, then N = r ^{m}, can be decimated into *r*-point sequences For each r point sequence, r-point DFT are often computed. Hence the DFT is of size r. if the number r is called the radix of the FFT algorithm while the number m indicates the number of stages in computation. If the value of r = 2, it is called radix-2 FFT.

**Steps involved in finding the FFT of the given sequence:**

- Let’s consider an input sequence as x[n].
- Let’s find the length for the input sequence using length command.
- In order to find FFT and IFFT by using the matlab commands fft and ifft.
- Let’s plot the magnitude and phase response for the given sequence.
- And finally display the results.

**Program:**

clc;

clear all;

close all;

x=input(‘Enter the sequence : ‘)

N=length(x)

xK=fft(x,N)

xn=ifft(xK)

n=0:N-1;

subplot (2,2,1);

stem(n,x);

xlabel(‘n—->’);

ylabel(‘amplitude’);

title(‘input sequence’);

subplot (2,2,2);

stem(n,abs(xK));

xlabel(‘n—->’);

ylabel(‘magnitude’);

title(‘magnitude response’);

subplot (2,2,3);

stem(n,angle(xK));

xlabel(‘n—->’);

ylabel(‘phase’);

title(‘Phase response’);

subplot (2,2,4);

stem(n,xn);

xlabel(‘n—->’);

ylabel(‘amplitude’);

title(‘IFFT’);

**Output:**

Enter the sequence: [3 2 4 5 1]

x = 3 2 4 5 1

N =5

xK =15.0000 + 0.0000i -3.3541 – 0.3633i 3.3541 – 1.5388i 3.3541 + 1.5388i -3.3541 + 0.3633i

xn =3.0000 2.0000 4.0000 5.0000 1.0000