DFT and FFT
Presented by
Asst.lecturer
Noora Hani
Matrix Formulation of The DFT and IDFT
The IDFT From The Matrix Form
Example: find the 4-point DFT of x(n)=[1, 2,1,0] using
matrix
Example: find the IDFT of X(k)=[4, -2j,0, 2j]
using DFT
SOL:
Given X(k) = {4, –j2, 0, j2}, X*(k) = {4, j2, 0, –j2}
The IDFT of X(k) is determined using matrix as shown below.
To find IDFT of X(k) first find X*(k), then find DFT of X*(k), then take
conjugate of DFT {X*(k)} and divide by N.
Example: find the IDFT of X(k)=[4,2,0,4]using
DFT
Fast Fourier Transform (FFT)
►Decimation-in-Time Algorithms: sequence s(n) will be broken
up into odd numbered and even numbered subsequences.
This algorithm was first proposed by Cooley and Tukey in 1965.
► Decimation-in-Frequency Algorithms: the sequence s(n) will
be. Broken up into two equal halves. This algorithm was first
proposed by Gentlemen and Sande in 1966.
Decimation-in-Frequency Algorithms (DIF)
►► In decimation-in-frequency FFT algorithm, the output DFT sequence X(K) is broken
into smaller and smaller subsequences. For the derivation of this algorithm, the number of
points or samples in a given sequence should be N = 2r where r > 0. For this purpose, we
can first-divide the input sequence into the first-half and the second-half of the points.
►► Flow graph of complete decimation-in-frequency (DIF)
decomposition of an N-point DFT computation (N = 8).
►► Complex multiplication in direct computation DFT is N*N
►► Complex multiplication in FFT algorithms is N/2 log2 N
►►the number of addition in FFT algorithms is N log2 N
►► the number of addition in DFT algorithms is N(N-1)
►► Basic butterfly to implement DIF FFT is
Example: find 4-point DFT of the sequence
x(n)=[2,1,4,3] using DIF FFT
SOL:
Example: an 8-point sequence is given by x(n)=
[2,2,2,2,1,1,1,1] compute DFT by radix-2 DIF FFT
algorithm
sol: