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There is no differennce between DFT and FFT. Both operations do the same thing, but FFT is faster in computation than DFT. Actually, in FFT, a divide and conquer approach is used to calculate the DFT. The algo for calculating FFT is as follows:
For a length N complex sequence , , the discrete Fourier transform (DFT) is defined by
X(f)=i/N∑xe^(-j2Πkn/N) where k=0,1,2----------N-1;
We are now in a position to have a full understanding of the transform kernel:
The kernel consists of samples of a complex sinusoid at discrete frequencies uniformly spaced between 0 and the sampling rate . All that remains is to understand the purpose and function of the summation over of the pointwise product of times each complex sinusoid. This can be interpreted as an inner product operation which computes the coefficient of projection of the signal onto the complex sinusoid . As such, , the DFT at frequency , is a measure of the amplitude and phase of the complex sinusoid which is present in the input signal at that frequency. This is the basic function of all linear transform summations (in discrete time) and integrals (in continuous time) and their kernels.
if we have to evaluate N point DFT, and if done in the obvious way clearly takes N^2 multiplications.
It is possible to calculate the DFT more efficiently than this, using the fast Fourier transform or FFT algorithm, which reduces the number of operations to O(NlogN).