Discrete Fourier Transform

Frequency analysis of discrete time signals is usually and most conveniently performed on a digital signal processor, which may be a general purpose digital computer or specially designed digital hardware. To perform frequency analysis on a discrete time signal {x(n)}, we convert the time domain sequence to an equivalent frequency domain representation. We know that such a representation is given by the Fourier transform X(w) of the sequence {x(n)}. However X(w) is a continuous function of frequency and therefore, it is not a computationally convenient representation of the sequence  {x(n)}.

Definition of Discrete Fourier Transform

When the Sequence x(n) has a finite length L, less than or equal to N than the discrete Fourier transform can be written as :

 Where k = 0,1,2,...,N - 1

Inverse Discrete Fourier Transform
  Where n = 0,1,2,...,N - 1

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