Signals and Systems

11.5 DISCRETE FOURIER TRANSFORM

We know that the Fourier transform of discrete sequence x(n) is given by

 

 

where X(ω) is the continuous function of ω and the range of ω is from −π to +π or 0 to 2π. Since X(ω) is the continuous function of ω, it is not possible to calculate it on a digital computer or a digital signal processor. Hence it is necessary to calculate it only for discrete values of ω. When Fourier transform is calculated at discrete points, it is called discrete Fourier transform (DFT) which is denoted by X(k) and is expressed as

 

 

In Eq. (11.19) X(k) is called DFT which is computed at k = 0, 1, 2, ……., N −1 i.e., N discrete ...