FFT algorithms are used to compute N point DFT for N samples of the sequence x(n). This requires N/2 log2N number of complex multiplications and N log2N complex additions. In some applications DFT is to be computed only at selected values of frequencies and selected values are less than log2N, then direct computations of DFT becomes more efficient than FFT. This direct computations of DFT can be realized through linear filtering of x(n). Such linear filtering for computation of DFT can be implemented using Goertzel algorithm.
By definition N point DFT is given as
Thus DFT can be obtained as the output of LSI system at n=N. Such systems can give X(k) at selected values of k. Thus DFT is computed as linear filtering operations by Goertzel Algorithm.
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