VoIP Voice and Fax Signal Processing

7.6 DFT OR FFT PROCESSING

The FFT efficiently calculates all possible frequency points in the DFT for an N-point DFT as given in reference [Madisetti and Williams (1998)]. A generic DFT calculation is given by

In the above equation, k = 0, 1, 2… N − 1 is the frequency index; ; and is for complex numbering representation. To get actual frequency mapping, frequency index “k” has to be multiplied by frequency resolution f_s/N, where f_s is the sampling frequency usually of 8000 Hz. For an 80 sample block (N) at an 8000-Hz sampling rate, the frequency resolution of analysis is 8000/80 = 100 Hz. For DTMF analysis, only selected DFT frequency bin values are required. The frequencies of interest are at four row values, four column values, and total of eight second harmonics of row and column frequencies. For selected frequency bins of computation, the DFT algorithm is more efficient than the FFT-based algorithm. A typical DTMF signal varies from 25 to 50 ms in duration, which gives few small blocks (about 10 ms) of DFT processing similar to short-time Fourier transform (STFT). DTMF tones do not appear at integer multiples of f_s/N, which calls for frequency-domain interpolation by padding several ...