The Goertzel algorithm is more efficient than the Fast Fourier Transform in computing an -point DFT if less than DFT coefficients are required . In DTMF detection, we only need 8 of, for example, 205 DFT coefficients to detect the first harmonics of the 8 possible tones, and then apply decision logic to choose the strongest touch tone. Since DTMF signals do not have second harmonics, we could compute another 8 DFT coefficients to compute the second harmonics to detect the presence of speech .
The Goertzel algorithm computes the th DFT coefficient of the input signal using a second-order filter
where . The th DFT coefficient is produced after the filter has processed samples: . The key in an implementation is to run for samples and then evaluate . The computation for takes one add () and one multiply-accumulate per sample. In DTMF detection, we are only concerned with the power of the th coefficient, :
The value of must be shorter than the samples in half of a DTMF signaling interval (), be large enough for good frequency resolution (), and meet the relative error specification 2(a). We used a conventional value of = 205  because it is roughly half of 400 samples. Decision logic can be added to give a valid DTMF signal if the same two DTMF tones are detected in a row to add robustness against noise.
Table 1: Chosen values to minimize the error in .