Overflow analysis in the fixed-point implementation of the first-order Goertzel algorithm for complex-valued input sequences

The first-order Goertzel algorithm has advantages over the second-order Goertzel algorithm for fixed-point implementations due in part to the small scaling factor required to avoid overflow. The first-order system can achieve better accuracy on the computed Fourier coefficients than the second-order system if same size multipliers and adders are used. And when it is implemented as a completely parallelized system, fewer resources are required. In this paper, it is demonstrated that for complex-valued input sequences the known scaling factor 1/N does not guarantee that overflows are avoided in fixed-point implementations of the first-order Goertzel algorithm. An analysis is carried out and a new scaling factor equal to 1/(4N/pi) is proposed. The use of the new scaling factor guarantees that overflow will never happen even for complex-valued input sequences.