Using a Scaling Factor in O(1/N) for the fixed-point implementation of the second-order goertzel filter

Overflows in the fixed-point implementation of the second-order Goertzel filter can be avoided by using either a non-uniform scaling factor or a very conservative one. According to a previously reported overflow analysis, where only real-valued input sequences of length N were considered, the required scaling factor can reach values in O(1/N²). In this paper, a method to use a scaling factor equal to π/(4N) is proposed based on a novel overflow analysis that considers also complex-valued input sequences. It is demonstrated that when a bank of Goertzel filters has to be implemented, the second-order version is more advantageous than the first-order version since it requires less hardware resources requirements.