Reply by spock April 5, 20082008-04-05
I have found fixed point implementation of Goertzel algorithm in book
“Real – Time Digital Signal Processing – Implementations and
Applications”. I will paste here a part of code responsible bor
computing recursive path.

void gFilter (short *d, short in, short coef)
  long AC0;
  d[0] = in >> 7;         // Get input with scale down by 7 bit
  AC0 = (long) d[1] * coef;
  AC0 >>= 14;
  AC0 -= d[2];
  d[0] += (short)AC0;
  d[2] = d[1];            // Update delay-line buffer
  d[1] = d[0];

Book explains that this operation (AC0 >>= 14) aligns multiplication
product to be stored in Q15 format. I have read in a few places about Q15
numbers multiplication and they all specified that after multiplication
you have to perform shift right by 15. I don’t understand why AC0 is
shifted by 14 not by 15. Similar operation is performed in non recursive
part of code responsible for computing Goertzel output.