Finite-precision Goertzel filters used for signal tone detection

The effect of coefficient quantization on the output of a second-order Goertzel filter tone-detector is analyzed in order that the Goertzel filter may be systematically designed for minimum complexity. Three alternative configurations are identified for the Goertzel filter's component second-order digital resonator, which are especially suited to VLSI implementation, and which complement each other on the zero-to-Nyquist frequency interval. The Goertzel filter's tone response is formulated using the Fourier summation transform (and shown to depend on both the frequency and phase of the incident tone). Hence analytic expressions are derived for the tone response of a fully finite-precision Goertzel filter when used with either a rectangular or half-end-point, input data window. These formulas are amenable to optimization, and are applied to improve a DTMF receiver design which is subject to the use of low-complexity second-order resonators.