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Quantizing the feedback coefficient in the Goertzel algorithm

Started by gretzteam 1 year ago2 replieslatest reply 1 year ago182 views

The value of the feedback coefficient in the Goertzel algorithm it given by:

c = 2*cos( 2*pi*k/N); 

where k is the bin number and N is the number of points.

Here Lyons shows that quantizing the coefficient to few bits does not affect the stability of the filter: it can be shown that the pole is guaranteed to lie on the unit circle no matter the number of bits used to represent the coefficient:

https://www.dsprelated.com/showarticle/796.php

At the same time, here the Goertzel algorithm is generalized for non-integer multiple of the fundamental frequency:

https://www.researchgate.net/publication/257879807...

Isn't the extra step needed for the generalization always required as soon as the coefficient is quantized? Of course quantizing to many bits likely pushes the error into don't care land, but curious about this in theory.

Thanks!

[ - ]
Reply by rbjApril 3, 2023

It's the denominator coefficients, a1 and a2, that are getting quantized.

The specific frequency might be off, due to quantizing a1, but the radii of the pair of poles is the sqrt(a2).  When a2=1, you're on the unit circle and, no matter how wide your coefficient word is, you can always get a2=1.

[ - ]
Reply by gretzteamApril 3, 2023

We agree that the pole is on the unit circle no matter what. 

The question is about 'The specific frequency might be off'. For quantized a2 it is off, so, doesn't this call for the 'generalized goertzel' all the time, with the extra rotation?

Thanks