The DSP Biquadratic Recursive Filter: A Fox in the Hen House
Fredric J Harris - Watch Now - Duration: 01:34:24
When we studied active analog filters we were taught that the biquadratic second order filter was the work horse of active filter design. What made it so was that fact we could form second order polynomials in both denominator and numerator with real coefficients. We also learned that when we performed sensitivity analysis reflecting root shifts with component value variation due to tolerance spreads that lower order polynomials had reduced sensitivity levels. We learned active filters should be implemented with multiple second order filters and possibly one first order filter. Control folks also learned this lesson. That was good perspective for a designer to have.
When we started to implement high order recursive filters in DSP land we followed the standard understanding that the sampled data biquadratic filter with decoupled second order denominator and second order numerators offered us the same capabilities, complex roots with real coefficients and low sensitivity to root shifts due to coefficient quantization. We were so pleased that the carryover from active analog filters to sampled data filters we failed to notice that it was not true! We let the Fox in the Hen house without realizing what we did.
The fox comes out to play when we try to form IIR low-pass filters with a large ratio of sample rate to bandwidth. What we learn is that it just doesn’t work! We need an alternate architecture or we should stop designing recursive filters with very small bandwidths relative to sample rate. One I see all the time is a 30 Hz wide low-pass or high-pass filter running at 48 kHz sample rate. Have you run into that? Did it take long for the hurt to go away when you found out your design didn’t work? We will discuss how to fix the problem and make the fox go away.