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## The DSP Biquadratic Recursive Filter: A Fox in the Hen House

Fredric J Harris - **Watch Now** - DSP Online Conference 2021 - 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.

**fred_h**Speaker

**remco_at_itsonlyaudio.com**

Some time in uni about 20 years ago I scored a used Spectral Dynamics SD375 (http://www.itsonlyaudio.com/ultranalog/sd375/).

It came with its own weight in documentation, amongst which several of your papers. It hooked me on signal processing and may even have been the reason I decided to finish my degree and thus, the foundation of my career.

Your presentation style is as elegant as the papers you write. Many, many thanks for the work you do to teach.

**dcomer**

Wonderful presentation! Where can I view the non-recursive presentation? Thanks.

**Stephane.Boucher**

Coming up later today actually! https://www.dsponlineconference.com/meeting/Live_Discussion_Green_FIR_Filters_with_Large_Ratio_of_Sample_Rate_to_Bandwidth

**RK**

As usual you are absolutely amazing in putting things into perspective.

**john.edwards**

Hi fred,

Sorry I couldn't attend your talk live but family called.

I just wanted to add a quick note to say thanks so much for the great presentation.

I came across the coefficient quantization problem many years ago. We solved the problem by changing the filter spec to still meet the application requirements but if I'd known your trick (slides 72 to 74) it would have saved a lot of time. This should be required study for all DSP students.

Its always a pleasure, my friend.

John

**fred_h**Speaker

i know,,, as they say, time goes by very quickly when you are having a good time....

thanks for the feedback!

**HardRealTime**

Great talk. The time flew by. It didn't feel like it was over an hour.

remco,

I helped design the spectrum analyzer for spectral dynamics as well as much of the signal processing they did in the two channel version that permitted all kinds of neat measurements... One you must recall is the coherency... (how much of what we see in the cross correlation came from different time intervals) i wrote a DSP application note for spectral DSP-005 (as I recall)... I'll send you a refresher copy!