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I've heard that you can stack biquads on top of each other to get the same (or similar) effect as higher order filters.

For instance a biquads is order 2, so putting something through 3 biquads is like an order 6 filter.

I've heard that doing this is more numerically stable than an actual order 6 filter.

I was wondering though, besides increased numerical robustness, are there any differences between doing this compared to just using a higher order filter?

Does it impact processing speed or frequency response or anything else?

For some context, this for audio synthesis for electronic music, on a modern desktop computer, using 32 bit floating point math.

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  • $\begingroup$ "I've heard" where? In what context? "same/similar effect": in which sense? at single-digit orders of filters, numerical stability for not overly recursive filters is usually not a concern at all. So: your question is a bit unclear, and I'm pretty sure you will get valuable answers only if you can describe what you actually need to achieve – specify the filter you want to build, and then we'll have something to discuss the advantages and disadvantages of a specific filter approach on. Is this hard- or software, floating or fixed point? Do you really need a biquad, or what is it that you want? $\endgroup$
    – Marcus Müller
    Jun 11, 2016 at 16:19
  • $\begingroup$ "audio synthesis": can you further specify? Why do you need a single biquad, let alone multiple ones? What is your need/application? $\endgroup$
    – Marcus Müller
    Jun 11, 2016 at 17:06
  • $\begingroup$ I'll try to make an analogy, please don't be offended: "I've heard I can use a cascade of biquads instead of a higher order filter; I work in digital music" to me is about as specific as "I've heard that I can use two cars instead of one truck, to similar effect. For background: I drive around on roads"; this might take a bit more background. $\endgroup$
    – Marcus Müller
    Jun 11, 2016 at 17:08
  • $\begingroup$ Does it help at all to mention I'm mainly wondering about a low pass filter? Both for subtractive synthesis as well as in the context of downsampling to remove frequencies above the "new Nyquist" before downsampling? $\endgroup$
    – Alan Wolfe
    Jun 11, 2016 at 17:12
  • $\begingroup$ well, why biquad then? $\endgroup$
    – Marcus Müller
    Jun 11, 2016 at 19:02

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A cascade of 2nd order biquadratic (biquad) sections is a commonly used approach for implementing higher order FIR/IIR filters in fixed point systems (Once you determine that you need an IIR filter, this would be a topology to consider). A big consideration for the biquad structure is stability: higher order IIR filters are prone to instability in fixed point systems due to coefficient rounding combined with long feedback delay paths (causing poles to go outside of the unit circle). The absence of any feedback between the individual cascaded sections (decoupling the poles and zeros between sections) results in a more stable structure that will be less sensitive to coefficient quantization.
I do not know of other advantages, but the numerical robustness is a very serious consideration especially in higher order structures (and fixed point systems).

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