Tuesday, November 27, 2012

Audio IIR v FIR EQs

Digital filters come in two flavors: IIR (or "Infinite Impulse Response") and FIR (or "Finite Impulse Response"). Those complex acronyms may confuse you, so let's shed a little light on the situation by defining both and explaining the differences.

Some people are interested in which is better. Unfortunately, as with many things, there is no easy answer to that question, other than "it depends", and sometimes what it depends on is your ears. I won't stray too deep into field of opinions, but I will try to mention why some people claim one is better than the other and what some of the advantages and disadvantages are in different situations.

How Filters Work

When you design a filter, you start with a set of specifications. To audio engineers, this might be a bit vague, like "boost 1 kHz by 3 dB", but electrical engineers are usually trained to design filters with very specific constraints. However you start, there's usually some long set of equations, and rules used to "design" the filter, depending on what type of filter you are designing and what the specific constraints are (to see one way you might design a filter, see this post on audio eq design). Once the filter is "designed" you can actually process audio samples.

IIR Filters

Once the filter is designed, the filter itself is implemented as difference equations, like this:

    y[i] = a0 * x[i] + a1 * x[i-1] ... + an * x[i-n] - b1 * y[i-1] ... - bm * y[i-m].

In this case, y is an array storing the output, and x is an array storing the input. Note that each output is a linear function of previous inputs and outputs, as well as the current input.

In order to know the current value of y, we need to know the last value of y, and to know that, you must know the value of still earlier values of y, and so on, all the way back until we reach our initial conditions. For this reason, this kind of filter is sometimes called a "recursive" filter. In principle, this filter can be given a finite input, and it will produce output forever. Because its response is infinite, we call this filter an IIR, or "Infinite Impulse Response" filter.

(To further confuse the terminology, IIR filters are often designed with certain constraints that make them "minimum phase." While IIR filters are not all minimum phase, many people use the terms "recursive", "IIR" and "minimum phase" interchangeably.)

Digital IIR filters are often modeled after analog filters. In many ways, analog-modled IIR filters sound like analog filters. They are very efficient, too: for audio purposes, they usually only require a few multiplies.

FIR Filters

FIR filters, on the other hand, are usually implemented with a difference equation that looks like this:

    y[i] = a0 * x[i] + a1 * x[i-1]  a2 * x[i-2] + ... an * x[i-n] + an * x[i-n-1] + ... + a1 * x[2i+1] + a0 * x[2i]

In this case, we don't use previous outputs: in order to calculate the current output, we only need to know the previous n inputs. This may improve the numerical stability of the filter because roundoff errors are not accumulated inside the filter. However, generally speaking, FIR filters are much more CPU intensive for a comparable response, and have some other problems, such as high latency, and both pass-band and stop-band ripple.

If an FIR filter can be implemented using a difference equation that is symmetrical, like the one above, it has a special property called "linear phase." Linear phase filters delay all frequencies in the signal by the same amount, which is not possible with IIR filters.

Which Filter?

When deciding which filter to use, there are many things to take into account. Here are some of those things:

  • Some people feel that linear phase FIR filters sound more natural and have fewer "artifacts".
  • FIR filters are usually much more processor intensive for the same response.
  • FIR filters have "ripple" in both the passband and stopband, meaning the response is "jumpy". IIR filters can be designed without any ripple.
  • IIR filters can be easily designed to sound like analog filters.
  • IIR filters require careful design to ensure stability and good numerical error properties, however, that art is fairly advanced.
  • FIR filters generally have a higher latency.

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