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Filter Types and Related Kyma Components

Contributions by: BenPhenix, DavidMcClain, CarlaScaletti, KurtHebel

Filters can be described by a number of different (somewhat orthogonal) characteristics:

  1. magnitude (or amplitude) response within the pass and stop bands
    • these characteristics give the filter its name (for example, Butterworth)
    • amount and location of ripple
    • width of transition band (between passing certain frequencies through the filter and stopping other frequencies)
  2. impulse response (output of the filter to an impulse input)
    • IIR: infinite impulse response (filter's output never stops)
    • FIR: finite impulse response (filter's output stops at some time)
  3. frequency response
    • lowpass, highpass, bandpass, band reject, allpass
    • the frequency response of any filter can be transformed (that is, a lowpass filter can be transformed into a highpass filter)
  4. phase response
    • the phase shift (or delay) imposed on each frequency as it goes through the filter
    • linear phase, constant group delay
  5. implementation
    • filter architecture
    • number of multiplications & additions
    • noise characteristics
    • relationship between coefficients and filter parameters like cutoff frequency

Generally, when speaking of filters, we speak of the characteristics of a prototypical lowpass filter (for example, a Butterworth filter), transformed into the frequency response of interest (Butterworth bandpass filter). These filters are almost always IIR filters; however, it is possible to approximate these as FIR filters.

The same filter can be implemented in a variety of ways. Each implementation will have different performance characteristics (amount of noise, numbers of multiplies, mappings between parameters and coefficients).

Magnitude Response

Butterworth

Butterworth (Kyma IIR HP/LP filters): maximally flat amplitude response, no consideration of phase linearity or group delay variations. Filter has no amplitude ripple in the passband or stopband.

Linkwitz Riley, serial Butterworth filters: A 1-pole Butterworth into 2nd 1-pole Butterworth gives you a 2-pole Linkwitz filter. 2 2-pole Butterworths give you a 4-pole Linkwitz, and so forth. Great for creating cross-overs where keeping phase alignment is critical, common in speaker and audio components.

Here's an interactive tool for designing Butterworth filters.

Another interactive tool for Butterworth, Bessel and Chebyshev filters.

Bessel

Bessel: most linear phase response of IIR filter class, no consideration of frequency response. constant group delay.

Bessel: retains nearly constant group delay through the passband with little consideration to amplitude response.

Chebyshev

Chebyshev: produces passband, or stopband, ripples constrained within fixed bounds, such as 1 dB, 2 dB, and 3 dB of ripple. Ripples are either in the passband (Type I) or stopband (Type II). Subject to a high degree of nonlinear phase response.

Elliptic

Elliptic (Cauer filters): very sharp roll-off, very poor phase linearity, and have ripples both the passpand and stopband.

Impulse Response

Finite Impulse Response (FIR)

Finite Impulse Response Filter (Kyma FIR): FIR filters are stable, and have linear phase responses if the coefficients of the filter are symmetric about their center point (for example, 0.5, 1, 0.5). (In DSP circles, this kind of filter is defined as a class of digital filters that has only zeros on the z-plane.)

The reason for the name is that if you feed an impulse (a single "1" followed by an infinite string of "0s") into an FIR, it eventually dies away. Its impulse response has a finite duration. An FIR is made up of delays and attenuators but no feedback. As anyone who's stuck a microphone in front of a speaker knows, feedback can "blow up" or grow without bounds (like a "pole" in an IIR filter). An FIR, not having feedback, can't blow up in the same way; hence its reputation for stability.

Infinite Impulse Response (IIR)

IIR filters require fewer multiplications and additions than FIR filters to achieve the same filtering action. (In DSP circles, this kind of filter is defined as a class of digital filters that has poles and zeros on the z-plane.)

They are built of combinations of delays, attenuators, mixers, with feedback. They cannot have linear phase (or constant group delay), but with enough work an approximation to linear phase can be obtained.

Frequency Response

All-Pass (Kyma IIR Allpass): passes all frequencies with equal gain. Kyma filters have a non-linear phase response see AllPass Discussion.

Implementation

Reson: to quote DavidMcClain, "Reson filters have the desirable property of exhibiting constant peak amplitude across the entire frequency range, and the peak frequency can be directly specified. They have a post-gain section to bring their outputs back up to where you want them." example no longer in the forum, perhaps David or someone can post the Reson filter example here.

Biquad, bi-quadratic: Dual path filter operating on the in-phase and quadrature-phase.

DavidMcClain 's microsound 2-pole filters: discussion surrounding his DirectIIR filters

 
 
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