The most common filter alignments are the Butterworth and the Bessel (AKA Thomson). Less common alignments are the Gaussian, Chebyshev and the elliptic (AKA Cauer). A recent addition to the palette of filters is the Linkwitz-Riley alignment (AKA Butterworth squared), which was specifically designed for use as loudspeakers crossover (More on this type of filter to follow.).

Common Audio Filter Types
   The Butterworth filter is the most commonly used filter type in audio applications. It has a fairly flat time response, a fairly crisp transition shape and the flattest passband response. In contrast, the Bessel filter offers a flatter time response, but not as sharp a transition shape or flat a passband response. The Chebyshev and the elliptic filter are seldom used in everyday audio gear, as these filters bring too many undesirable characteristics such as ripples in the passband or wild phase shifts. 
    The Linkwitz-Riley filter improves on the Butterworth when feeding loudspeakers. Odd order crossovers (6 dB and 18 dB per octave filters) are marked by of 90 degree multiples in phase differences between outputs. The sum of two 90 degree equal amplitude signals is +3 dB boost in output. Thus the -3 dB point of the filter works to yield a flat transition region.
   Even-order crossovers (12 dB and 24 dB per octave filters) are marked by 180 degree multiples of phase differences between outputs. The sum of two 180 degree phase shifted, but equal amplitude, signals is a deep null in the output. The solution is to reverse the phase of one of the loudspeaker drivers to eliminate the phase difference at the crossover frequency. However, when two loudspeakers play the exact same signal (no amplitude or phase differences), the result is a twofold increase (+6 dB boost) in volume. So if we wish to reconstruct the outputs of one lowpass and one highpass filter, we must tailor the attenuation at the crossover frequency to match the intended summing device.

   If only low frequencies are passed, we call this filter a low-pass filter, as that is what it passes. If only high frequencies are passed, we call this filter a high-pass filter. If only a band of frequencies, say 500 to 5000 Hz, are passed, we call this filter a band-pass filter. If all frequency are passed save for a very narrow band of frequencies, we call this filter a notch filter.
    Beyond these four basic divisions, a further classification is applied based the steepness of the rejection of undesired frequencies.
    Since the slope's steepness is marked by the "order" in a filter's design. (The number of poles is also sometimes used as a short hand description of the steepness.) For example, a single order filter, also called a single pole filter, has a crossover slope of -20 dB per decade, which equals -6 dB per octave. Thus a second order filter attenuates at -12 dB per octave; a third order filter, -18 dB per octave; a fourth order filter, -24 dB per octave; a fifth order filter, -30 dB per octave; and a sixth order filter, -36 dB per octave.

   The next taxonomic division is made based on the filters alignment. The alignment is based on the relative position of the poles in the filter, which in turn gives rise to the different shapes each filter exhibits across its frequency range. Some filters offer a sharp transition from passband to stopband, but at the cost of some ripple and even ringing in the frequency response. Other filters offer a soft transition that droops at the transition frequency that is ripple free. In short, an analog filter's design hopes to optimize one or two parameters, but always at the cost of some other parameters.

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