Analog Allpass Filters

It turns out that analog allpass filters are considerably simpler mathematically than digital allpass filters (discussed in §10.2). In fact, when working with digital allpass filters, it can be fruitful to convert to the analog case using the bilinear transform (§G.3.1), so that the filter may be manipulated in the analog plane rather than the digital plane. The analog case is simpler because analog allpass filters may be described as having a zero at for every pole at , while digital allpass filters must have a zero at for every pole at . In particular, the transfer function of every first-order analog allpass filter can be written as

This simplified rule works because every complex pole is accompanied by its conjugate for some .

Multiplying out the terms in Eq. (C.14), we find that the numerator polynomial is simply related to the denominator polynomial :

As an example of the greater simplicity of analog allpass filters
relative to the discrete-time case, the graphical method for computing
phase response from poles and zeros (§8.2) gives immediately
that the phase response of every real analog allpass filter is equal
to *twice* the phase response of its numerator (plus when
the frequency response is negative at dc). This is because the angle
of a vector from a pole at to the point along the
frequency axis is minus the angle of the vector from a zero at
to the point .

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