presence filter

transition band

axis while maintaining the "shape" of the frequency response. Such transforma-

tion is a so-called conformal transformation [62] of the complex plane onto itself.

In this section we are interested in conformal transformations that map the unit

circumference (instead of the imaginary axis) onto itself, in such a way that, if

applied to a discrete-time filter, they give a new discrete-time filter having the

same stability properties.

onto itself. Moreover, if the transformation (53) is applied to a discrete-time filter

described by a transfer function in z, it preserves the filter order in the variable

.

a presence filter it is convenient to start from a second-order resonant filter pro-

totype having center frequency at /2 and tunable bandwidth and boost. Then,

it is possible to compute the coefficient of the conformal transformation (53)

in such a way that the resonant peak gets moved to the desired position [62].

Conformal transformations of order higher than the first are often used to de-

sign multiband filters starting from the design of a lowpass filter, or to satisfy

demanding specifications on the slope of the transition band that connects the

pass band from the attenuated band.

optimize some psychoacoustic-based quality measure. Namely, the warping of

the frequency axis can be tuned in such a way that it resembles the frequency

distribution of critical bands in the basilar membrane of the ear [99]. Similarly

to what we saw in section 1.5.2 for the bilinear transformation, it can be shown

that a first-order conformal map is determined by setting the correspondence

in three points, two of them being = 0 and = . The mapping of the third

point is determined by the coefficient a to be used in (53). Surprisingly enough, a

simple first-order transformation is capable to follow the distribution of critical

bands quite accurately. Smith and Abel [99], using a technique that minimizes

the squared equation error, have estimated the value that has to be assigned

to a for sampling frequencies ranging from 1Hz to 50KHz, in order to have a

ear-based frequency distribution. An approximate expression to calculate such

coefficient is

the plane to the frequencies in the z plane can be drawn and compared to