Implementing IIR/FIR Filters

Although this transformation was developed using a
first-order system, it holds, in general, for an N th -order
system (see Reference 14). By letting s = σ + jΩ and
z = re jθ , it can be shown that the left-half plane in the
s-domain is mapped inside the unit r = 1 circle in the
z-domain under the bilinear transformation. More importantly,
when r = 1 and σ = 0, the frequencies in the
s-domain and the z-domain are related by:
Ω
or equivalently:
2 θ
= -tan-- T 2
θ 2tan 1 – ΩT
=
-------
2
where: θ is the digital domain normalized
frequency equal to 2πf/f s
Eqn. 2-14
Eqn. 2-15
Ω is the analog domain frequency used
in the analysis of the previous section
On the jΩ axis or equivalently along the frequency
axis, the scale has been changed nonlinearly. The
gain and phase values depicted on the vertical axis
of Figure 1-2, Figure 1-4, Figure 1-6, and Figure 1-8
remain exactly the same in the digital domain (or zplane).
The horizontal (frequency) axis is modified
so that an infinite frequency in the analog domain
maps to one-half of the sample frequency, f s /2, in
the digital domain; whereas, for frequencies much
less than f s /2, the mapping is approximately 1:1
2-6 MOTOROLA