Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 115

This relation is called the bilinear transform. It should be noted that the
approximation in used above is true only when z is near 1, which means that the
low frequency region maps quite reliably from the analog domain to the digital
domain. At higher frequencies, however, there is considerable error in the
approximation. It will be seen subsequently that the bilinear transform actually
introduces a warping of the analog frequency axis, and the warping is most severe
at high frequencies. This warping causes the analog frequency range 0 ? ? to be
mapped into the digital frequency range 0 ? p.
To gain further insights into the nature of the bilinear transform one can put
z ¼ re jX [recall that the frequency response of any digital system H(z) can be
found by substituting z ¼ re jX ]. Substituting the polar form for z ð¼ re jX Þ in the
bilinear transform s ð 2 = T s Þð z 1 Þ=ð z þ 1 Þ gives:

s 2
T s

re jX 1
re jX þ 1

¼ 2
T s

½ r cos ð X Þ 1 þ jr sin ð X Þ
½ r cos ð X Þþ 1 þ jr sin ð X Þ

:

¼ 2
T s

r 2 1
1 þ r 2 þ 2r cos ð X Þ þ j

2r sin ð X Þ
1 þ r 2 þ 2r cos ð X Þ

Since s ¼ r þ jx :

r 2 1
1 þ r 2 þ 2r cos ð X Þ and

r ¼ 2
T s

x ¼ 2
T s

2r sin ð X Þ
1 þ r 2 þ 2r cos ð X Þ

Now for r = 0, |z| = r = 1 and so z ¼ e jX : Hence, the s = jx axis in the analog
domain is transformed into the z ¼ e jX

unit circle, where the new relation between

X and x is given by:

x ¼ 2
T s 1 þ cos ð X Þ ¼ 2

sin ð X Þ X

2 T s tan

ð 2 : 32 Þ

) X ¼ 2 tan ð xT s = 2 Þ:

The above equation is non-linear, as illustrated graphically in Fig. ( 2.34 ). The
infinite analog frequency domain (x = 0 ? ?) is mapped onto the principle
digital frequency range X ¼ 0 ! p :
If r \ 0, then r \ 1 (hence the LHS of the s-plane is mapped inside the unit
circle), and for r [ 0, r [ 1.
It can be shown that filter design using the bilinear transform is independent of
T s [ 2 ]; hence, one can put T s = 2 (for convenience) and use the following trans-
form instead:

:

s ¼ z 1
z þ 1

X ¼ 2 tan 1 ð x Þ¼ 2 tan 1 ð 2pf Þ

with ð 2 : 33 Þ