Exercise [08.05] b

vasco · 11 Mar 2009, 17:47

The first solution to this exercise is incorrect. I have explained why I think this in a reply to the original solution

The three transformations to be carried out in order are:

$z\rightarrow Az+B$
$z\rightarrow 1/z$
$z\rightarrow Cz+D$

If you do a sequence of transformations in the following way you are less likely to get the wrong answer:

$w_{_1}=Az+B$ $\{z$ -plane to $w_{_1}$ -plane $\}$

$\displaystyle{w_{_2}=\frac{1}{w_{_1}}}$ $\{w_{_1}$ -plane to $w_{_2}$ -plane $\}$

$w_{_3}=Cw_{_2}+D$ $\{w_{_2}$ -plane to $w_{_3}$ -plane $\}$

So $\displaystyle{w_{_2}=\frac{1}{w_{_1}}=\frac{1}{Az+B}$ and $\displaystyle {w_{_3}=Cw_{_2}+D=C\frac{1}{w_{_1}}+D=C\frac{1}{Az+B}+D}$

Finally $\displaystyle{w_{_3}=\frac{ADz+DB+C}{Az+B}=\frac{A^\prime z+B^\prime}{C^\prime z+D^\prime}}$

which is a bi-linear transformation, where $A^\prime=AD$ , $B^\prime=DB+C$ , $C^\prime=A$ and $D^\prime=B$

Exercise [08.05] b

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