P$-10\mathrm{.}21$ The IIR filter described by the difference equation

$$

$[$

$$

$]$

$=$

$0\mathrm{.}8$

$$

$[$

$$

$$

$1$

$]$

$$

$0\mathrm{.}8$

$$

$[$

$$

$]$

$+$

$$

$[$

$$

$$

$1$

$]$

is an all$-$pass filter, because it is possible to show that

$|$

$$

$($

$$

$$

$$

$^$

$)$

$|$

$2$

$=$

$1$

for all

$$

$^$

$.$

Determine the system function H$($z$)$ for this system. Express H$($z$)$ as a ratio of polynomials in z $1$ and as a ratio of polynomials in z$.$

For this first$-$order system, determine the pole and zero locations. Then verify that the zero and the pole are reciprocals.

For H$($z$)$ in part $($a$),$ show that

$|$

$$

$($

$$

$$

$$

$^$

$)$

$|$

$2$

$=$

$1$

$.$

If the input to the system is

$$

$[$

$$

$]$

$=$

$7$

$+$

$8$

$$

$$

cos

$($

$0\mathrm{.}2$

$$

$$

$)$

$+$

$9$

$$

$$

cos

$($

$0\mathrm{.}7$

$$

$$

$)$

explain why no calculation is required to write the output as

$$

$[$

$$

$]$

$=$

$7$

$+$

$8$

$$

$$

cos

$($

$0\mathrm{.}2$

$$

$$

$+$

$$

$1$

$)$

$+$

$9$

$$

$$

cos

$($

$0\mathrm{.}7$

$$

$$

$+$

$$

$2$

$)$

Determine the value of the two phase terms $1$ and $2$ above, which does require some calculations.