A Partial fraction expansion questions.

Question

I am learning signals and systems. I solve the problem and reach the equation (5.164). How can I work out the value of $A$ and $B$?

I do the partial fraction expansion and yields

Answer 1

We are trying to find $A$ and $B$ such that

$\dfrac{1}{[1-(re^{j\theta})e^{-j\Omega}][1-(re^{-j\theta})e^{-j\Omega}]} = \dfrac{A}{1-(re^{j\theta})e^{-j\Omega}}+\dfrac{B}{1-(re^{-j\theta})e^{-j\Omega}}$

holds for all values of $\Omega$.

Multiply both sides by $[1-(re^{j\theta})e^{-j\Omega}][1-(re^{-j\theta})e^{-j\Omega}]$ to get:

$1 = A[1-(re^{-j\theta})e^{-j\Omega}] + B[1-(re^{j\theta})e^{-j\Omega}]$

$1 = (A+B) - (re^{-j\theta}A+re^{j\theta}B)e^{-j\Omega}$

In order for this to hold for all values of $\Omega$, we need $A+B = 1$ and $re^{-j\theta}A+re^{j\theta}B = 0$.

Solving this system of equations yields $A = \dfrac{e^{j\theta}}{2j\sin\theta}$ and $B = -\dfrac{e^{-j\theta}}{2j\sin\theta}$.