Inverse Z Transform with $2-z^{-2}$

Question

How do I find the inverse z transform for:

$$Y[z]=\frac{2-z^{-2}}{(1-.5z^{-1})(1-.2z^{-1})}$$

I believe I need to use partial fractions but everytime I try it I get stuck.

Thanks for help.

Answer 1

Hint:

Setting $z^{-1}=x,$

we can write $$Y[z]=a+\dfrac b{1-.5z^{-1}}+\dfrac c{1-.2z^{-1}}$$

Now $Z\{b^n\}=?$

Answer 2

$$Y(z) = \frac{10(2z^2 - 1)}{(2z-1)(5z-1)}$$

Here you have to see that Degree of Numerator and Denominator are same hence during partial fraction you will have to consider a constant term A

$$Y(z) = A + \frac{B}{2z-1} + \frac{C}{5z-1}$$

Now find the A,B and C

$$A = 2 ; B = -\frac{10}{3} ; C = \frac{46}{3}$$

$$Y(z) = 2 - \frac{5}{(3)(z-\frac{1}{2})} + \frac{46}{(15)(z - \frac{1}{5})}$$

$$y(n) = 2\delta(n) - [ \frac{5(.5^n-1)}{3} - \frac{46(.2^n-1)}{15}] u(n-1)$$