Question

If f(k)=4^k U(k)and g(k)=5^k U(k),
then find the z-transform of f(k).g(k)

I know that
\( Z[a^k U(k)] = 1/(1-az^{-1})\)
|z|>|a|

Answer 1

if it was f(k) * g(k)
* = convolution
then it would have been easy

Answer 2

Just so we're on the same page... Is it
\[U[k-c]=\begin{cases}1&\text{for }k\ge c\\0&\text{for }k<c\end{cases}~~?\]

Answer 3

I suppose we can say this:
\[\left(4^kU[k]\right)\left(5^kU[k]\right)=20^k\left(U[k]\right)^2=20^kU[k]\]

Answer 4

yes, U is that only....
oh, and is that multiplication directly possible ?

Answer 5

I would think so. Wolfram would agree:
\[Z[a^kU[k]]=\frac{1}{1-\dfrac{a}{z}}=\frac{z}{z-a}\]
http://www.wolframalpha.com/input/?i=ZTransform%5B%284%5Ek*Heaviside%5Bk%5D*5%5Ek*Heaviside%5Bk%5D%29%2Ck%2Cz%5D