Z-domain transfer function question

Question

For the difference equation below:

$y(k)=1/2{x(k)+x(k-1)}$

How i can find he z-domain transfer function?

Thanks.

Answer 1

I assume this is homework, but transforming a difference equation to the $z$-domain is simple; just recall the time-shifting property of the transform.

$$ x[n] \Leftrightarrow X(z) \rightarrow x[n-k] \Leftrightarrow z^{-k}X(z) $$

So then we have:

$$ y[n] = \frac{1}{2}x[n] + x[n-1] $$

$$ Y(z) = \frac{1}{2}X(z) + z^{-1}X(z) $$

The transfer function can be written as:

$$ H(z) = \frac{Y(z)}{X(z)} = \frac{1}{2} + z^{-1} $$