For the difference equation below:


How i can find he z-domain transfer function?



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:

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

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  • 2
    $\begingroup$ The region of convergence is missing from the result (you need both the analytic expression plus the ROC in order to fully identify $h[n]$. In this case, ROC is the whole complex plane (except 0). $\endgroup$ – Juancho Feb 21 '13 at 15:52

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