# Computing frequency response of a filter given Z-transform

I am currently working on a project that involves analyzing the frequency response $$H(e^{j\omega})$$ of the filter $$H(z)= \frac{1}{2} (1+z^{-1})$$. However, I am unsure about the specific steps and formulas required to compute the frequency response. I have a basic understanding of trigonometry and suspect that trigonometric formulas might be involved in this process.

The result is $$H(e^{j\omega}) = e^{-j\frac{\omega}{2}}\cos(\frac{\omega}{2})$$ Could someone please guide me on how to compute the frequency response of a filter? Are there any specific formulas or techniques that I should be aware of? Additionally, if there are any alternative methods or resources that can aid in this calculation, I would greatly appreciate any recommendations.

1. To get the frequency response $$H(e^{j\omega})$$ from the transfer function $$H(z)$$, evaluate $$H(z)$$ at $$z=e^{j\omega}$$: $$H(e^{j\omega}) = \frac{e^{-j\omega} + 1}{2}\tag{1}$$
2. Use the following identity: $$\cos(\omega) = \frac{e^{j\omega} + e^{-j\omega}}{2}\tag{2}$$ Can you see how to factor $$e^{-j\omega} + 1$$ using $$(2)$$?
Hint: it involves $$e^{-j\omega/2}$$