Computing frequency response of a filter given Z-transform

Question

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.

Thank you in advance for your assistance!

Answer 1

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 $e^{-j\omega/2}$