Magnitude and Phase of frequency response

Question

I solved the impulse response above as follows. $H(e^{j\hat\omega})=H_1(e^{j\hat\omega})H_2(e^{j\hat\omega})H_3(e^{j\hat\omega})$ $H(e^{j\hat\omega})=e^{-j3\hat\omega}(e^{-j\hat\omega}+e^{-j5\hat\omega})(3e^{-j\hat\omega}+e^{-j4\hat\omega})=3e^{-j3\hat\omega}+4e^{-j6\hat\omega}+e^{-j9\hat\omega}$

But I want to use this value to get magnitude$(\vert H(e^{j\hat\omega})\vert)$ and phase$(\angle H(e^{j\hat\omega}))$, but I don't know how to change the expression.

Answer 1

It is often easier to use the properties of $\measuredangle$ and $\|\cdot\|$ directly on $H$, i.e.,

$$\measuredangle H(e^{j\omega}) = \measuredangle H_1(e^{j\omega}) + \measuredangle H_2(e^{j\omega}) + \measuredangle H_3(e^{j\omega}),$$ $$\|H(e^{j\omega})\| = \|H_1(e^{j\omega})\|\, \|H_2(e^{j\omega})\|\, \| H_3(e^{j\omega})\|$$

and evaluate the frequency response of the individual transfer functions.