# FInd the amplitude function to a system

I'm studying a course in signal analysis and have come across an exercise where I find the analytical part a bit tricky.

I am to find the amplitude function $$|H( f )|$$ for a system with the impulse response $$h(n)=\delta(n)+0.9\delta(n-D)$$

Where $$D=500$$

I did some transforms:

$$h(n)=\delta(n)+0.9\delta(n-D)\\ H(z)=1+0.9z^{-D}\\ z=e^{j\omega}\\ H(\omega)=1+0.9e^{-j\omega D}$$

My plan was to factor out $$e^{-jxD\omega}$$ for some factor x and then use Euler's formula. But the $$0.9$$ term is holding me back!

I'd greatly appreciate any help or tips!

• Have you tried $|H(\omega)|^2 = H(\omega)\cdot H'(\omega)$ ? You get the amplitude squared by multiplying with the complex conjugate. Aug 3 '21 at 17:36

$$|1+z|^2=1+2\,\textrm{Re}\{z\}+|z|^2,\qquad z\in\mathbb{C}\tag{1}$$
and take the square root to obtain the amplitude function. There is no simpler expression when $$|z|\neq 1$$.
• I tried with $|H(\omega)|^2=H(\omega)\cdot H'(\omega)$ $H(\omega)\cdot H'(\omega)=(1+0.9e^{-j\omega D})(1+0.9e^{j\omega D})$ $|H(\omega)|^2=1+1,8\cos(\omega D)+0.81$ $|H(\omega)|=\sqrt{1.81+1.8\cos(\omega D)}$ Not sure if this is correct though Aug 4 '21 at 13:24