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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!

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  • $\begingroup$ Have you tried $|H(\omega)|^2 = H(\omega)\cdot H'(\omega)$ ? You get the amplitude squared by multiplying with the complex conjugate. $\endgroup$
    – Hilmar
    Aug 3 '21 at 17:36
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As mentioned in a comment, just use the fact that

$$|1+z|^2=1+2\,\textrm{Re}\{z\}+|z|^2,\qquad z\in\mathbb{C}\tag{1}$$

and take the square root to obtain the amplitude function. There is no simpler expression when $|z|\neq 1$.

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  • $\begingroup$ 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 $\endgroup$
    – Aedrha
    Aug 4 '21 at 13:24
  • $\begingroup$ @Aedrha: Looks good I guess. There are many ways to check this. E.g., just use some software (Matlab or similar) to compute the frequency response numerically, and compare with your analytical result. $\endgroup$
    – Matt L.
    Aug 4 '21 at 14:00

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