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My teacher of signals and systems gave us a review problem as following:

given a DT rightsided LTI system with transfer function

$$\frac{1-a^*z}{z-a}, \left | a \right |<1 $$ show that the system's magnitude response is in unity for all frequencies using graphical method.

Can anyone give me any suggestion of solving this problem? I've been going back and forth between the textbook and course video for hours, but still very clueless. I am particularly confused about the indication of using graphical method, because it seems no matter how I manipulate the fraction, there's always vector multiplication instead of simple addition.

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The frequency response of a given filter is calculated evaluating $H(z)$ in the unit circle, i.e. $z=e^{j\omega}$.

Hint 1:

Note that you can express your transfer functions as follows: $$H(e^{j\omega})=\frac{1-a^*e^{j\omega}}{e^{j\omega}-a}=\frac{1}{e^{j\omega}}\cdot\frac{1-a^*e^{j\omega}}{1-ae^{-j\omega}}$$

Hint 2:

There is a property that states that given any complex number $z$, the following equality holds:

$$|z|=|z^{*}|$$

Can you solve it now?

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  • $\begingroup$ Thank you very much sir. With you hints, I've cleared some misunderstanding of the question and sorted it out finally. Thanks again!! $\endgroup$
    – Timothy Hannon
    Jun 6, 2017 at 15:01
  • $\begingroup$ @TimothyHannon Glad to help! $\endgroup$
    – Tendero
    Jun 6, 2017 at 16:50

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