Topic: Review of Complex Numbers

## Question

Compute the Magnitude of the following discrete-time signals

a) $x[n]=e^{2n}$

b) $x[n]=e^{2jn}$

c) $x[n]=j^n$

What properties of the complex magnitude can you use to check your answer?

a) $|e^{2n}|=\sqrt{(e^{2n})^2}=e^{2n}$

Instructor's comments: Yes, this is correct, but I would like to propose an alternative justification:
${\color{green}|e^{2n}|=\sqrt{(e^{2n})(e^{2n})^*}=\sqrt{e^{2n}e^{2n}}=e^{2n}}$
where ${\color{green}~^*}$ denotes the complex conjgate.-pm

b) $|e^{2jn}|=|cos(2n)+jsin(2n)|=\sqrt{cos(2n)^2+sin(2n)^2}=\sqrt{1}=1$

Instructor's comments: Again, the answer and justification are correct. But can somebody propose a different justification? One that is similar to the one I proposed above? -pm

c) $|j^n|=|e^{nj\pi/2}|=|cos(n\pi/2)+jsin(n\pi/2)|=\sqrt{cos(n\pi/2)^2+sin(n\pi/2)^2}=\sqrt{1}=1$

b) $|e^{2jn}| = \sqrt{(e^{2jn})(e^{2jn})^*} =\sqrt{(e^{2jn})(e^{-2jn})} = 1$

Instructor's comments: Good, this is what I was talking about above. I would like to argue that this approach 1. is more general, and 2. is oftentimes more straightforward (i.e. easier computations). -pm