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Practice Problem: 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?
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
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 $
Answer 2
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
Answer 3
write it here.
