Practice Question on "Signals and Systems"
Topic: Review of Complex Numbers
Question
Compute the Magnitude of the following continuous-time signals a) $ x(t)=e^{2t} $
b) $ x(t)=e^{2jt} $
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^{(2t)}| = \sqrt{(e^{(2t)})^2} = \sqrt{e^{(4t)}} = e^{(2t)} $ (cmcmican 10:59, 10 January 2011 (UTC))
b) $ |e^{(2jt)}| = |(cos(2t) + j*sin(2t))| = \sqrt{(cos(2t))^2 + (sin(2t))^2} = \sqrt{1} = 1 $ (cmcmican 10:59, 10 January 2011 (UTC))
- Instructor's comments: Both answers and justifications are correct. Note that an alternative method to obtain the complex magnitude of the signal in b) is to multiply the signal value by its complex conjugate and taking the square root of the result. (This is basically what you are doing in a), but since the signal is real, it is equal to its conjugate.) A quick note though on the symbol $ * $: we will be using it to denote the convolution operation later on, so it will be important not to use it to denote multiplication anymore. -pm
Answer 2
write it here.
Answer 3
write it here.
