(New page: =ECE 301 Fall 2008, Professor C.C. Wang = ==Use this area to post your questions== I am trying to figure out how to compute the norm of the DT signal <math> x[n]= e^{j 2 \pi n} </math>. ...) |
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which is clearly wrong, because my high school teacher showed us the graph of <math> e^{j 2 \pi t} </math> and it was oscillating. So what am I doing wrong? | which is clearly wrong, because my high school teacher showed us the graph of <math> e^{j 2 \pi t} </math> and it was oscillating. So what am I doing wrong? | ||
| − | === | + | === [[MA181]] to the rescue! === |
The identity <math>(e^a)^t=e^{ta}</math> is true for real numbers, but it is not always true for complex numbers, even when <math>a</math> is complex and <math>t</math> is real. When <math>z</math> is complex and <math>t</math> is real, <math>z^t</math> stands for <math>e^{t\log z}</math>, where <math>\log z</math> is the complex logarithm of <math>z</math>, which has infinitely many possible values. | The identity <math>(e^a)^t=e^{ta}</math> is true for real numbers, but it is not always true for complex numbers, even when <math>a</math> is complex and <math>t</math> is real. When <math>z</math> is complex and <math>t</math> is real, <math>z^t</math> stands for <math>e^{t\log z}</math>, where <math>\log z</math> is the complex logarithm of <math>z</math>, which has infinitely many possible values. | ||
Revision as of 11:46, 26 April 2010
Contents
ECE301 Fall 2008, Professor C.C. Wang
Use this area to post your questions
I am trying to figure out how to compute the norm of the DT signal $ x[n]= e^{j 2 \pi n} $. According to the solutions, the answer is
$ \left| e^{j 2 \pi n} \right| = 1 $.
I don't get it. Should'nt the answer be a function of n???
Response
Since we are dealing with complex numbers, we can make the following substitution:
$ \left| e^{j 2 \pi n} \right| = \left| \cos{2 \pi n} + j \sin{2 \pi n} \right| $
The magnitude of this expression will always be one since $ \cos^2\theta+\sin^2\theta\equiv 1 $.
Related Question
I see your point. But if what you say is true, then we also have
$ e^{j 2 \pi t} = \left( e^{j 2 \pi} \right)^t= \left( cos{2 \pi } + j sin{2 \pi } \right)^t= 1^t =1, $
which is clearly wrong, because my high school teacher showed us the graph of $ e^{j 2 \pi t} $ and it was oscillating. So what am I doing wrong?
MA181 to the rescue!
The identity $ (e^a)^t=e^{ta} $ is true for real numbers, but it is not always true for complex numbers, even when $ a $ is complex and $ t $ is real. When $ z $ is complex and $ t $ is real, $ z^t $ stands for $ e^{t\log z} $, where $ \log z $ is the complex logarithm of $ z $, which has infinitely many possible values.
By the way, the constant function is a function. The modulus of $ e^{j\theta} $ is equal to one for any value of $ \theta $ because that complex number represents a point on the unit circle.
I saw the plea on our page and thought I'd check it out. While I can't claim to have much understanding of what is going on, it seems like that solution is valid because $ n $ represents either the natural numbers or integrers, not the real numbers. Therefore, it would be like taking the value of the sine function at whole-number products of $ 2\pi $. Even though the function oscillates, the sequence is constant. Hope I haven't made a fool of myself. --Jmason
