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[[Category:complex numbers]] | [[Category:complex numbers]] | ||
| + | [[Category:Complex Number Magnitude]] | ||
| + | [[Category:Euler's formula]] | ||
[[Category:ECE438]] | [[Category:ECE438]] | ||
[[Category:ECE438Fall2011Boutin]] | [[Category:ECE438Fall2011Boutin]] | ||
[[Category:problem solving]] | [[Category:problem solving]] | ||
| − | = | + | |
| + | |||
| + | <center><font size= 4> | ||
| + | '''[[Digital_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' | ||
| + | </font size> | ||
| + | |||
| + | Topic: Review of complex numbers | ||
| + | |||
| + | </center> | ||
| + | ---- | ||
| + | ==Question== | ||
After [[Lecture7ECE438F11|class today]], a student asked me the following question: | After [[Lecture7ECE438F11|class today]], a student asked me the following question: | ||
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===Answer 1=== | ===Answer 1=== | ||
| − | By Euler's | + | By [[More_on_Eulers_formula|Euler's formula]] |
<math> e^{j \omega} = cos( \omega) + i*sin( \omega) </math> | <math> e^{j \omega} = cos( \omega) + i*sin( \omega) </math> | ||
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<math>\left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1 </math> | <math>\left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1 </math> | ||
| + | |||
| + | :<span style="color:green">TA's comments: Is this true for all <math>\omega \in R</math>? The answer is yes.</span> | ||
| + | |||
| + | :<span style="color:purple">Instructor's comment: I would like to propose a more straightforward way to compute this norm using the fact that <math>|z|^2=z \bar{z}</math>. Can you try it out? -pm </span> | ||
| + | |||
===Answer 2=== | ===Answer 2=== | ||
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<math>| e^{j \omega}|=|cos(\omega) + i*sin(\omega)|=\sqrt{cos(\omega)^2 +sin(\omega)^2}=1</math> | <math>| e^{j \omega}|=|cos(\omega) + i*sin(\omega)|=\sqrt{cos(\omega)^2 +sin(\omega)^2}=1</math> | ||
| + | |||
| + | :<span style="color:green">TA's comments: The point here is to use [[More_on_Eulers_formula|Euler's formula]] to write a complex exponential as a complex number. Then the norm(magnitude) and angle(phase) of this complex number can be easily computed.</span> | ||
| + | |||
| + | :<span style="color:purple">Instructor's comment: Again, I would argue that using the fact that <math>|z|^2=z \bar{z}</math> is more straightforward. Can you try it out? -pm </span> | ||
===Answer 3=== | ===Answer 3=== | ||
| − | + | <math> e^{j \omega} = cos( \omega) + i*sin( \omega) </math> | |
| + | |||
| + | |||
| + | <math>\left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1 </math> | ||
| + | |||
| + | :<span style="color:purple">Instructor's comment: Can you think of a way to compute this norm without using [[More_on_Eulers_formula|Euler's formula]]? -pm </span> | ||
---- | ---- | ||
[[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | [[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | ||
[[ECE301|Back to ECE438]] | [[ECE301|Back to ECE438]] | ||
Latest revision as of 13:24, 21 April 2013
Practice Question on "Digital Signal Processing"
Topic: Review of complex numbers
Question
After class today, a student asked me the following question:
$ \left| e^{j \omega} \right| = ? $
Please help answer this question.
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
$ e^{j \omega} = cos( \omega) + i*sin( \omega) $
hence,
$ \left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1 $
- TA's comments: Is this true for all $ \omega \in R $? The answer is yes.
- Instructor's comment: I would like to propose a more straightforward way to compute this norm using the fact that $ |z|^2=z \bar{z} $. Can you try it out? -pm
Answer 2
becasue: $ e^{jx} =cos(x)+ jsin(x) $
$ | e^{j \omega}|=|cos(\omega) + i*sin(\omega)|=\sqrt{cos(\omega)^2 +sin(\omega)^2}=1 $
- TA's comments: The point here is to use Euler's formula to write a complex exponential as a complex number. Then the norm(magnitude) and angle(phase) of this complex number can be easily computed.
- Instructor's comment: Again, I would argue that using the fact that $ |z|^2=z \bar{z} $ is more straightforward. Can you try it out? -pm
Answer 3
$ e^{j \omega} = cos( \omega) + i*sin( \omega) $
$ \left| e^{j \omega} \right| = \left|cos( \omega) + i*sin( \omega) \right| = \sqrt{cos^2( \omega) + sin^2( \omega)} = 1 $
- Instructor's comment: Can you think of a way to compute this norm without using Euler's formula? -pm
