Difference between revisions of "Norm of a complex exponential ECE438F11" - Rhea

Latest revision as of 14:24, 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.

Share your answers below

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

By Euler's formula

$e^{j \omega} = cos( \omega) + i*sin( \omega)$

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

Back to ECE438 Fall 2011 Prof. Boutin

Back to ECE438

@@ Line 1: / Line 1: @@
 [[Category:complex numbers]]
+[[Category:Complex Number Magnitude]]
+[[Category:Euler's formula]]
 [[Category:ECE438]]
 [[Category:ECE438Fall2011Boutin]]
 [[Category:problem solving]]
-= What is the norm of a complex exponential?=
+<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:
@@ Line 14: / Line 26: @@
 ----
 ===Answer 1===
-Write it here.
+By [[More_on_Eulers_formula|Euler's formula]]
+<math> e^{j \omega}  = cos( \omega) + i*sin( \omega) </math>
+hence,
+<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===
-Write it here.
+becasue: <math>  e^{jx} =cos(x)+ jsin(x) </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===
-Write it here
+<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]]
 [[ECE301|Back to ECE438]]