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I am new to DSP, and I'm self studying. I am confused about the magnitude of $e^{j\omega}$ - where $\omega$ is the normalized angular frequency - when we are on the unit circle.

According to the text book, it's $|e^{j\omega}|=1$. But why is that? I'd really appreciate it is someone could please kindly explain this to me.

robert bristow-johnson
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Shady
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  • My first blog article titled "The Exponential Nature of the Complex Unit Circle" starts with the definition of $i$ (aka $j$) and uses simple algebra to work up to an understanding of Euler's equation which is the answer you are looking for. You can find my article here: https://www.dsprelated.com/showarticle/754.php – Cedron Dawg Nov 23 '18 at 02:30

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Warning: $|e^{j\omega}|$ is equal to $1$ if and only if $\omega$ is a real number. More generally, for $z$ complex, $|e^{z}|=e^{\Re{z}}$.

This really depends in your prior knowledge, because functions like exponentials, sines or cosines, can be developed in different ways. Here, I'll assume you know about sines and cosines, as you refer to angular frequency. Then, an historical way to define the exponential is via Euler formula, for instance:

$$e^{j\omega} = \cos \omega+j\sin \omega\,.$$

Then, through Argand's complex plane interpretation, you can interpret a complex number as a point with 2D coordinates $(x,y) \leftrightarrow x+jy$: the real part refers to the $X$-axis, the imaginary one to the $Y$-axis. Thus, obviously, $e^{j\omega}$ has the coordinates of a point on the $(0,0)$-center circle with radius $1$, hence the magnitude $|e^{j\omega}|$ is equal to the radius, or $(\cos \omega)^2+(\sin \omega)^2=1^2$.

In more modern versions, $e^z$ is defined by a series (see Proof of complex conjugate symmetry property of DFT), and $\cos$ and $\sin$ are derived for it, so they don't come first.

Laurent Duval
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By Euler's formula, assuming $\omega$ is a real number: $$ e^{j \omega} = \cos(\omega) + j \sin(\omega) $$

The definition of magnitude for a complex number $z = x + jy$ is: $$ |z| = \sqrt{x^2 + y^2}, $$ therefore: $$ \left|e^{j \omega} \right| = \sqrt{\cos^2\omega + \sin^2\omega} = 1 $$ by trigonometric identity.

Robert L.
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  • So it's always true, right? Doesn't matter if we are moving on the unit circle or not. – Shady Nov 21 '18 at 20:47
  • @Shady correct. The second link shows why – Robert L. Nov 21 '18 at 20:48
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    Always, as long as $\omega$ is a real number – Laurent Duval Nov 21 '18 at 20:54
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    @LaurentDuval eventhough you are spot on your warning on the type of variable being used in the exponential, as you can remember at the very beginning of Diff.Eq., Signals&Systems, etc. the quantitty was first introduced as the general complex exponential : $e^s$ with $s = \sigma + j \omega$ with real $\sigma,\omega$ and then $e^{s} = e^{\sigma + j\omega}$ was the natural consequent, upon which the interpretation of $e^{j \omega}$ needed no more re-definition for $\omega$ to be real. hence I think un-defining $\omega$ as complex would be against the Occam's Razor principle. ;-) – Fat32 Nov 21 '18 at 22:35
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    even though it was not explicitly given as an axiom, i had always considered $\Im{\omega}=0$ as an implicit axiom. – robert bristow-johnson Nov 22 '18 at 07:31
  • @Fat32 Complex eigenvalues remain at the core of any fully-real LTI systems. So, I kinda believe that complex numbers are fundamentally more central than real ones, which are mere shadows of the formers. – Laurent Duval Nov 22 '18 at 19:38
  • @robert-bristow-johnson Mathematically, yes, of course. In discrete practice, some meets floats sometimes, and the imaginary part is not always really zero, and this should be taken care of – Laurent Duval Nov 22 '18 at 19:41
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    @LaurentDuval, i still don't understand where the presumption that $\omega$ might not be real in $e^{j\omega}$ comes from. – robert bristow-johnson Nov 23 '18 at 09:00
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    // "complex numbers are fundamentally more central than real..." // i think that complex numbers are mathematically more general than real numbers, but i don't really perceive reality in terms of complex numbers. all of the physical quantities that i perceive are real. – robert bristow-johnson Nov 23 '18 at 09:02
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    I was worried that somebody might think that $e^{ia}$ as always unit modulus, without checking that this $a$ is real – Laurent Duval Nov 23 '18 at 09:06
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    I understand the need to be complete, but I think it's clear from context that $\omega$ is angular frequency (real quantity). If it was $\sigma$, or some other variable, I would agree that more explanation would be needed. I'll go ahead and edit my answer to put this to bed – Robert L. Nov 23 '18 at 18:31
  • @LaurentDuval you are a successful genius at shadowing your fllaws behind your brilliant (yet irrelevant) explanations.. ;-)) – Fat32 Nov 23 '18 at 21:32
  • I just call this unconscious intuition :) – Laurent Duval Nov 24 '18 at 00:40
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$Ke^{j\theta}$ is a mathematical way of describing a phasor with magnitude K and angle $\theta$. I think knowing this simple relationship takes a lot of the mystery away; which is proven by Euler's identity as Carlos has shown.

What is interesting is if you raise ANY real number to the power of j the magnitude will be 1!

You can see this by solving for the magnitude of a complex number which is the square root of the complex conjugate multiplication:

$\sqrt{x^j x^{-j}} = \sqrt{x^0} = 1$

Dan Boschen
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