More Notation and Terminology

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. More Notation and Terminology

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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More Notation and Terminology

It’s already been mentioned that the rectilinear coordinates of a complex number $z = x + jy$ in the complex plane are called the real part andimaginary part, respectively.

We also have special notation and various names for the radius and angle of a complex number $z$ expressed in polar coordinates $(r,\:

\


The complex conjugate of $z$ is denoted $\ and is defined by

\

where, of course, $z\. Sometimes you’ll see the notation$z^\ in place of $\, but we won’t use that here.

In general, you can always obtain the complex conjugate of any expression by simply replacing $j$ with $-j$. In the complex plane, this is a vertical flip about the real axis; in other words, complex conjugation replaces each point in the complex plane by its mirror image on the other side of the $x$ axis.

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