Free Books    Mathematics of the DFT  

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 and imaginary part, respectively.

We also have special notation and various names for the polar coordinates $ (r,\theta)$ of a complex number $ z$:

\begin{eqnarray*} r &\isdef & \left\vert z\right\vert = \sqrt{x^2 + y^2}\\ &=&... ...!argument, angle, or phase\vert textbf}, or \emph{phase} of $z$} \end{eqnarray*}

The complex conjugate of $ z$ is denoted $ \overline{z}$ (or $ z^\ast$) and is defined by

$\displaystyle \zbox {\overline{z} \isdef x - j y} $

where, of course, $ z\isdef x+jy$.

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; i.e., complex conjugation replaces each point in the complex plane by its mirror image on the other side of the $ x$ axis.

Next Section:
Elementary Relationships
Previous Section:
The Complex Plane