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The Complex Plane

Figure 2.2: Plotting a complex number as a point in the complex plane.

We can plot any complex number $ z = x + jy$ in a plane as an ordered pair $ (x,y)$ , as shown in Fig.2.2. A complex plane (or Argand diagram) is any 2D graph in which the horizontal axis is the real part and the vertical axis is the imaginary part of a complex number or function. As an example, the number $ j$ has coordinates $ (0,1)$ in the complex plane while the number $ 1$ has coordinates $ (1,0)$ .

Plotting $ z = x + jy$ as the point $ (x,y)$ in the complex plane can be viewed as a plot in Cartesian or rectilinear coordinates. We can also express complex numbers in terms of polar coordinates as an ordered pair $ (r,\theta)$ , where $ r$ is the distance from the origin $ (0,0)$ to the number being plotted, and $ \theta$ is the angle of the number relative to the positive real coordinate axis (the line defined by $ y=0$ and $ x>0$ ). (See Fig.2.2.)

Using elementary geometry, it is quick to show that conversion from rectangular to polar coordinates is accomplished by the formulas

r &=& \sqrt{x^2 + y^2}\\
\theta &=& \tan^{-1}(y,x).

where $ \tan^{-1}(y,x)$ denotes the arctangent of $ y/x$ (the angle $ \theta$ in radians whose tangent is $ \tan(\theta)=y/x$ ), taking the quadrant of the vector $ (x,y)$ into account. We will take $ \theta$ in the range $ -\pi$ to $ \pi $ (although we could choose any interval of length $ 2\pi $ radians, such as 0 to $ 2\pi $ , etc.).

In Matlab and Octave, atan2(y,x) performs the ``quadrant-sensitive'' arctangent function. On the other hand, atan(y/x), like the more traditional mathematical notation $ \tan^{-1}(y/x)$ does not ``know'' the quadrant of $ (x,y)$ , so it maps the entire real line to the interval $ (-\pi/2,\pi/2)$ . As a specific example, the angle of the vector $ (x,y)=(1,1)$ (in quadrant I) has the same tangent as the angle of $ (x,y)=(-1,-1)$ (in quadrant III). Similarly, $ (x,y)=(-1,1)$ (quadrant II) yields the same tangent as $ (x,y)=(1,-1)$ (quadrant IV).

The formula $ r = \sqrt{x^2 + y^2}$ for converting rectangular coordinates to radius $ r$ , follows immediately from the Pythagorean theorem, while the $ \theta = \tan^{-1}(y,x)$ follows from the definition of the tangent function itself.

Similarly, conversion from polar to rectangular coordinates is simply

x &=& r\,\cos(\theta)\\
y &=& r\,\sin(\theta).

These follow immediately from the definitions of cosine and sine, respectively.

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``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
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University