Complex Numbers

What does the ‘complex’ in the complex logarithmic map refer to? The two formulas relating the cortical coordinates \(X\) (eqn. eqn_mapeqn2) and \(Y\) to the visual field coordinates \(r\) and \(\phi\) can be combined by using complex numbers: \(Z=X+iY\), with \(i\) the square root of -1. One reason this is useful, is because we can relate the Cartesian (rectangular) coordinates \(X\) and \(Y\) to the polar coordinates \(r\) and \(phi\). You can represent complex numbers as vectors in a 2-dimensional (complex) plane (Fig. Complex coordinate system. From rectangular to polar coordinates.). This shows that you can easily convert from one notation to another:

\[\begin{split}\begin{cases} r= \quad \sqrt{X^2+Y^2}\\ \phi= \quad \arctan (\frac{Y}{X})\\ \end{cases}\end{split}\]

and also:

\[\begin{split}\begin{cases} X= \quad r\cos{\phi}\\ Y= \quad r\sin{(\phi)}\\ \end{cases}\end{split}\]

where \(X\) and \(Y\) can also be regarded as the real and imaginary part of the proper complex number: \(X+iY\). All this follows from simple geometry and trigonometry (Pythagorean Theorem and SOSCASTOA…), in which \(r\) is the magnitude of the vector and \(\phi\) its angle.

Fig. 87 Complex coordinate system. From rectangular to polar coordinates.

We can therefore convert the proper complex number from rectangular to polar representation as follows:

\[X+iY=r(\cos{\phi}+i \sin{\phi})\]

As before, on the left side, a proper complex number is represented in “rectangular coordinates” (X and Y, or cos and sin), on the right side in “polar coordinates” (eccentricity and direction). Now, one of the most important equations in mathematics (equivalent to \(e=mc^2\)) is Euler’s relation:

\[e^{(ix)}=\cos{x}+i \sin{x}\]

which relates \(e\), \(i\), and the harmonic functions cos and sin. With this relation, we can now express a complex number in polar coordinates as a complex exponential:

\[X+iY=re^{(i\phi)}\]

Complex numbers in this form are the backbone of digital signal processing and analysis. It would be good to memorize these equations as it will help you in your further career (but unfortunately, this is not a learning goal of this course; so, fortunately, you do not have to memorize them for the exam).

Putting this together, and using a visual point stimulus as an example, using the symbols \(x\) and \(y\) for the rectangular coordinates of a visual stimulus in the visual field, and \(r\) and \(phi\) its polar coordinates, we can define the complex number \(\mathbf{z}\):

(32)\[\mathbf{z} = x+iy=re^{i\phi}\]