By Sneha Das and Raghavendra S

“Complex”, the word by itself has an inherent sense which would make one expect complications rather than clear ideas! Blame the English.

And the “i” for imaginary associated with the complex numbers adds further to the disconnect aggravating the already persistent Mathematics reluctance amongst many.

In general, a word of caution about the beauty in Math is in that it is revealed in a slow and seductive process. It certainly requires a little topping of imagination to be able to interpret and appreciate its subtleties.

In this article, in an attempt to present the wonder and beauty of a mathematical nuance we look at complex sinusoids.

Power of the “i”

The profound Euler’s identity and the polar form representation of complex numbers in the Euler’s formula are some wonders of Mathematics possible because of “i” Complex Plane:

Rectangular and Polar representations of Complex numbers fig 1 & fig 2

Simply put, a Complex plane is the mathematical visualization of space with a real and imaginary axis in the Rectangular form. In the Rectangular coordinate system, the preferred convention is that the horizontal axis is the real axis and the vertical axis to be the Imaginary axis. Complex numbers have their extensive applications because of their ability toggle from rectangular to another form called the “Polar form” with amplitude and angle parameters defining points in the plane.

The two polar coordinates r and θ can be converted to the two Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

while the Cartesian coordinates x and y can be converted to the polar coordinates r and θ by:

A visualization of rectangular to polar form is comprehensively demonstrated in this animation. Rectangular to polar conversion Fig 3

Further, another aspect of the polar and rectangular forms of representation is that:

cosine of the angle ‘theta’ made by the point is the projection one should be expecting on the Real axis (horizontal) of the complex plane, whereas sine of theta on the Imaginary axis (vertical).

If that already does not make sense, we’ll look at these claims in a more convincing manner in a little while; before that let us try and visualize the exponential form of complex numbers

Consider a complex exponential of the form exp(i*2*pi*f*t), and let us try visualizing this function in GNU Octave:

The 2-D view of this function for frequency f=1 Hz and 1000 time samples in the interval [0,1], when only the amplitude of this function is plotted and the notion of the 2-pi revolutions on a circle is clear from the plot in fig 1.

octave:1> f=1;

octave:2> t=0:0.001:1;

octave:3> c=exp(2*pi*i*f*t);

octave:4> plot(c)

The 2-D view of this function for frequency f=1 Hz and 1000 time samples in the interval [0,1], when the amplitude of this function is plotted with respect to time, the same function yields a different, yet known plot. As shown in the figure below, when observed along the varying time axis,

the 2-pi circle is seen as a 2-pi period cosine waveform!

octave:1> f=1;

octave:2> t=0:0.001:1;

octave:3> c=exp(2*pi*i*f*t);

octave:4> plot(t,c)

Now, if it hasn’t dawned upon the reader that the same function exp(i*2*pi*f*t) yields the above two plots and the ramification of this duality, the following animation will clarify! [observe the transitions from a circle to a cosine and then a sine of the 3-D complex sinusoid] fig 6

So, what we were looking at in the first two plots fig 4 and fig 5 were the front view and side view of sorts of the 3-D plot shown in fig 6

Complex plane is a 3-D space where the ‘phasor’ of the quantities carve out the 2-pi phase as time varies, which can be interpreted as circular revolutions or the 2-pi periodic sinusoids.

Another example with f=5 Hz. Sine and Cosine as projections from a complex sinusoid

As shown in figure 6 sine and cosine are projections of the complex sinusoid on the imaginary (vertical) and real (horizontal) planes

Now, the pure sine and pure cosine can be obtained from complex sinusoids using simple arithmetic operations.

So, let us try it with GNU Octave:

octave:1> f=1;

octave:2> t=0:0.001:1;

octave:3> c=exp(2*pi*i*f*t);

octave:4> c1=exp(-(2*pi*i*f*t));

octave:5> cosine= (c+c1)/2;

octave:6> sine= (c-c1)/2;

octave:7> figure, plot3(cosine)

octave:8> figure, plot3(sine) Figure for cosine : Projection on real (horizontal) plane Figure for sine : Projection on imaginary (vertical) plane

Another profound implication that can be observed from the above figure is the orthogonality (perpendicularity) of cosine and sine functions! In calculus denoted as the common area between sine and cosine function in a period of 2-pi is zero. An important property in signal processing, which takes care of uniqueness of information carried by sine and cosine functions.

These are some of the important special attributes that the complex sinusoid in Complex plane ushers with making analysis of real world comprehensible and awesome!

References

https://en.wikipedia.org/wiki/Polar_coordinate_system

Special thanks to Mr.Saneesh for providing extra insights into the Complex plane and complex sinusoids