Desmos Graphing Demonstration

Desmos Graphing & Signal Visualization Demonstration

This note demonstrates the full capabilities of the obsidian-desmos plugin. It is structured around the mathematical and signal concepts relevant to the 2-1 Fourier Series and Systems syllabus, illustrating how to plot basic waveforms, show mathematical symmetries, construct Fourier series approximations, and represent piecewise functions.

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1. Basic Trigonometric Waveforms

To plot basic continuous waveforms, we can use simple LaTeX functions. Here, we plot a fundamental sine wave along with its second harmonic. We restrict the viewport to focus on two periods.

left = -2\pi; right = 2\pi;
top = 2; bottom = -2;
grid = true
---
y = \sin(x) | RED | SOLID
y = \sin(2x) | BLUE | DASHED

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2. Symmetry Analysis (Even vs. Odd Functions)

Symmetry simplifies Fourier coefficient calculations.

• Even functions ($f(-x)=f(x)$) are symmetric about the y-axis.
• Odd functions ($f(-x)=-f(x)$) are symmetric about the origin.

The following graph plots a quadratic function (Even, green) and a cubic function (Odd, orange). We also place a point to label the origin symmetry transition.

left = -4; right = 4;
top = 8; bottom = -8;
---
y = 0.5x^2 | GREEN | SOLID
y = 0.2x^3 | ORANGE | SOLID
(0,0) | open | BLACK | label:Origin Symmetry Point

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3. Piecewise Functions and Signal Restrictions

Piecewise signals are very common in communications. We can use domain restrictions to plot a piecewise function such as: $f(x)=\{\begin{array}{ll}-1& -\pi <x<0\\ 1& 0\le x<\pi \end{array}$

We also use open and closed markers to indicate interval endpoints at $x=0$.

left = -4; right = 4;
top = 2; bottom = -2;
grid = true
---
y = -1 | -\pi < x < 0 | BLUE | SOLID
y = 1 | 0 \le x < \pi | RED | SOLID
(0,-1) | open | BLUE
(0,1) | POINT | RED | label:Discontinuity

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4. Fourier Series Square Wave Approximation

A square wave can be synthesized using a sum of odd harmonics: $x(t)=\frac{4}{\pi }{\sum }_{n=1,3,5,\dots }^{\infty }\frac{\sin (n{\omega }_{0}t)}{n}$

In this graph, we define a hidden base frequency ${\omega }_0=1$ and intermediate harmonics, then plot the resulting 1st, 3rd, and 5th harmonic partial sum approximations to visualize convergence.

left = -10; right = 10;
top = 2; bottom = -2;
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w = 1 | hidden
f_1(x) = \frac{4}{\pi} \sin(w \cdot x) | hidden
f_3(x) = f_1(x) + \frac{4}{3\pi} \sin(3 \cdot w \cdot x) | hidden
f_5(x) = f_3(x) + \frac{4}{5\pi} \sin(5 \cdot w \cdot x) | hidden
 
f_1(x) | DOTTED | CYAN
f_3(x) | DASHED | MAGENTA
f_5(x) | SOLID | PURPLE

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5. Gibbs Phenomenon Peak Detection

When approximating a signal with jump discontinuities, the Fourier series exhibits overshoot (about $9\%$) near the discontinuities. Here, we plot a high-order Fourier sum and label the overshoot peak.

left = -1; right = 4;
top = 1.5; bottom = -1.5;
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g(x) = \frac{4}{\pi} ( \sin(x) + \frac{\sin(3x)}{3} + \frac{\sin(5x)}{5} + \frac{\sin(7x)}{7} + \frac{\sin(9x)}{9} ) | hidden
 
g(x) | BLUE | SOLID
(0.28, 1.18) | cross | RED | label:Gibbs Overshoot (~18%)