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The Problem:

You encounter the following wave when examining a digital switching circuit. enter image description here

You need to create a mathematical model so that you can examine changes in the circuit’s behavior. Explain how you would build a series approximation for the square wave. How would you decide how far to extend the model to ensure that the approximate values it produces are accurate enough to be useful?

Does my work make sense? How else can I explain the differences between the 3 different series?

My Work:

There are three different ways to model an expression, Taylor Series, Maclaurin Series and the Fourier series. The Fourier series is primarily responsible for breaking down periodic functions and signals like sin and cos waves. The Fourier series is the only method that can be used to graph piecewise functions.
The Taylor series is set up to map a function by taking the derivatives of the given function then evaluating those derivatives at a given point. The Taylor series can be thought of as looking at the circuit in a small region of the function whereas the Fourier series looks at a function over a greater area. When looking at the series approximation we are looking at the infinite sum of the approximation.
The Maclaurin series is a branch off the Taylor series but is centered around 0. The Maclaurin series can’t be used when the function involves taking the ln. Like the Taylor series the Maclaurin series is when looking at the definition of the function we are looking at taking the infinite sum of the function.
When analyzing this circuit and looking at the 3 different methods we are given, we would use the Fourier series to find the series approximation for the square wave since the Fourier series is the only one that can approximate a piecewise function. From the screen shot of the circuit we can observe that over one period of the function

Therefore we can derive our needed functions to solve the Fourier series.

$$f(x) = \begin{cases} -1 & 0\leq x\leq 0.5\\ 1 & 0.5\leq x\leq 1 \end{cases}\\\\ \begin{align}A_0 &= \frac{1}{\pi}\int_0^1 2\,\mathrm dt\\ A_k &= \frac{1}{\pi}\int_0^1 2\cos(kx)\,\mathrm dx\\ B_k &= \frac{1}{\pi}\int_0^1 2\sin(kx)\,\mathrm dx \end{align}$$

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