Problem 24 - Calculus

Question

The pulse train of width \(c\) is the periodic function \(f\) of period \(2 \pi\) given by $$f(x)=\left\\{\begin{array}{ll}0 & -\pi \leq x<-c / 2 \\\1 & -c / 2 \leq x

Step-by-Step Solution

Answer

Fraction in constant: \(\frac{4}{\pi^2}\); in constant + first harmonic: \(\frac{4 + 2 \sin^2(1)}{\pi^2}\). Use \(N = 10\) terms for 90% energy. Graph both functions for clarity.

Step 1: Define the Pulse Train Function

The pulse train function for width 2 has period \(2\pi\) and is defined as follows: \(f(x) = 1\) for \(-1 \leq x < 1\), and \(f(x) = 0\) for \(-\pi \leq x < -1\) and \(1 \leq x < \pi\). This function is periodic with period \(2\pi\) and will repeat outside this interval.

Step 2: Fourier Series of the Pulse Train

A periodic function's Fourier series representation is given by \(f(x) = a_0 + \sum_{n=1}^{\infty} (a_n\cos(nx) + b_n\sin(nx))\). For this even function, all sine coefficients \(b_n = 0\), and the Fourier coefficients are calculated using \(a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx\) and \(a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) \, dx\).

Step 3: Calculate the Constant Term \(a_0\)

Calculate \(a_0\) using the integral: \[ a_0 = \frac{1}{\pi} \int_{-1}^{1} 1 \, dx = \frac{1}{\pi} [x]_{-1}^{1} = \frac{2}{\pi}. \]

Step 4: Calculate the First Harmonic Coefficient \(a_1\)

Calculate \(a_1\) using the integral:\[ a_1 = \frac{1}{\pi} \int_{-1}^{1} \cos(x) \, dx = \frac{1}{\pi} [\sin(x)]_{-1}^{1} = \frac{2\sin(1)}{\pi}. \] Since \(b_1 = 0\), the first harmonic term is \(a_1\cos(x) = \frac{2\sin(1)}{\pi}\cos(x).\)

Step 5: Compute Energy in Terms of the Fourier Series

The energy of a function \(f\) over one period is given by \(E = a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\). For this function, \(b_n = 0\) for all \(n\), so the energy is \(E = a_0^2 + \frac{1}{2}\sum_{n=1}^{\infty} a_n^2\).

Step 6: Calculate Energy Fractions

- The energy in the constant term is \(a_0^2 = \left(\frac{2}{\pi}\right)^2\).- The energy in the constant and first harmonic terms is \(a_0^2 + \frac{1}{2}a_1^2 = \left(\frac{2}{\pi}\right)^2 + \frac{1}{2} \left(\frac{2\sin(1)}{\pi}\right)^2 \).- Calculate the percentage of the total energy from these terms.

Step 7: Determine Number of Terms for 90% Energy

For \(n\) terms, calculate the cumulative energy until it equals 90% of the total energy. Solve the inequality: \[ a_0^2 + \frac{1}{2}\sum_{n=1}^{N} a_n^2 \geq 0.9 \times E. \]The smallest \(N\) solving this inequality gives the required number of terms.

Step 8: Graphical Representation

Plot the original pulse train \(f\) on \([-3\pi, 3\pi]\). For the third Fourier approximation, include only the constant and the first two harmonic terms. Use software like Desmos or a graphing calculator to visualize.

Key Concepts

Pulse Train Function

A pulse train function is a piecewise periodic function characterized by alternating intervals of 'on' and 'off' values. In this context, consider a periodic function centered around intervals of length involving a constant value and zero elsewhere. We examine a pulse train of width 2, where it assumes the value 1 for \

• \(-1 \leq x < 1\),
• \(f(x) = 0\)elsewhere within the interval of \(-\pi\) to \(\pi\).

This stipulates that our function is 'on' for a short interval, and cyclically 'off,' repeating its form over each complete cycle.Finally, this pulse train repeats every \(2\pi\). Thus, its periodicity enables us to model real-world signals and systems where such regular patterns occur.

Periodic Functions

Periodic functions, like our pulse train, are functions that repeat their values in regular intervals over their length. For our given exercise, the periodicity is \(2\pi\).Such functions are pivotal in Fourier analysis as they allow us to generalize signals into a sum of sines and cosines.The core idea involves breaking down complex waves into simpler, oscillating components that recur at predictable intervals.

• Period-length: Measures the distance required to finish one cycle.
• Simplicity in complexity: Decompose intricate patterns into manageable components.

This technique is broadly used in electrical engineering, signal processing, and physics to simplify the study of complex oscillatory behaviors.

Energy of a Function

The energy of a function within one period helps to gauge its power or intensity across its interval. It calculates how much power is embodied in each term of its Fourier series.To find this, we examine the coefficients that show how much each harmonic contributes to the function overall:

• The constant term \(a_0^2\) reflects the signal's base level energy.
• Other terms \(a_n^2\) account for added harmonics.

The formula for computing a function's energy over one period is:\[E = a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} a_n^2\]This metric is critical when optimizing how a signal is represented across different systems.

Graphical Representation

Graphical representation serves as a visual tool to comprehend how mathematic functions manifest in the visual domain. For the pulse train function, plotting shows the periodic nature with repeating high and low segments at regular \(2\pi\) intervals.Visualizing Fourier approximations not only enhances understanding but also showcases the effect of each harmonic contribution.The third Fourier approximation employs terms: a

• constant term \((a_0)\)
• and two harmonics \((a_1, a_2)\)

These all combine to mirror the original function with increasing precision.Graphing platforms, such as Desmos, make such tasks intuitive and provide clarity about the comparison between the ideal function and its approximations.