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14.7. Worked Examples
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Today, we're going to apply Parseval's Theorem to a square wave function. Can anyone remind me of the definition of a square wave?
Isn’t it a function that has equal durations of positive and negative values, like 1 and -1?
Exactly! Now, what can we say about its Fourier coefficients?
For an odd function like a square wave, the cosine coefficients are zero.
Correct! So we just calculate the sine coefficients, which we can find using the integral. Let's set it up.
What’s the formula for the sine coefficients again?
Great question! The coefficient b_n is given by the integral of f(x) multiplied by sin(nx). Now, let’s calculate it.
I see that it results in b_n = 4/nπ for odd n and b_n = 0 for even n, right?
Exactly! Now let's apply Parseval's Theorem to relate the total energy of the square wave to these coefficients.
So, we integrate the square of the function and set it equal to the sum of the squares of the b_n coefficients?
That's right! By confirming that these values are equal, we validate Parseval's Theorem.
In summary, we've shown how energy calculations for periodic functions like the square wave can be validated using Parseval's Theorem.
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Now, let’s shift gears and look at the triangular waveform. Who can describe this function?
It’s a function that has linearly increasing or decreasing segments, I think.
That's right! We'll define f(x) = x on (-π, π) that extends into an odd function. What are our next steps?
We need to find the sine coefficients again, right?
Yes, and remember, how do we actually compute those coefficients?
We integrate x * sin(nx) from 0 to π, and we should use integration by parts.
Excellent! Let's perform that integration. Who can explain what we expect from our final answer?
We will end up with a formula for b_n that involves even and odd n, showing that their squares sum up to the function's total energy.
Exactly, and this will show us how energy is conserved between different forms! Let's calculate and check this against Parseval’s identity.
In summary, we’ve applied Parseval's Theorem to the triangular waveform, reinforcing the theory's practical application.
Overview
Short Summary
This section provides practical worked examples demonstrating the application of Parseval's Theorem in determining the energy of periodic functions.
Medium Summary
In the Worked Examples section, we calculate the energy of a square wave and a triangular waveform using Parseval's Theorem. These examples illustrate the process of deriving Fourier coefficients and applying the theorem in real scenarios.
Detailed Summary
Worked Examples
In this section, we examine practical applications of Parseval's Theorem through two worked examples: the energy of a square wave and a triangular waveform. Parseval's Theorem relates the total energy of a function to the sum of the squares of its Fourier coefficients, providing a critical tool in engineering mathematics, particularly in signal processing.
Example 1: Energy of a Square Wave
We define a periodic square wave function of period 2π, characterized by values of 1 in the interval (-π, 0) and -1 in (0, π). Its nature as an odd function ensures that all cosine coefficients (a_n) are zero. The sine coefficients (b_n) are calculated by integrating the sine function, resulting in:
- For n odd: b_n = 4/nπ
- For n even: b_n = 0.
Using Parseval's Theorem, we verify that the total energy computed from the integral of the square wave equals the sum of the squares of the b_n coefficients, confirming the theorem's validity.
Example 2: Energy of a Triangular Waveform
Next, we consider a triangular waveform defined by the function f(x) = x over the interval (-π, π) extended to be odd. Again, we derive the sine coefficients (b_n) through integration by parts. The final result, calculated with Parseval's identity, reinforces our understanding of how to balance energy calculated through integration with that calculated through Fourier coefficients, confirming that the derived values hold the necessary equivalence.
These examples are pivotal for engineers in understanding how to analyze energy in different signal types and enhance their skills in applying Fourier analysis in real-world problems.
Reference YouTube Videos
Audio Book
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Create a free accountExample 1: Compute the energy of a square wave using Parseval’s Theorem Let f(x) be a periodic square wave function of period 2π, defined on (−π,π) as:
f(x) = \begin{cases} 1, & -\pi < x < 0 \\ -1, & 0 < x < \pi \end{cases}$$ This is an odd function, so all $a_n = 0$, and only sine terms appear. The Fourier coefficients are:b_n = \frac{1}{\pi} \int_{0}^{\pi} f(x) \sin(nx) dx = \frac{1}{\pi} \int_{0}^{\pi} (-1)\sin(nx)dx = \frac{1 - (-1)^n}{n}$$
So,
b_n = \begin{cases} \frac{4}{n\pi}, & n \text{ odd} \\ 0, & n \text{ even} \end{cases}$$ Now using Parseval’s Theorem: $$\frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx = \sum_{n=1}^{\infty} \frac{b_n^2}{2}$$ Since $f(x)^2 = 1$ always, $$\frac{1}{\pi} \int_{-\pi}^{\pi} 1 dx = \frac{1}{\pi} \cdot 2\pi = 2$$ Therefore: $$\sum_{n \text{ odd}} \frac{4^2}{(n\pi)^2} = 2 \Rightarrow \sum_{n \text{ odd}} \frac{1}{n^2} = \frac{\pi^2}{8}$$ This identity confirms Parseval’s Theorem.Detailed Explanation
In this example, we compute the energy of a periodic square wave function using Parseval's Theorem, which relates the function's energy in the time domain to the sum of the squares of its Fourier coefficients. A periodic square wave alternates between 1 and -1, leading to simplifications since its even Fourier coefficients vanish. We find the odd Fourier coefficients are non-zero, and then apply Parseval's Theorem to establish a connection to the total energy computed through integration.
Examples & Analogies
Imagine a light switch that toggles between being 'on' (1) and 'off' (-1). The energy consumed over time can be viewed as the area covered by this switch being 'on'. Parseval’s Theorem in this context allows us to quantify the total energy used not just by summing moments in time but by evaluating the energy on a frequency spectrum, which tells us how these moments contribute overall.
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Create a free accountExample 2: Energy of a triangular waveform Let f(x)=x on (−π,π), extended as an odd function.
Being odd, only sine terms remain:
b_n = \frac{2}{\pi} \int_{0}^{\pi} x \sin(nx) dx$$ Using integration by parts, we find: $$b_n = \frac{2(−1)^{n+1}}{n}$$ Then: $$\frac{1}{\pi} \int_{-\pi}^{\pi} x^2 dx = \sum_{n=1}^{\infty} \frac{b_n^2}{2}$$ We know $\int_{-\pi}^{\pi} x^2 dx = \frac{2\pi^3}{3}$, so: $$\frac{2\pi^3}{3} \cdot \frac{1}{\pi} = \sum_{n=1}^{\infty} \frac{4(−1)^{n+1}}{n^2}$$ This classic result is verified via Parseval’s identity.Detailed Explanation
This example focuses on a triangular wave, defined as a linear function on the interval. Again, because the function is odd, only sine terms contribute to the Fourier series. Through integration by parts, we derive the Fourier coefficients. We then equate the computed energy, via integration of the square of the function, to the sum of the squares of the Fourier coefficients, verifying Parseval's theorem for this specific waveform.
Examples & Analogies
Think of a triangular wave as a series of peaks resembling mountains along a hiking trail. The height of each peak represents energy at various frequencies. By using Parseval’s theorem, we can calculate the total energy of this trail by analyzing the peaks’ heights instead of evaluating every single point on the path. Just like hiking, measuring the overall effort gives a clearer picture than checking each step.
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Key concepts
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
Calculating the energy of a square wave function results in using Parseval's theorem and obtaining a numerical equivalence.
Using a triangular waveform to derive behavior in the Fourier domain illustrates practical application of signal energy concepts.
Memory aids
Imagine a barbell with equal weights on both sides representing the balance of energy in the time and frequency domains, showcasing Parseval's theorem in action.
E-C-F: Energy Conservation Formula (to remember the connection of energy in time and frequency domains).
Flash Cards
Glossary
Fourier Series
A way to represent a periodic function as a sum of sine and cosine functions.
Parseval’s Theorem
A theorem that relates the total energy of a function in time domain to the sum of the squares of its Fourier coefficients.
Sine Coefficient
The coefficients of the sine terms in a Fourier series, denoted as b_n.
Energy of a Function
The integral of the square of the function over its period, representing the total energy.
Odd Function
A function f(x) that satisfies f(-x) = -f(x), leading to zero cosine coefficients in Fourier series.