Calculate the integral using the Fourier coefficients

• MHB
• mathmari
In summary, the conversation discusses the calculation of $\int_0^{T_0}x^2(t)\, dt$ for a periodic signal with period $T_0=2$ using the given Fourier coefficients. The speaker wonders if their calculation is correct and checks their work. The second speaker corrects a mistake and clarifies that since the signal is real, the integral can be written without the absolute value.
mathmari
Gold Member
MHB
Hey!

A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$.
I want to calculate $\int_0^{T_0}x^2(t)\, dt$.

I have done the following:

It holds that $$\frac{1}{T_0}\int_{T_0}|x(t)|^2\, dt=\sum_{k=-\infty}^{+\infty}|X_k|^2$$ right? (Wondering)

Then do we get $$\int_{T_0}|x(t)|^2\, dt=2\sum_{k=-\infty}^{+\infty}|X_k|^2=2\left [\left(\frac{2}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{4}\right )^2+1\right ]$$ But the result that I get is not one of the choices. So have I done something wrong? (Wondering)

Hey mathmari!

Shouldn't it be:
$$\frac{1}{T_0}\int_{T_0}|x(t)|^2\, dt =\sum_{k=-N}^{+N}|X_k|^2 \\ \int_{T_0}|x(t)|^2\, dt =T_0\sum_{k=-N}^{+N}|X_k|^2 =2\left\{\left(\frac{2}{3}\right )^2 + 2\left [\left(\frac{1}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{4}\right )^2+1\right ]\right\}$$
(Wondering)

Oh, and since it's given that $x(t)$ is a real signal, we can write $\int_{T_0}|x(t)|^2\, dt = \int_{T_0}x(t)^2\, dt$, can't we? (Wondering)

1. What is the purpose of calculating the integral using Fourier coefficients?

The purpose of calculating the integral using Fourier coefficients is to decompose a function into a combination of simple trigonometric functions, making it easier to analyze and solve complex mathematical problems.

2. How do you calculate the Fourier coefficients?

The Fourier coefficients can be calculated using the Fourier series formula, which involves integrating the function over one period and multiplying it by specific trigonometric functions. Alternatively, they can also be calculated using the Fast Fourier Transform algorithm.

3. What is the significance of the Fourier coefficients?

The Fourier coefficients represent the amplitude and phase of each individual trigonometric function in the Fourier series. They provide valuable information about the frequency components of the original function and can be used to reconstruct the function.

4. Can the Fourier coefficients be used to approximate any function?

Yes, the Fourier coefficients can be used to approximate any continuous function, as long as it has a finite number of discontinuities and is periodic. However, the accuracy of the approximation depends on the number of terms used in the Fourier series.

5. Are there any limitations to using Fourier coefficients to calculate integrals?

One limitation is that the function must be periodic, which means it repeats itself over a certain interval. Additionally, the Fourier series may not converge for some functions, making it impossible to accurately calculate the integral using Fourier coefficients.

• General Math
Replies
1
Views
1K
• General Math
Replies
7
Views
1K
• Topology and Analysis
Replies
4
Views
442
• General Math
Replies
3
Views
326
• General Math
Replies
3
Views
1K
• General Math
Replies
1
Views
773
• Topology and Analysis
Replies
1
Views
463
• General Math
Replies
5
Views
3K
• General Math
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
297