Calculating Fourier Coefficients

In summary, the conversation discusses taking the integral of the function f(t) multiplied by one of its components, involving a sigma and orthogal properties of sine and cosine. The confusion lies in isolating the terms due to the presence of a sigma. The response clarifies that the sigma can be separated into two components, acos(nwt) and bsin(nwt), and the integral of the sum can be taken as the sum of the integrals due to the improving rate of convergence.
  • #1
cscott
782
1
Yes, another thread... lab due tomorrow :tongue2:

We take the integral of the function f(t) times one of the components:

integral(0->T) of [a0 + sigma(n=1->N) acos(nwt) + bsin(nwt)]sin(nwt)

Now, in order to evaluate this is it correct to say we multiple sin(nwt) through then take the integral of each term seperately, and given the orthogal properties of sine/cosine we say the first two terms are 0 but the bsin^2(nwt) becomes bT/2?

I guess I should say what's confusing me: the sigma; because shouldn't the acos and bsin be in brackets and then we couldn't isolate the terms in the same way.

Is it because this is true?:

integral(0->T) of [a0 + sigma(n=1->N) acos(nwt) + sigma(n=1->N) bsin(nwt)]sin(nwt)
and then we can multiply through by sin(nwt) to get the three terms?

Sorry for the shotty math.
 
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  • #2
Yupp,

sigma(n=1->N)[acos(nwt) + bsin(nwt)] =
sigma(n=1->N) [acos(nwt)] + sigma(n=1->N)[bsin(nwt)]

if that was what you wondered.

Then you need to understand why you can take the sum of the integrals as the integral of the sum.
Thats because integration improves the rate of convergence for the sum.
This is not true for the derivative, where you need to know that the sum converge uniformly before you can differentiate each term separately.
 
  • #3
Thanks for your reply.
 

What is the purpose of calculating Fourier coefficients?

The purpose of calculating Fourier coefficients is to represent a periodic function as a sum of simple sinusoidal functions. This can be useful in analyzing and understanding the behavior of complex signals or in solving differential equations.

How do you calculate Fourier coefficients?

Fourier coefficients can be calculated using the Fourier series formula, which involves integrating the periodic function with respect to time and multiplying it by a complex exponential function. The resulting coefficients represent the amplitudes and frequencies of the sinusoidal functions that make up the original function.

What is the difference between Fourier series and Fourier transform?

Fourier series is used to represent a periodic function as a sum of simple sinusoidal functions, while Fourier transform is used to analyze a non-periodic function by decomposing it into its frequency components. Fourier transform is a generalization of Fourier series for non-periodic functions.

How many Fourier coefficients do you need to accurately represent a function?

The number of Fourier coefficients needed to accurately represent a function depends on the complexity of the function and the desired level of accuracy. Usually, a larger number of coefficients will result in a more accurate representation of the original function.

What are some common applications of Fourier coefficients?

Fourier coefficients are commonly used in signal processing, image processing, and data analysis. They are also used in solving differential equations, analyzing vibrations in mechanical systems, and in the study of periodic phenomena in physics and engineering.

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