Fourier Series/Transformations and Convolution

In summary, the problem involves evaluating en, the Fourier coefficients for the convolution of two functions, in terms of cn and dn, the Fourier coefficients for the two individual functions. One possible approach is to use the theorem mentioned in the relevant equations section, which states that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the individual functions. Another possible approach is to substitute the definition of convolution into the integral for en and work it as a double integral.
  • #1
Yosty22
185
4

Homework Statement



(f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy
f(x) = ∑cneinx
g(x) = ∑dneinx

en is defined as the Fourier Coefficients for (f*g) {the convolution} an is denoted by:

en = 1/(2pi) integral from -pi to pi of (f*g)e-inx dx

Evaluate en in terms of cn and dn

Hint: somewhere in the integral, the substitution z = x - y might be helpful.

Homework Equations



For convolution, if C(x) = the integral from -infinity to infinity of (f(y)g(x-y))dx, the Fourier transform of C(x) is equal to the Fourier transform of f times the Fourier transform of g.

The Attempt at a Solution



I'm really not too sure where to begin with this other than understanding the definition in the Relevant Equations section. Before this problem, all we were told was this definition. I do not see any way this is related to the Fourier series mentioned in the problem, let alone how to incorporate just the coefficients of the series into the integral.

Any help to point me in the right direction would be greatly appreciated.
 
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  • #2
Do you understand how that result, the theorem you mention in the relevant equations section, was obtained? If not, start there. You might try something similar in solving the problem.
 
  • #3
I am pretty confused. The only thing jumping out to me on what to do is to substitute substitute (f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy into the integral for en and work it like a double integral. I am not exactly too sure if this would work though because I cannot think of where cn and dn would come in though (since he wants the answer in terms of those coefficients).
 

FAQ: Fourier Series/Transformations and Convolution

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function using a sum of sine and cosine functions. It allows us to break down a complex function into simpler trigonometric components, making it easier to analyze and manipulate.

What is a Fourier transform?

A Fourier transform is a mathematical tool used to convert a function from its time domain to its frequency domain. It decomposes a function into its individual frequency components, providing information about the frequency and amplitude of each component.

What is the difference between a Fourier series and a Fourier transform?

A Fourier series is used for periodic functions, while a Fourier transform is used for non-periodic functions. A Fourier series represents a function as a sum of sine and cosine functions, while a Fourier transform represents a function as a sum of complex exponential functions.

What is the significance of convolution in Fourier analysis?

Convolution is a mathematical operation used to combine two functions into a third function. In the context of Fourier analysis, convolution allows us to combine the frequency components of two functions to obtain the frequency components of their convolution. This is useful in signal processing and image filtering applications.

How are Fourier series/transformations and convolution used in real-world applications?

Fourier series/transformations and convolution have numerous applications in various fields such as signal processing, image processing, and data analysis. They are used in audio and video compression, radar and sonar signal processing, and medical imaging, to name a few. They also have applications in solving differential equations and in quantum mechanics.

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