Fourier series - by Ray Betz

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Fourier Series

If

  1. Dirichlet conditions are satisfied

then we can write

The above equation is called the complex fourier series. Given , we may determine by taking the inner product of with . Let us assume a solution for of the form . Now we take the inner product of with .

If then,

If then,

We can simplify the above two conclusion into one equation.

So, we may conclude

Orthogonal Functions

The function and are orthogonal on if and only if .

The set of functions are orthonormal if and only if .

Linear Systems

Let us say we have a Linear time Invarient System, where is the input and is the output. What outputs do we get as we put different inputs into this system?

File:Linear System.jpg

INPUT OUTPUT REASON

Given

Fourier Series (indepth)

I would like to take a closer look at in the Fourier Series. Hopefully this will provide a better understanding of .

We will seperate x(t) into three parts; where is negative, zero, and positive.

Now, by substituting into the summation where is negative and substituting into the summation where is positive we get:

Recall that

If is real, then . Let us assume that is real.

Recall that Here is further clarification on this property

So, we may write:

Fourier Transform

Fourier transforms emerge because we want to be able to make Fourier expressions of non-periodic functions. We can take the limit of those non-periodic functions to get a fourier expression for the function.

Remember that:


So,

From the above limit we define and .

We can take the derivitive of and then put in terms of the reverse fourier transform.

What happens if we just shift the time of ?

In the same way, if we shift the frequency we get:

What would be the Fourier transform of ?


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