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?

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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