Assignment

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Summery of the class notes from Oct. 5:

What if a periodic signal had an infinite period? We would no longer be able to tell the difference between it and a non periodic signal. We can use this property to look at signals that do not have a period (an observable one at least).

Begining with a Fourier Series:

(1)

where

We then take the limit of a Fourier series as its period T approaches infinity:

(2)

In order to evaluate this limit we need the following relationships:

We can now write out the following:

(3)

which can also be written as:

(4)

using,

we now have

(5)

We can now relate a signal in the time domain to a signal in the frequency domain. Using vector notation we can show this relationship as such:


Where,

Is the Fourier transform, or


Is the inverse Fourier transform, or

Now we can use our new tool, the Fourier transform on equation (2) to give us the following:


Notice

Similarly,

Again, notice

Both the time-domain and frequency domain have non-zero integrals when respectively.