Week Three - Day Two

Today, I continued learning about Fourier Transform, which is what I started reading about between Day One and Day Two. This is to help with the python project that analysis the data collected from the EEG. The signals collected, which are just functions on a graph, will be turned into functions of frequencies to be analyzed by the python code.

Also: Found out that Emotiv EPOC only runs for 800 seconds before crashing.

The next few pieces of data are also in the PowerPoint shown below but will be summarized. Fourier Transform is linear since it is both homogeneous and additive.
For homogeneous systems, amplifying the input will likewise amplify the output. S{af(t)} = aS{f(t)}
For additive systems, the response of the sum is the sum of the responses. S{f1(t) + f2(t)} = S{f1(t)} + S{f2(t)}
All linear systems produce an output of zero when the input is zero. S{0} = 0

Euler's Identity:
e^i*2pi*t = cos(2pi*t) + isin(2pi*t)

Fourier analysis of a periodic function refers taking the sine and cosine components of the overall "weird" function and separating them into simplified pieces of a whole. An example would be if you played piano, the keys in a cord are played at the same time and therefore you hear one sound, but when you use fourier analysis, the frequencies of each key are taken into account and therefore you get three different frequencies, which means three different waves. This is what transforming a function of time into a function of frequency means. And if you take the inverse, you are transforming a function of frequency into a function of time.

Fast Fourier Transform is an algorithm to compute the Discrete Fourier Transform and its inverse. A transform is a mapping between two sets of data/domains (time domain, frequency domain, or even a space domain) The real component would be an even function and the imaginary component would be an odd function on the real (x-axis) and imaginary (y-axis) plane.