“How can we take a function in the time domain and put it into the frequency domain?”
When dealing with signals, we are sometimes only given information about the time domain or frequency domain, even though it would be nice to see the other side. So how can we transform this information to suit our need? Well, let’s think about it. We know that we can decompose a continuous signal into multiple sine waves of varying frequency.
If we wanted to convert from frequency to time, what if we were to go through all of the frequencies, take the area under the curve to be an amplitude and multiply it by a sine wave with its prescribed period? Well, this is the fundamental idea behind an Inverse Fourier transform and can be represented by the equation f(s) = 1/(2pi) * (integral from -infinity to +infinity)f(omega)*e^(i*omega*t)d omega
The normal Fourier Transform simply goes in reverse and can be represented by the equation f(t) = (integral from -infinity to +infinity)f(omega)*e^(-2pi*i*omega*t)dt
Fourier transforms are the bedrock foundation of signal processing, making it possible for complex control systems to exist