Input data files
This answer is for storing input data files for testing solutions to the problem.
Here are the 120 $x,y$ coordinates for the $\pi$ shape, by @Chris. Save as pi.csv:
Complex Fourier series of a piece-wise linear waveform tracing the desired shape
Instead of using discrete Fourier transform (DFT) / fast Fourier transform (FFT), a more direct approach is to define a piece-wise linear continuous-time waveform that traces the desired shape on the complex plane, and to directly calculate its Fourier series. Bezier curves or ...
The product $x(t)y(t)$ of two periodic signals with fundamental periods $T_x$ and $T_y$ is not a periodic signal unless $T_x$ and $T_y$ are rational multiples of one another; that is, $T_x = aT_y$ where $a$ is a rational number. Thus, except when such a relationship holds, $x(t)y(t)$ does not have a Fourier series.
When $T_x$ is a rational multiple of $T_y$,...
I know that I am not really being listened to, Chris, but I know exactly what you're trying to do. I know exactly what the problem is. I know exactly what the mathematics are. And I know exactly what you should do and exactly how you should think about it. You're starting to move in the right direction with an ordered set of $N$ points with a horizontal ...
I'm not understanding the comments.
Of course you can do this. It is simply a matter of understanding what a DFT means, how to calculate DFT bin values, and how to interpret those bin values as continuous fourier series coefficients.
First off, the plane you are looking at is the complex plane. Your points are a set of $N$ discrete samples. Each sample ...