fourier_transform | A Fourier Transform example in Ruby | Video Utils library

I have been reading about the Fourier Transform and its importance. To get a feel by visualizing it myself, I tried a few things.

• Plotted one signal and its Fourier transform
• Plotted 2 signals with a different frequency (but same amplitudes) and its Fourier transform
• Plotted 3 signals with different frequencies (but same amplitudes) and its Fourier transform

Here is what I got.

One signal and its Fourier transform:

Two signals and its Fourier transform:

Three signals and its Fourier transform:

Note: Orange lines represent the Fourier transform and blue lines the signal being processed.

I understand that Fourier transform helps to decompose the signals to give the ingredients of which the main signal is composed. In other words, we get the number of different frequencies of which the main signal is composed of. But I do not understand the above outputs. If we consider the first signal where there is a single signal and its Fourier transform is calculated. Why there are 4 peaks? What does each peak represent? I had assumed the output to be a single peak for a single signal, 2 peaks for 2 signals and 3 peaks for 3 signals.

Here is the code that I used to generate the above images.

I'm trying to demonstrate a concept like Fourier Transform. While searching the web, I encountered an image in Wikipedia:

Is that possible to plot this figure in Python or MATLAB?

So I am trying, like many others before me, to sift through all the celestial alignments of phase shifts and normalization coefficients necessary to make differentiation via Fourier transform work.

I am trying to use as little code as possible, making total use of numpy functionality to keep the code clean while using its array operations, which, I was told, are faster than explicit python loops.

From line 106 of the table of Functional relationships, One-dimensional, in the section about Important Fourier Transforms of the Wikipedia article on Fourier Transforms, I get that the Fourier transform has this property where differentiation in real space is equivalent to multiplication in frequency space such that d^nf/dx^n = ifft[F*(i*w)^n], where F is the fourier transform of f.

Looking for a more solid reference to define the what the w would be in d^nf/dx^n = ifft[F*(i*w)^n], I found a PDF paper on the mathematics of differentiating using DFT. In the paper, the author discusses that there are infinitely many of the so-called trigonometric interpolations for a given function and shows a way to obtain to "less wiggly" solution, which is unique.

So I used his definitions and wrote a small python code to do the differentiations. The code I came up is this one (pay attention to the last two lines, since they will change)

I am using sympy to calculate the Fourier-transform of

I'm studying the IFFT in Matlab by applying it to a Gaussian. According to Wikipedia tables, the Fourier transform pair would be

I have adapted the code in Comparing FFT of Function to Analytical FT Solution in Matlab for this question. I am trying to do FFTs and comparing the result with analytical expressions in the Wikipedia tables.

My code is:

I have an expression in the time domain

We know that the Fourier Transform of a derivative is

where k is the fourier variable. Explanation here

My question is, why doesn't sympy use this knowledge? For example: