Generates numpy/fft_plot.svg, the plot of

2 sin (t) + cos (4 t)

with 25 points and its DFT.

The output was also uploaded to: commons.wikimedia.org/wiki/File:DFT_2sin(t)_%2B_sin(4t).svg and added to en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&oldid=1176616763 only to be later removed of course: Deletionism on Wikipedia.

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ with 25 points
. Source code at: numpy/fft_plot.py.

@cirosantilli/_file/numpy/numpy/fft.py

0 0

Output:

sin(t)
fft
real 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10
rfft
real 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 0 0 0 0 0 0 0

sin(t) + sin(4t)
fft
real 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 -10 0 0 0 0 0 0 0 0 0 0 0 10 0 0 10
rfft
real 0 0 0 0 0 0 0 0 0 0 0
imag 0 -10 0 0 -10 0 0 0 0 0 0

With our understanding of the discrete Fourier transform we see clearly that:

the signal is being decomposed into sinusoidal components
because we are doing the Discrete Fourier transform of a real signal, for the fft, $X_{k} = \overset{ˉ}{X_{N - k}}$ so there is redundancy in the. We also understand that rfft simply cuts off and only keeps half of the coefficients