How to center the FFT of sine wave in matplotlib?

Question

I want to plot the FFT of a sine wave using matplotlib and I want to plot a single line at a frequency where the sine wave belongs.

here's my code

import numpy as np
import matplotlib.pyplot as plt


f = 3
t = np.arange(-np.pi, np.pi, 0.01)
y = np.sin(2*np.pi*f*t)


plt.subplot(2, 1, 1)
plt.plot(t, y)


ft = np.fft.fft(y)

plt.subplot(2, 1, 2)
plt.plot(ft.real)

plt.show()

And, here's the output of the above code.

I want to plot the FFT as a single line, and this single line will be straight up at the given sine wave's frequency. Suppose, if I have two sine waves of 20Hz and 40Hz added together like sine(2*pi*20*t) + sine(2*pi*40*t) then their FFT should have two straight lines at 20Hz and 30Hz.

How do I fix my code to get what I wanted? I've seen many questions here on StackOverflow but I did not understand them. Any help would be appreciated.

Answer 1

You have two peaks because it is the right result. The Fourier transform decomposes a signal with a series of complex exponential function rather than (real) sinusoidal function. A single frequency sine wave can be written in the form of $$ \sin(\omega_0t) = \frac{\exp(j\omega_0t)-\exp(-j\omega_0t)}{2j} $$ Hence the two peaks represent the magnitudes at positive frequency $\omega_0$ and negative frequency $-\omega_0$, respectively. If you only need the positive frequency components, just discard the negative results as

ft = np.fft.fft(y)[:nfft//2]

where nfft is the FFT size which is equal to length of y if not specified.