Plot periodic function with Fourier coefficients

[schniefen]

Homework Statement

Consider the function $p(t)=\sin{(t/\tau)}$ for $0\leq t <2\pi \tau$ and $p(t)=0$ for $2\pi \tau \leq t < T$, which is periodically repeated outside the interval $[0,T)$ with period $T$. Plot this function for $\tau=T/30$ using a restricted set of Fourier coefficients.

Relevant Equations

The complex Fourier series: $\sum_{j=-\infty}^{\infty} a_j e^{i2\pi jt/T}$. Since the $p(t)$ is real-valued, we have $a_{-j}=\overline{a_j}$.

I have plotted the function for $T=15$ and $\tau=T/30$ below with the following code in Python:

import numpy as np
import matplotlib.pyplot as plt

def p(t,T,tau):
    n=np.floor(t/T)
    t=t-n*T
    if t<(2*np.pi*tau):
        p=np.sin(t/tau)
    else:
        p=0
    return p
tdata=np.linspace(-5*np.pi,5*np.pi,500)
pdata=[]
for i in tdata:
  pdata.append(p(i,15,1/2))

plt.plot(tdata,np.array(pdata),label='$T=15$, $\u03C4=1/2$')
plt.legend(loc='lower right')
tick_pos= [-5*np.pi,-4*np.pi,-3*np.pi, -2*np.pi , -np.pi, 0, np.pi , 2*np.pi,3*np.pi,4*np.pi,5*np.pi]
labels = ['$-5\pi$','$-4\pi$','$-3\pi$','$-2\pi$','$-\pi$','0', '$\pi$', '$2\pi$','$3\pi$','$4\pi$','$5\pi$']
plt.xticks(tick_pos, labels)
plt.xlabel('$t$')
plt.ylabel('$p(t)$');

Output:

I would now like to plot an approximation to this function using a restricted set of Fourier coefficients. How can I do that in python?

 

[Baluncore]

Perform an FFT to identify the amplitude and phase of the harmonics present in the signal.

 

[schniefen]

Perform an FFT to identify the amplitude and phase of the harmonics present in the signal.

Yes, I was thinking along those lines, that is, to find the coefficients through FFT. I am very new to this, so I have had little experience with FFT.

 

[Baluncore]

Take a data record that includes an integer number of full cycles, or exactly one full cycle of the waveform.
Avoid a step where the ends of the waveform wrap around.
An FFT works best on 2^n samples, so generate the function with a sampling rate that results in maybe 64, 256 or 1024 samples.
Calibrate your analyser with some known sine and cosine waves.