Square wave example

Can we express basically any periodic function/signal as a sum of just cosines and sines with different frequencies?
Yes! This notebook will demonstrate that a square wave function of 1 Hz can be constructed as a sum of sine functions in the form below:

\[ x(t) = \sin(2\pi t) + \frac{\sin(3 \cdot 2\pi t)}{3} + \frac{\sin(5 \cdot 2\pi t)}{5} + \dots \frac{\sin(N \cdot 2\pi t)}{N} \]

\(x(t)\) as a function of time t , and with odd number N.

import numpy as np
from matplotlib import pyplot as plt

%config InlineBackend.figure_formats = ['svg']

First, create the square wave function and the first sine term in the above equation

from scipy.signal import square
L = 1
t = np.linspace(- L, L, 200) # 200 time steps from -1 to 1 seconds
ysquare = square(2 * np.pi * t) * np.pi / 4
x1 = np.sin(2 * np.pi* t)
plt.plot(t, ysquare, 'r--', label='square')
plt.plot(t, x1, 'b-', label='sine')
plt.axhline(0, color='k', alpha=0.5)
plt.axvline(0, color='k', alpha=0.5)
plt.legend();

Now, we add the second sine term x2

x2 = 1 / 3 * np.sin(2 * np.pi * 3 * t)
plt.plot(t, ysquare, 'r--', label='square')
plt.plot(t, x1 + x2, 'b-', label='sine')
plt.axhline(0, color='k', alpha=0.5)
plt.axvline(0, color='k', alpha=0.5)
plt.legend();

Then, the third sine term x3. We see that a square wave starts to take shape as more terms are included.

x3 = 1 / 5 * np.sin(2 * np.pi * 5 * t)
plt.plot(t, ysquare, 'r--', label='square')
plt.plot(t, x1 + x2 + x3, 'b-', label='sine')
plt.axhline(0, color='k', alpha=0.5)
plt.axvline(0, color='k', alpha=0.5)
plt.legend();

To observe the effects of adding more sine terms, we define a function to do so.

# Define variables 
x = np.zeros(len(t)) # create x array that is the same length as t 
odds = np.arange(1,1000,2) # array consists of odd numbers from 1,3,5, to 999

# Define a function that takes certain length of odds array as inputs to construct the x series
def transform(number, plot=False, print_value=False):
    """
    number: the length of the odds array that is used for constructing signal x(t) in array x
    """
    for i in range(len(t)):
        term = []
        for j in range(len(odds[0:number])):
            term.append(1 / odds[j] * np.sin(2 * np.pi * odds[j] *t[i]))
        x[i] = np.sum(term)
    if plot==True:
        plt.plot(t,x, label=f'N={odds[number-1]}')
        plt.xlabel('time (sec)')
        plt.ylabel('x[t]')
        plt.legend()
    if print_value==True:
        return x

# Take a look at the x series given by different number of sine terms
# Test number of terms N = 1,9,19,49

plt.figure()
plt.plot(t, ysquare,'r--', label='square')
plt.axhline(0, color='k', alpha=0.5)
plt.axvline(0, color='k', alpha=0.5)
for number in (np.array([1, 5, 9, 25]) + 1) / 2:
    transform(int(number), True, False)
# You can replace any odd number with np.array([1, 5, 9, 25]) to see the effect of using different number of sine terms (N)

Using SymPy!

The calculation of Fourier series is (among others) implemented in SymPy, see documentation. The calculations above are repeated using sympy:

import sympy as sym

x = sym.symbols('x')
N = sym.symbols('N')
f = sym.Heaviside(x+1) - sym.Heaviside(x+1/2)*2+sym.Heaviside(x)*2 - sym.Heaviside(x-1/2)*2
s = sym.fourier_series(f, (x, 0, 1))
s1  = s.truncate(1)
s5  = s.truncate(5)
s9  = s.truncate(9)
s25 = s.truncate(25)
display(s5)

\[\displaystyle \frac{4 \sin{\left(2 \pi x \right)}}{\pi} + \frac{4 \sin{\left(6 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(10 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(14 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(18 \pi x \right)}}{9 \pi}\]

p = sym.plot(f, s1, s5, s9, s25, (x, -1, 1), show=False, legend=True,ylabel='x[t]',xlabel=('time (sec)'))
p[0].line_color = 'red'
p[0].label = 'square'
p[1].label = 'N = 1'
p[2].label = 'N = 5'
p[3].label = 'N = 9'
p[4].label = 'N = 25'
p.show()

Conclusion

We can observe that the created function becomes more like a square wave function as the number of terms (N) increases. So, a square wave function can be simulated by a sum of harmonically related sine functions.

Attribution

This chapter is written by Christiaan Tiberius. Find out more here.