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Symbolic evaluation of Fourier coefficients

from sympy import cos, integrate, pi, sin, sqrt, symbols
from sympy.plotting import plot
# init_printing(pretty_print=True,use_latex=True)
# %matplotlib inline
# import matplotlib.pyplot as plt
from numpy import arange

f,t,T,G = symbols('f t T G')

f_1 = G * t - G / 2
print('f =', f_1)

f = G*t - G/2

p = plot((f_1.subs(G, 25), (t, 0, 1)), ((f_1 - 25).subs(G, 25), (t, 1, 2)), ((f_1 - 50).subs(G, 25), (t, 2, 3)), ylabel='$f = 25 t$ [V]', xlabel='$t$ [sec]', xlim=(0, 3), figsize=(10, 8))

c0 = integrate(f_1, (t, 0, T)) * (1 / T)
print(c0)
print(c0.subs([(G, 25), (T, 1)]))

(G*T**2/2 - G*T/2)/T
0

a1 = 2 / T * integrate(f_1 * sin(2 * pi * 1 * t), (t, 0, T))

print(a1)
print( a1.subs([(T,1),(pi, 3.14),(G,25)]))

2*(-G*T*cos(2*pi*T)/(2*pi) + G*sin(2*pi*T)/(4*pi**2) + G*cos(2*pi*T)/(4*pi) - G/(4*pi))/T
-7.96178343949045

a, b = ([], [])
for _n in range(10):
    a.append(2 / T * integrate(f_1 * sin(_n * pi * t), (t, 0, T)))
    b.append(2 / T * integrate(f_1 * cos(_n * pi * t), (t, 0, T)))

a[9], b[9]
  • 0: ||[ \frac{2 \left(- \frac{G T \cos{\left(9 \pi T \right)}}{9 \pi} + \frac{G \sin{\left(9 \pi T \right)}}{81 \pi^{2}} + \frac{G \cos{\left(9 \pi T \right)}}{18 \pi} - \frac{G}{18 \pi}\right)}{T} ||]
  • 1: ||[ \frac{2 \left(\frac{G T \sin{\left(9 \pi T \right)}}{9 \pi} - \frac{G \sin{\left(9 \pi T \right)}}{18 \pi} + \frac{G \cos{\left(9 \pi T \right)}}{81 \pi^{2}} - \frac{G}{81 \pi^{2}}\right)}{T} ||]
c = []
for _n in range(1, 10):
    print(a[_n])
    expr = sqrt(a[_n] ** 2 + b[_n] ** 2)
    expr = expr.subs([(T, 1), (pi, 3.14), (G, 25)])
    c.append(expr)  # print expr  # val = lambdify(t,expr,'numpy')

2*(-G*T*cos(pi*T)/pi + G*sin(pi*T)/pi**2 + G*cos(pi*T)/(2*pi) - G/(2*pi))/T
2*(-G*T*cos(2*pi*T)/(2*pi) + G*sin(2*pi*T)/(4*pi**2) + G*cos(2*pi*T)/(4*pi) - G/(4*pi))/T
2*(-G*T*cos(3*pi*T)/(3*pi) + G*sin(3*pi*T)/(9*pi**2) + G*cos(3*pi*T)/(6*pi) - G/(6*pi))/T
2*(-G*T*cos(4*pi*T)/(4*pi) + G*sin(4*pi*T)/(16*pi**2) + G*cos(4*pi*T)/(8*pi) - G/(8*pi))/T
2*(-G*T*cos(5*pi*T)/(5*pi) + G*sin(5*pi*T)/(25*pi**2) + G*cos(5*pi*T)/(10*pi) - G/(10*pi))/T
2*(-G*T*cos(6*pi*T)/(6*pi) + G*sin(6*pi*T)/(36*pi**2) + G*cos(6*pi*T)/(12*pi) - G/(12*pi))/T
2*(-G*T*cos(7*pi*T)/(7*pi) + G*sin(7*pi*T)/(49*pi**2) + G*cos(7*pi*T)/(14*pi) - G/(14*pi))/T
2*(-G*T*cos(8*pi*T)/(8*pi) + G*sin(8*pi*T)/(64*pi**2) + G*cos(8*pi*T)/(16*pi) - G/(16*pi))/T
2*(-G*T*cos(9*pi*T)/(9*pi) + G*sin(9*pi*T)/(81*pi**2) + G*cos(9*pi*T)/(18*pi) - G/(18*pi))/T

import matplotlib.pyplot as plt
fig,ax = plt.subplots(figsize=(10,8))
ax.bar(-.5+arange(1,10),c,alpha=.5)
ax.set_xticks([1,2,3,4,5,6,7,8,9,10])
ax.set_xlabel('$n$',fontsize=16)
ax.set_ylabel('Fourier coefficients $a_n^2 + b_n^2$',fontsize=16)