This question is inspired by the following video:
I own a sign, a drawing of a square with 200 points in total:
import numpy as np import matplotlib.pyplot as plt from scipy.integrate import quad, quad_vec POINTS_EACH_SIDE = 50 x = list(np.linspace(-np.pi, np.pi, POINTS_EACH_SIDE )) + [np.pi] * POINTS_EACH_SIDE + list(np.linspace(np.pi, -np.pi, POINTS_EACH_SIDE )) + [-np.pi] * POINTS_EACH_SIDE y = [np.pi] * POINTS_EACH_SIDE + list(np.linspace(np.pi, -np.pi, POINTS_EACH_SIDE )) + [-np.pi] * POINTS_EACH_SIDE + list(np.linspace(-np.pi, np.pi, POINTS_EACH_SIDE )) t_list = np.linspace(0, 2*np.pi, 200) fig, ax = plt.subplots() ax.set_aspect('equal') ax.plot(x, y, 'b') xlim_data = plt.xlim() ylim_data = plt.ylim() plt.show()
In the Fourier Transform, one of the parameters is the function
The original signal is not a function, so it was necessary to build 2 new graphs, where the
x-axis goes from 0 to 2pi, which is the interval of the integral, and the
y-axis of the two graphs comes one from the
x-axis of the signal, and the other from the
y-axis of the signal
This 13s video is very explanatory:
Here is the code that will generate the 2 graphics mentioned.
x_of_function = np.linspace(0, 2*np.pi, 200) fig2, ax2 = plt.subplots(1,2, figsize=(10, 10)) ax2.set_aspect('equal') ax2.set_aspect('equal') ax2.plot(x_of_function, x) ax2.plot(x_of_function, y)
In this way, it is possible to make the
linear interpolation of each graph.
I will have it at the end of the process 2 interpolations.
def f(t, x_0_2pi, x_of_signal, y_of_signal): y_graphic_1 = np.interp(t, x_0_2pi, np.array(x_of_signal)) y_graphic_2 = np.interp(t, x_0_2pi, np.array(y_of_signal)) resp = y_graphic_1 + 1j* y_graphic_2 #???? return resp
My question is, why can I convert one of the interpolations as the imaginary part and join it with the another interpolation that will be the real part?
The code works, I can make drawings using circles, my question is precisely this, why can I do this?
y_graphic_1 + 1j* y_graphic_2
What is the mathematical basis that allows me to use one of the axes of my sign as an imaginary number?