I have a window signal, which i calculate it's Fourier coefficients but then in the output i get a small complex value [3 order of magnitude less in the origin and the same order of magnitude in the edges of my sampling (from -1000 to 1000)] where the output should be purely real, if it was just a floating point approx error it wouldn't have the pattern of sin like i found, so what could cause this? It's all done in Python (through Google Colab) with the libraries cmath, numpy, and matplotlib alone. import numpy as np import cmath import matplotlib.pyplot as plt D=1000 a=np.zeros(2*D+1) for i in range(-100,100): a[i+D] = 1 This code creates the following window signal: and then to get the fourier coefficients: j=complex(0,1) pi=np.pi N=2001 ak = np.zeros(N, dtype=complex) for k in range(-D, D + 1): for n in range(-D, D + 1): ak[k + D] += (1/N) * a[n + D] * cmath.exp(-j * 2 * pi * k * n / N) which gives me the following image: when i do the math in paper i get the coeficients to be: where omega0 is 2*pi/N and a[n]=u(n+100)-u(n-100) (where u(n) is the Heaveside function). but it's weird as when i plot the imaginary part of ak i get that it's (at a very good approximation) equal to 1/N *sin(201*pi/N where N=2001 and is my entire sampling area (from -1000 to 1000 as i mentioned), which if it was just a rounding error because of floating point it wouldn't be in this form. here it's the plot of the imaginary part of ak, not freq of ak or bk. Continue reading...
