spectral stuff

week1

import numpy as np
from numpy.fft import fft, fftshift, ifft, fftfreq

L = 10
n = 128

x2 = np.linspace(-L, L, n + 1)
x = x2[1:]

# k = (2 * np.pi / L) * np.concatenate((np.arange(0, n/2), np.arange(-n/2 + 1, 1)))
k = fftfreq(n) * L

u = np.exp(-x**2)
ut = fft(u)

# plot these on same graph
#plot(k, fftshift(ut))  # whacky
#plot(k, np.abs(fftshift(ut)))
#plot(fftshift(k), np.abs(fftshift(ut)))  # the spectrum?


u2 = sech = 1 / np.cosh(x)
u2d1 = -sech * np.tanh(x)  # sech'
u2d2 = sech - 2 * sech**3   # sech''

u2t = fft(u2)

u2d1s = ifft(1j * k * u2t)  # spectral approximation, 1j is (0+i)
u2d2s = ifft((1j * k)**2 * u2t) 

# for some reason these are waaay out
plot(x, u2d1, 'b', label='"actual" derivative') # "actual" derivative
plot(x, u2d1s, 'g', label='spectral estimate') # spectral approximation







import matplotlib.pyplot as plt
t = np.arange(256)
sp = fft(np.sin(t))
freq = np.fft.fftfreq(t.shape[-1])
plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
plt.show()