import numpy as np
import matplotlib.pyplot as plt

#%%
L = 10000  # Length of Signals
F = 100    # Frame location
N = 16     # Frame Size = DFT Sample Count
#%%
P = np.zeros(L)  # Position
D = np.zeros(L)  # Difference

for n in range(L):
    P[n] = 1.2 + 0.3 * n + 0.045 * n * n
P[F - 1] = 42069

for n in range(1, L):
    D[n] = P[n] - P[n-1]
#%%
x = P[F:F+N]
y = D[F:F+N]

X = np.fft.fft(x) / N
Y = np.fft.fft(y) / N
#%%
Z = np.zeros(N, dtype=complex)
C = (x[N-1] - P[F-1]) / N

for k in range(N):
    Z[k] = X[k] * (1 - np.exp(-1j * (2.0 * np.pi / N) * k)) + C
#   coef = coef * (1 - np.exp(-1j * 2 * np.pi / N * np.arange(N)))
#%%
for n in range( N ):
    print("%2d %10.6f %10.6f   %10.6f %10.6f" % \
          (n, Y[n].real, Y[n].imag, Z[n].real, Z[n].imag ))
print(np.sum(np.abs(Y - Z)))  # just use this
#%%
fig, axes = plt.subplots(1, 2, figsize=(14, 6))
axes[0].plot(x)
axes[1].plot(y)
axes[0].annotate("x[n]",  fontsize=20, xy=(.85, .05), xycoords='axes fraction')
axes[1].annotate("x'[n]", fontsize=20, xy=(.85, .05), xycoords='axes fraction')