I analyzed the sunspots.dat data (below) using fft which is a classic example in this area. I obtained results from fft in real and imaginery parts. Then I tried to use these coefficients (first 20) to recreate the data following the formula for Fourier transform. Thinking real parts correspond to a_n and imaginery to b_n, I have
import numpy as np
from scipy import *
from matplotlib import pyplot as gplt
from scipy import fftpack
def f(Y,x):
total = 0
for i in range(20):
total += Y.real[i]*np.cos(i*x) + Y.imag[i]*np.sin(i*x)
return total
tempdata = np.loadtxt("sunspots.dat")
year=tempdata[:,0]
wolfer=tempdata[:,1]
Y=fft(wolfer)
n=len(Y)
print n
xs = linspace(0, 2*pi,1000)
gplt.plot(xs, [f(Y, x) for x in xs], '.')
gplt.show()
For some reason however, my plot does not mirror the one generated by ifft (I use the same number of coefficients on both sides). What could be wrong ?
Data: