#!/usr/bin/env python
# coding: utf-8

# In[1]:


import numpy as np
from scipy.fftpack import fft, ifft,fftshift,fftfreq
np.set_printoptions(threshold=15)


# ### Cosine function
# 
# $$y(t)=cos(2πAt+φ)=cos(ωt+φ)$$
# 
# And it's Fourier Transform:
# 
# $$F_{transform}\left[y(t)\right](k)=1/2[\delta(f-A)+\delta(f+A)]$$
# 
# where $\delta(x)$ is the delta function.
# 

# In[2]:


# let's generate cosine signal
A = 2
t = np.linspace(0, 1, 100)
mySignal = np.cos(2.*np.pi*A*t) 
print('time', t)
print('signal', mySignal)


# Plot your signal versus time.

# In[3]:


get_ipython().run_line_magic('matplotlib', 'notebook')
import matplotlib.pyplot as plt

fig = plt.figure()
ax = plt.subplot(111)

ax.plot(t, mySignal, "k*-")

plt.xlabel("Time")
plt.ylabel("Magnitude")
plt.show()


# ### Take FFT of the signal
# 
# Pay attention these FFTs, one using correct periodic signal, the other one is not, results has significant difference. 

# In[4]:


# calculate spectrum
myfft1  = fft(mySignal) 
myfft2  = fft(mySignal[0:-1]) 
print('fft1 real', myfft1.real)
print('fft1 imag', myfft1.imag)
print('fft2 real', myfft2.real)
print('fft2 imag', myfft2.imag)


# ### Plot real and imag parts of the results
# Let's plot real and imag. parts of the 2 fft seperately. 

# In[9]:


get_ipython().run_line_magic('matplotlib', 'notebook')
import matplotlib.pyplot as plt

# Real Parts

fig = plt.figure()
ax = plt.subplot(111)

ax.plot(myfft1.real, "*-")
ax.plot(myfft2.real, "o-")

plt.title("Real Parts")
plt.xlabel("Index")
plt.ylabel("Spectrum")
plt.show()

# Imag. Parts

fig = plt.figure()
ax = plt.subplot(111)

ax.plot(myfft1.imag, "*-")
ax.plot(myfft2.imag, "o-")

plt.title("Imaginary Parts")
plt.xlabel("Index")
plt.ylabel("Spectrum")
plt.show()


# ### Conclusion
# **Only myfft2 has the correct results here. Why?** 
# 
# _FFT is a FT, only fully periodic signal can be used, see definition of the FT._
# 

# In[ ]: