``````

In [2]:

import numpy as np
import matplotlib.pyplot as plt

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``````

In [48]:

#funções

x = np.array(x, dtype=np.float)
len_x = len(x)
N = 1
if (len_x % 2 != 0):
while N < len_x:
N *= 2
x_new = np.zeros((N, 1))
for i in range(0, len_x):
x_new[i] = x[i]
x = x_new
return x

#transformação discreta de Fourier no modo convencional
def DFT(x):
N = x.shape[0] #quantidade de elementos
n = np.arange(N)
k = n.reshape((N, 1)) #transposição de n
M = np.exp(-2j * np.pi * k * n / N)
return np.dot(M, x)

#def FFT(x):
#fonte https://jakevdp.github.io/blog/2013/08/28/understanding-the-fft/
def FFT(x):
"""A vectorized, non-recursive version of the Cooley-Tukey FFT"""
x = np.asarray(x, dtype=float)
N = x.shape[0]

if np.log2(N) % 1 > 0:
N = x.shape[0]

# N_min here is equivalent to the stopping condition above,
# and should be a power of 2
N_min = min(N, 32)

# Perform an O[N^2] DFT on all length-N_min sub-problems at once
n = np.arange(N_min)
k = n[:, None]
M = np.exp(-2j * np.pi * n * k / N_min)
X = np.dot(M, x.reshape((N_min, -1)))

# build-up each level of the recursive calculation all at once
while X.shape[0] < N:
X_even = X[:, :X.shape[1] / 2]
X_odd = X[:, X.shape[1] / 2:]
factor = np.exp(-1j * np.pi * np.arange(X.shape[0])
/ X.shape[0])[:, None]
X = np.vstack([X_even + factor * X_odd,
X_even - factor * X_odd])

return X.ravel()

#calcular as séries de fourier
def serieFourier(A_n, n, x):
len_x = len(x)
L = (x[len_x - 1] - x[0]) / 4
y = np.zeros((len_x, 1))
for i in range(0, len_x):
y[i] = np.sum(A_n * np.sin(n * x[i] * np.pi / L))
return [x, y]

def transfFourier(x, y, fft=1):
dx = x[1] - x[0]
len_x = len(x)
p = np.arange(0, len_x)
len_p = len(p)
k_p = 2 * np.pi * p / (dx * len_x)
Y = np.zeros((len_p, 1), dtype=np.complex_)
if fft == 0:
Y = FFT(y)
elif fft == 1:
Y = DFT(y)
elif fft == 2:
Y = np.fft.rfft(y)
return [k_p, Y]

def plotSerieFourier(x, y):
plt.plot(x, y, '-+')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Séries de Fourier')
plt.show()

def plotTransfFourier(k_p, Y):
Ysq = np.square(np.abs(Y))
plt.plot(k_p, Ysq, '-+')
plt.xlabel('\$k_p\$')
plt.ylabel('\$|Y(k_p)^2|\$')
plt.show()

``````
``````

In [49]:

#condições iniciais
x0 = 0
xf = 2.00
dx = 0.01
n = np.array([1])
A_n = np.array([1.0])

#---
L = xf - x0
x = np.arange(x0, xf+dx, dx)
len_x = len(x)
ser_fourier = serieFourier(A_n, n, x)
trans_fourier = transfFourier(*ser_fourier, fft=1)

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In [50]:

(x, y) = ser_fourier

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In [51]:

(k_p, Y) = trans_fourier

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``````

In [52]:

plt.plot(x, y)
plt.show()

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``````

In [53]:

plt.plot(k_p, Y)
plt.xlim(0, 200)
plt.show()

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``````

/usr/lib/python3/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
return array(a, dtype, copy=False, order=order)

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``````

In [54]:

plt.plot(k_p, np.fft.fft(y))
plt.xlim(0, 200)
plt.show()

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``````

/usr/lib/python3/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
return array(a, dtype, copy=False, order=order)

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``````

In [57]:

x = np.linspace(0, 2*np.pi, 1e2)
y = np.sin(x)

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``````

In [ ]:

plt.plot(np.fft.fft(y))
plt.xlim(0, 200)
plt.show()

``````
``````

/usr/lib/python3/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
return array(a, dtype, copy=False, order=order)

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``````

In [ ]:

``````