如何用python寫傅里葉級數

最近需要用到傅里葉級數來擬合曲線，了解到有三種庫的寫法

• 使用scipy
• 使用sympy
• 使用lmfit

首先是標准的傅里葉級數的數學表達式

$周期為2l的傅式展開\\\\ f(x)\sim S(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}{(a_{n}\cos\frac{n\pi x}{l}+b_{n}\sin\frac{n\pi x}{l})}\\\\ 其中\\\\ \begin{cases} a_{0}=\frac{1}{l}\int_{-l}^{l}f(x)\ {\tt d}x\\\\ a_{n}=\frac{1}{l}\int_{-l}^{l}f(x)\cos\frac{n\pi x}{l} \ {\tt d} x\quad(n=0,1,\dots)\\\\ b_{n}=\frac{1}{l}\int_{-l}^{l}f(x)\sin\frac{n\pi x}{l} \ {\tt d} x\quad(n=0,1,\dots) \end{cases}$

可以化簡為

$f(x) = a_0+\sum_{n=1}^{\infty}(a_n\cos nwx + b_n\sin nwx)$

其中參數 $a_0$、 $w$、 $a_n$、 $b_n$為待求參數，關於傅里葉函數的更多介紹，可以上網搜一搜，網上有很多講解的文章，這里就不細說了。

1. 使用Scipy庫

其中用到的函數是scipy.optimize.curve_fit

From:StackOverflow

使用變長參數和for循環定義傅里葉函數，tau可以自定義，也可以從參數中獲取。

tau = 0.045
def fourier(x, *a):
    ret = a[0] * np.cos(np.pi / tau * x)
    for deg in range(1, len(a)):
        ret += a[deg] * np.cos((deg+1) * np.pi / tau * x)
    return ret

提供8個變量

popt, pcov = curve_fit(fourier, z, Ua, [1.0] * 8)

現在可以使用該函數擬合了。

# Fit with 15 harmonics
popt, pcov = curve_fit(fourier, z, Ua, [1.0] * 15)

# Plot data, 15 harmonics, and first 3 harmonics
fig = plt.figure()
ax1 = fig.add_subplot(111)
p1, = plt.plot(z,Ua)
p2, = plt.plot(z, fourier(z, *popt))
p3, = plt.plot(z, fourier(z, popt[0], popt[1], popt[2]))
plt.show()

2. 使用sympy庫

From:StackOverflow
這位仁兄推薦用sympy庫寫

from symfit import parameters, variables, sin, cos, Fit
import numpy as np
import matplotlib.pyplot as plt

def fourier_series(x, f, n=0):
    """ Returns a symbolic fourier series of order `n`. :param n: Order of the fourier series. :param x: Independent variable :param f: Frequency of the fourier series """
    # Make the parameter objects for all the terms
    a0, *cos_a = parameters(','.join(['a{}'.format(i) for i in range(0, n + 1)]))
    sin_b = parameters(','.join(['b{}'.format(i) for i in range(1, n + 1)]))
    # Construct the series
    series = a0 + sum(ai * cos(i * f * x) + bi * sin(i * f * x)
                     for i, (ai, bi) in enumerate(zip(cos_a, sin_b), start=1))
    return series

x, y = variables('x, y')
w, = parameters('w')
model_dict = {y: fourier_series(x, f=w, n=3)}
print(model_dict)

輸出的模型為：

{y: a0 + a1*cos(w*x) + a2*cos(2*w*x) + a3*cos(3*w*x) + b1*sin(w*x) + b2*sin(2*w*x) + b3*sin(3*w*x)}

下面是簡單的使用方法

# Make step function data
xdata = np.linspace(-np.pi, np.pi)
ydata = np.zeros_like(xdata)
ydata[xdata > 0] = 1
# Define a Fit object for this model and data
fit = Fit(model_dict, x=xdata, y=ydata)
fit_result = fit.execute()
print(fit_result)

# Plot the result
plt.plot(xdata, ydata)
plt.plot(xdata, fit.model(x=xdata, **fit_result.params).y, color='green', ls=':')

輸出為：

Parameter Value        Standard Deviation
a0        5.000000e-01 2.075395e-02
a1        -4.903805e-12 3.277426e-02
a2        5.325068e-12 3.197889e-02
a3        -4.857033e-12 3.080979e-02
b1        6.267589e-01 2.546980e-02
b2        1.986491e-02 2.637273e-02
b3        1.846406e-01 2.725019e-02
w         8.671471e-01 3.132108e-02
Fitting status message: Optimization terminated successfully.
Number of iterations:   44
Regression Coefficient: 0.9401712713086535

圖像為：

3. 使用lmfit庫

From:StackOverflow

import numpy as np
from lmfit import Minimizer, Parameters, report_fit

# create data to be fitted
x = np.linspace(0, 15, 301)
data = (5. * np.sin(2*x - 0.1) * np.exp(-x*x*0.025) +
        np.random.normal(size=len(x), scale=0.2))

# define objective function: returns the array to be minimized
def fcn2min(params, x, data):
    """Model a decaying sine wave and subtract data."""
    amp = params['amp']
    shift = params['shift']
    omega = params['omega']
    decay = params['decay']

    return model(x,amp,shift,omega,decay) - data
def model(x,amp,shift,omega,decay):
    return amp * np.sin(x*omega + shift) * np.exp(-x*x*decay)

# create a set of Parameters
params = Parameters()
params.add('amp', value=10, min=0)
params.add('decay', value=0.1)
params.add('shift', value=0.0, min=-np.pi/2., max=np.pi/2,)
params.add('omega', value=3.0)

# do fit, here with leastsq model
minner = Minimizer(fcn2min, params, fcn_args=(x, data))
result = minner.minimize()
result.params

Parameters([('amp',
             <Parameter 'amp', value=5.04765633375743 +/- 0.0375, bounds=[0:inf]>),
            ('decay',
             <Parameter 'decay', value=0.025679131083170995 +/- 0.000435, bounds=[-inf:inf]>),
            ('shift',
             <Parameter 'shift', value=-0.11625028574097462 +/- 0.00949, bounds=[-1.5707963267948966:1.5707963267948966]>),
            ('omega',
             <Parameter 'omega', value=2.0042576410770288 +/- 0.00308, bounds=[-inf:inf]>)])

# calculate final result
final = data + result.residual
p = result.params
final = model(x,p['amp'],p['shift'],p['omega'],p['decay'])

# write error report
report_fit(result)

[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 63
    # data points      = 301
    # variables        = 4
    chi-square         = 10.4860614
    reduced chi-square = 0.03530660
    Akaike info crit   = -1002.47608
    Bayesian info crit = -987.647634
[[Variables]]
    amp:    5.04765633 +/- 0.03747673 (0.74%) (init = 10)
    decay:  0.02567913 +/- 4.3514e-04 (1.69%) (init = 0.1)
    shift: -0.11625029 +/- 0.00949363 (8.17%) (init = 0)
    omega:  2.00425764 +/- 0.00307570 (0.15%) (init = 3)
[[Correlations]] (unreported correlations are < 0.100)
    C(shift, omega) = -0.784
    C(amp, decay)   =  0.585
    C(amp, shift)   = -0.118

# try to plot results
try:
    import matplotlib.pyplot as plt
    plt.plot(x, data, 'k+')
    plt.plot(x, final, 'r')
    plt.show()
except ImportError:
    pass