How to calculate a Fourier series in Numpy?

Question

I have a periodic function of period T and would like to know how to obtain the list of the Fourier coefficients. I tried using fft module from numpy but it seems more dedic

Answer 1

This is an old question, but since I had to code this, I am posting here the solution that uses the numpy.fft module, that is likely faster than other hand-crafted solutions.

The DFT is the right tool for the job of calculating up to numerical precision the coefficients of the Fourier series of a function, defined as an analytic expression of the argument or as a numerical interpolating function over some discrete points.

This is the implementation, which allows to calculate the real-valued coefficients of the Fourier series, or the complex valued coefficients, by passing an appropriate return_complex:

def fourier_series_coeff_numpy(f, T, N, return_complex=False):
    """Calculates the first 2*N+1 Fourier series coeff. of a periodic function.

    Given a periodic, function f(t) with period T, this function returns the
    coefficients a0, {a1,a2,...},{b1,b2,...} such that:

    f(t) ~= a0/2+ sum_{k=1}^{N} ( a_k*cos(2*pi*k*t/T) + b_k*sin(2*pi*k*t/T) )

    If return_complex is set to True, it returns instead the coefficients
    {c0,c1,c2,...}
    such that:

    f(t) ~= sum_{k=-N}^{N} c_k * exp(i*2*pi*k*t/T)

    where we define c_{-n} = complex_conjugate(c_{n})

    Refer to wikipedia for the relation between the real-valued and complex
    valued coeffs at http://en.wikipedia.org/wiki/Fourier_series.

    Parameters
    ----------
    f : the periodic function, a callable like f(t)
    T : the period of the function f, so that f(0)==f(T)
    N_max : the function will return the first N_max + 1 Fourier coeff.

    Returns
    -------
    if return_complex == False, the function returns:

    a0 : float
    a,b : numpy float arrays describing respectively the cosine and sine coeff.

    if return_complex == True, the function returns:

    c : numpy 1-dimensional complex-valued array of size N+1

    """
    # From Shanon theoreom we must use a sampling freq. larger than the maximum
    # frequency you want to catch in the signal.
    f_sample = 2 * N
    # we also need to use an integer sampling frequency, or the
    # points will not be equispaced between 0 and 1. We then add +2 to f_sample
    t, dt = np.linspace(0, T, f_sample + 2, endpoint=False, retstep=True)

    y = np.fft.rfft(f(t)) / t.size

    if return_complex:
        return y
    else:
        y *= 2
        return y[0].real, y[1:-1].real, -y[1:-1].imag

This is an example of usage:

from numpy import ones_like, cos, pi, sin, allclose
T = 1.5  # any real number

def f(t):
    """example of periodic function in [0,T]"""
    n1, n2, n3 = 1., 4., 7.  # in Hz, or nondimensional for the matter.
    a0, a1, b4, a7 = 4., 2., -1., -3
    return a0 / 2 * ones_like(t) + a1 * cos(2 * pi * n1 * t / T) + b4 * sin(
        2 * pi * n2 * t / T) + a7 * cos(2 * pi * n3 * t / T)


N_chosen = 10
a0, a, b = fourier_series_coeff_numpy(f, T, N_chosen)

# we have as expected that
assert allclose(a0, 4)
assert allclose(a, [2, 0, 0, 0, 0, 0, -3, 0, 0, 0])
assert allclose(b, [0, 0, 0, -1, 0, 0, 0, 0, 0, 0])

And the plot of the resulting a0,a1,...,a10,b1,b2,...,b10 coefficients:

This is an optional test for the function, for both modes of operation. You should run this after the example, or define a periodic function f and a period T before running the code.

# #### test that it works with real coefficients:
from numpy import linspace, allclose, cos, sin, ones_like, exp, pi, \
    complex64, zeros


def series_real_coeff(a0, a, b, t, T):
    """calculates the Fourier series with period T at times t,
       from the real coeff. a0,a,b"""
    tmp = ones_like(t) * a0 / 2.
    for k, (ak, bk) in enumerate(zip(a, b)):
        tmp += ak * cos(2 * pi * (k + 1) * t / T) + bk * sin(
            2 * pi * (k + 1) * t / T)
    return tmp


t = linspace(0, T, 100)
f_values = f(t)
a0, a, b = fourier_series_coeff_numpy(f, T, 52)
# construct the series:
f_series_values = series_real_coeff(a0, a, b, t, T)
# check that the series and the original function match to numerical precision:
assert allclose(f_series_values, f_values, atol=1e-6)

# #### test similarly that it works with complex coefficients:

def series_complex_coeff(c, t, T):
    """calculates the Fourier series with period T at times t,
       from the complex coeff. c"""
    tmp = zeros((t.size), dtype=complex64)
    for k, ck in enumerate(c):
        # sum from 0 to +N
        tmp += ck * exp(2j * pi * k * t / T)
        # sum from -N to -1
        if k != 0:
            tmp += ck.conjugate() * exp(-2j * pi * k * t / T)
    return tmp.real

f_values = f(t)
c = fourier_series_coeff_numpy(f, T, 7, return_complex=True)
f_series_values = series_complex_coeff(c, t, T)
assert allclose(f_series_values, f_values, atol=1e-6)