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16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow How to calculate a Fourier series in Numpy? Asked 10 years ago 26 Active 4 months ago Viewed 52k times 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 dedicated to Fourier transforms than series. Maybe it a lack of mathematical knowledge, but I can't see how to calculate the Fourier coefficients from fft. Help and/or examples appreciated. 13 python numpy fft edited May 12 '13 at 21:43 1 Saullo G. P. Castro Mermoz 46.9k 12.1k 21 152 216 15 52 80 numpy does not provide options for Fourier series, whereas Sympy does (sympy fouries documentation). Here are some examples where I used sympy to work out some problems for Krezig Advanced engineering mathematics! – Pradeep Padmanaban C May 20 '18 at 6:41 5 Answers 24 asked Nov 23 '10 at 16:10 Active Oldest Votes In the end, the most simple thing (calculating the coefficient with a riemann sum) was the most portable/efficient/robust way to solve my problem: import numpy as np def cn(n): c = y*np.exp(-1j*2*n*np.pi*time/period) return c.sum()/c.size def f(x, Nh): f = np.array([2*cn(i)*np.exp(1j*2*i*np.pi*x/period) for i in range(1,Nh+1)]) return f.sum() y2 = np.array([f(t,50).real for t in time]) plot(time, y) plot(time, y2) By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 1/7 16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow gives me: edited Jul 22 at 22:34 answered Nov 24 '10 at 11:59 zabop 3,118 Mermoz 2 12 42 12.1k 15 52 80 Thanks for posting this solution. It saved me some time :) – zega Jun 28 '13 at 15:44 Thank you. Exactly what I wanted – Thoth19 Dec 29 '14 at 18:15 4 15 Any chance you could include a commented minimum working example? – SeSodesa Jun 22 '18 at 15:23 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 theacknowledge complex valued coefficients, by understand passing anour appropriate return_complex : , and By usingseries, our site,oryou that you have read and Cookie Policy , Privacy Policy our Terms of Service. https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 2/7 16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow 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,you n2,acknowledge n3 = 1., 4.that , 7.you # have in Hz, or and nondimensional forCookie the matter. By using our site, read understand our Policy, Privacy Policy, and a0, a1, b4, a7 = 4., 2., -1., -3 our Terms of Service. return a0 / 2 * ones_like(t) + a1 * cos(2 * pi * n1 * t / T) + b4 * sin( https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 3/7 16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow 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( By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and 2 * pi * (k + 1) * t / T) our Terms of Service . tmp return https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 4/7 16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow 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) edited May 12 '16 at 18:00 answered Dec 31 '14 at 11:12 gg349 18.1k 11 2 43 56 Numpy isn't the right tool really to calculate fourier series components, as your data has to be discretely sampled. You really want to use something like Mathematica or should be using fourier transforms. To roughly do it, let's look at something simple a triangle wave of period 2pi, where we can easily calculate the Fourier coefficients (c_n = -i ((-1)^(n+1))/n for n>0; e.g., c_n = { -i, i/2, -i/3, i/4, -i/5, i/6, ... } for n=1,2,3,4,5,6 (using Sum( c_n exp(i 2 pi n x) ) as Fourier series). import numpy x = numpy.arange(0,2*numpy.pi, numpy.pi/1000) y = (x+numpy.pi/2) % numpy.pi - numpy.pi/2 fourier_trans = numpy.fft.rfft(y)/1000 If you look at the first several Fourier components: array([ -3.14159265e-03 +0.00000000e+00j, 2.54994550e-16 -1.49956612e-16j, 3.14159265e-03 -9.99996710e-01j, 1.28143395e-16 +2.05163971e-16j, By using our site, you acknowledge that you have read ,and understand our Cookie Policy, Privacy Policy, and -3.14159265e-03 +4.99993420e-01j 5.28320925e-17 -2.74568926e-17j , our Terms of Service. 3.14159265e-03 -3.33323464e-01j, https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 5/7 16/12/2020 python - How to calculate a Fourier series in Numpy? - Stack Overflow 7.73558750e-17 -3.14159265e-03 1.73758496e-16 3.14159265e-03 -1.74044469e-16 -3.14159265e-03 -1.02291982e-16 3.14159265e-03 1.96729377e-17 -3.14159265e-03 -7.50516717e-17 3.14159265e-03 -1.27900121e-16 -3.14159265e-03 -3.41761974e-16j, +2.49986840e-01j, +1.55882418e-17j, -1.99983550e-01j, -1.22437710e-17j, +1.66646927e-01j, -2.05092972e-16j, -1.42834113e-01j, +5.35550532e-17j, +1.24973680e-01j, +3.33475329e-17j, -1.11081501e-01j, -3.32193126e-17j, +9.99670992e-02j, First neglect the components that are near 0 due to floating point accuracy (~1e-16, as being zero). The more difficult part is to see that the 3.14159 numbers (that arose before we divide by the period of a 1000) should also be recognized as zero, as the function is periodic). So if we neglect those two factors we get: fourier_trans = [0,0,-i,0,i/2,0,-i/3,0,i/4,0,-i/5,0,-i/6, ... and you can see the fourier series numbers come up as every other number (I haven't investigated; but I believe the components correspond to [c0, c-1, c1, c-2, c2, ... ]). I'm using conventions according to wiki: http://en.wikipedia.org/wiki/Fourier_series. Again, I'd suggest using mathematica or a computer algebra system capable of integrating and dealing with continuous functions. edited Nov 23 '10 at 17:59 answered Nov 23 '10 at 17:12 dr jimbob 15.2k 1 5 4 52 73 Excellent, excellent point about having to put some effort into understanding the result. +1. – mtrw Nov 23 '10 at 17:29 As other answers have mentioned, it seems that what you are looking for is a symbolic computing package, so numpy isn't suitable. If you wish to use a free python-based solution, then either sympy or sage should meet your needs. answered Nov 24 '10 at 9:40 DaveP 6,094 1 21 37 here is the reference for fourier series using sympy: docs.sympy.org/dev/modules/mpmath/calculus/…. It requires mpmath which is not even in my sympy distribution. Although a good hint, i won't choose this the sake of portability of theread code. – understand Mermoz Nov '10 at 10:35 By using our solution site, youfor acknowledge that you have and our24Cookie Policy, Privacy Policy, and 2 our Terms of Service. https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 6/7 16/12/2020 4 python - How to calculate a Fourier series in Numpy? - Stack Overflow Do you have a list of discrete samples of your function, or is your function itself discrete? If so, the Discrete Fourier Transform, calculated using an FFT algorithm, provides the Fourier coefficients directly (see here). On the other hand, if you have an analytic expression for the function, you probably need a symbolic math solver of some kind. answered Nov 23 '10 at 16:49 mtrw 28.6k 7 54 68 By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. https://stackoverflow.com/questions/4258106/how-to-calculate-a-fourier-series-in-numpy 7/7