Fourier series question, in regards to integration and plotting finite sums

f(t) = 1 : 0 <t<pi/2 and 0: -pi/2 <t <0, f(t+pi) = f(t)

[Mathieu NOE]

and what is the question ?

[Thomas]

How I would use matlab to generate the A_n and B_n integrals and then plot the finite sums.

Answer 1

[Sam Chak]

Hi @Thomas

I also struggled with calculating the integrations (by hand) while learning Fourier series many years ago. A friend taught me this 'dirty and fast trick' to find the coefficients of the Fourier series using MATLAB. Please note that linear and constant functions, as well as discontinuities, cannot be perfectly expressed by the Fourier series in finite terms. The example below includes up to eight terms of both sines and cosines.

However, if you wish to calculate the Fourier integrals 'fair and square', please provide the formulas for the coefficients , , , and the angular frequency ω.

%% Set heaviside(0) to 1, because OP didn't specify whether f(0) is 1 or 0.
sympref('HeavisideAtOrigin', 1);

%% the signal
t   = linspace(-pi, pi, 3601);
f   = heaviside(sin(2*t));
plot(t/pi, f, 'linewidth', 1.5), grid on, ylim([-0.5, 1.5])
xlabel('t/\pi'), ylabel('f(t)'), title('The signal, f(t)')

●

%% the fourier series
[fc, gof] = fit(t', f', "fourier8")

fc = 
     General model Fourier8:
     fc(x) = 
               a0 + a1*cos(x*w) + b1*sin(x*w) + 
               a2*cos(2*x*w) + b2*sin(2*x*w) + a3*cos(3*x*w) + b3*sin(3*x*w) + 
               a4*cos(4*x*w) + b4*sin(4*x*w) + a5*cos(5*x*w) + b5*sin(5*x*w) + 
               a6*cos(6*x*w) + b6*sin(6*x*w) + a7*cos(7*x*w) + b7*sin(7*x*w) + 
               a8*cos(8*x*w) + b8*sin(8*x*w)
     Coefficients (with 95% confidence bounds):
       a0 =      0.5002  (0.4967, 0.5037)
       a1 =   0.0003409  (-0.004617, 0.005299)
       b1 =      0.6361  (0.6312, 0.641)
       a2 =   0.0003408  (-0.004617, 0.005299)
       b2 =  -0.0002442  (-0.005136, 0.004648)
       a3 =   0.0003405  (-0.004617, 0.005298)
       b3 =      0.2106  (0.2057, 0.2155)
       a4 =   0.0003401  (-0.004616, 0.005297)
       b4 =  -0.0004827  (-0.005389, 0.004423)
       a5 =   0.0003395  (-0.004616, 0.005295)
       b5 =      0.1247  (0.1197, 0.1296)
       a6 =   0.0003389  (-0.004615, 0.005293)
       b6 =  -0.0007099  (-0.005651, 0.004231)
       a7 =   0.0003381  (-0.004614, 0.005291)
       b7 =     0.08723  (0.08225, 0.09221)
       a8 =   0.0003372  (-0.004614, 0.005288)
       b8 =  -0.0009204  (-0.006035, 0.004194)
       w =       1.976  (1.973, 1.978)

gof = struct with fields:
           sse: 40.5713
       rsquare: 0.9549
           dfe: 3583
    adjrsquare: 0.9547
          rmse: 0.1064

plot(fc, t, f), grid on, axis([-pi,  pi, -0.5, 1.5])
xlabel('t'), ylabel('f(t)'), title('Eight-term Fourier Series Model')