exponential fourier series coefficients square wave example

) are needed to approximate the function this is because of the symmetry of the function.

As before, only odd harmonics (1, 3, 5.There is no discontinuity, so no Gibb's overshoot.Even with only the 1st few harmonics we have a very good approximation to the original function. The vowel signal and the square-wave are both examples that suggest the idea of approximating a periodic signal with a sum of complex exponentials. Conceptually, this occurs because the triangle wave looks much more like the 1st harmonic, so the contributions of the higher harmonics are less. The amplitudes of the harmonics for this example drop off much more rapidly (in this case they go as 1/n 2 (which is faster than the 1/n decay seen in the pulse function Fourier Series (above)).As you add sine waves of increasingly higher frequency, the approximation gets better and better, and these higher frequencies better approximate the details, (i.e., the change in slope) in the original function.Note: this is similar, but not identical, to the triangle wave seen earlier. If x T(t) is a triangle wave with A=1, the values for a n are given in the table below (note: this example was used on the previous page).

During one period (centered around the origin) T = 0.0:5.0e-5:4.0e-3 % it's already given in text (0 to 4ms with interval of 0.The periodic pulse function can be represented in functional form as Π T(t/T p). % Now, we have a row vector " a2" with total values " 81" į1 = a2' % Here, we have final values " f1" (total 81 points) as transpose of a2 computed above S1(n,m) = (4/pi)*(1/(2*n - 1))*sin((2*n - 1)*2*pi*f*dt*(m-1)) % Approximate Fourier Series g(t)Ī1 = s1(:,m) % VERY IMPORTANT ! Here, we are assigning a1 for each single column (total 81)Ī2(m) = sum(a1) % Here, we are summing up the whole column to one single value (adding all 12 values in one column) Tpts = (4.0e-3/5.0e-5) + 1 % Total points " (final point-initial point)/Interval+1%įor n = 1: 12 % Values we are considering to approximate Fourier Seires instead of infinity as given in original function x(t)įor m = 1: tpts % Here, we'll consider all " t" points to cover " from 0 to 4ms interval" % Fourier Series Expansion for Square Waveĭt = 5.0e-05 % Interval between teo time steps Now, we will write a Matlab code for g(t) between 0 and 4ms with an interval of 0.05 ms to demonstrate that g(t) is a decent approximation of original function x(t). Let’s assume we have a square wave with following characteristics:

Exponential fourier series coefficients square wave example