# Fourier Series of Square Wave.

Cuthbert Nyack
 A Square wave is shown opposite. Because of the discontinuities, f(t) cannot be expressed as a single function and 3 pieces must be used. These are:- f(t) = -1 for -T/2 < t < -T/4 f(t) = + 1 for -T/4 £ t £ T/4 f(t) = - 1 for T/4 < t < T/2

 Since the square wave is an even function, then bn = 0 for all n and only an as given in the equation opposite needs to be calculated.

 Because of the symmetries in the square, it is only necessary to integrate from 0 to T/4. However here the long approach is used. Substituting the expression for f(t) into that for an produces the equation opposite.

 Carrying out the integrations { òcos nwt = (1/nw) sin nwt } give the expression shown opposite.

 Inserting the limits into the result of the integration produces:-

 Using the odd property of the sin (sin(-x) = - sin(x)) and wT = 2p then the resulting expression for an is:-

 The Equation for the Square wave when expressed as a Fourier Series is given opposite:-