Fourier Series of Triangular Wave.

Cuthbert Nyack
The triangular wave is shown opposite. Because of the discontinuities in the derivative, then the function has to be expressed as 3 pieces. These are:-
f(t) = - 4t/T - 2 for -T/2 £ t £ -T/4
f(t) = 4t/T for -T/4 £ t £ T/4
f(t) = - 4t/T + 2 for T/4 £ t £ T/2

Since the Function id odd an = 0 for all n and only the coefficients bn occur in the Fourier Series for f(t). Expression to evaluate bn is shown opposite.

Substituting the Expressions for f(t) into that for bn gives the result Þ

The Expression for bn can be simplified by replacing t with - t in the first integral. This gives Þ

The first 2 integrals contain the term tsinnwt and may be integrated by parts using the result Þ

Evaluating the integrals and using wT = 2p result in the expression opposite for bn.

The triangle wave can now be represented by the Fourier Series Þ.

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COPYRIGHT © 1996 Cuthbert Nyack.