as small as one likes by including more and more terms in the summa- (2.20) tion term. Thus in the limit, the mean squared error is zero when Equation 2.12 is satisfied. Substituting these coefficients into the general expression for the (2.12) expansion of f(t) in terms of these orthonormal basis functions, we find that the Fourier series for the signal, or function, f(t) can be expressed as in Equations 2.21 and 2.22, which may be written as If this relationship holds, then the infinite sum of the weighted in Equation 2.23, where A, A and B are appropriately defined. basis functions is said to converge in the mean to f(t), an equality known as Parseval's relation, and if it holds for all f(t) of a certain class, then the set 4, (t) is said to be complete for that class of functions. If the basis functions are complex valued functions of a real argument t, orthogonality is defined as in Equation 2.13 where 4: (2.21) is the complex conjugate of 4. and the generalised Fourier coeffi- cients are given by Equation 2.14. (2.13) dt sin (nt) <fl4;> (2.14) (2.22) <4,14 > = 1, the set 4, is called an orthonormal set of basis (2.23) functions. For example, the trigonometric system of functions with period forms an orthogonal set over the region -, and it can be What we have shown is that if a function f(t) can be represented shown that the following forms an orthonormal set over this region: by a Fourier expansion, then the coefficients of the expansion may be calculated by the methods described. However, the question of convergence of the expansion has not yet been addressed. The necessary and sufficient conditions for the convergence of a Fourier expansion are well known, but it is useful to state a sufficient Clearly, other orthonormal sets or trigonometric functions can be condition, known as the Dirichlet condition, which states that if f(t) defined over other intervals, e.g. for the interval (0,), the set is bounded and of period T and if f(t) has at most a finite number of maxima and minima in one period and a fini