te number of disconti- nuities, then the Fourier series for f(t) converges to f(t) at all points where f(t) is continuous, and converges to the average of the right hand and left hand limits of f(t) at each point where f(t) is discon- is orthonormal. tinuous. If 4, is an orthonormal set, the generalised Fourier coefficients for In many applications, such as the design of filters, it may be the representation of a function f, are given by Equation 2.15 and as necessary to use only a finite number of terms of the Fourier series an example, we can consider the orthonormal trigonometric system to approximate a function f(t) over (0, and it is therefore of of period 2 for which Equation 2.16 to 2.20 are satisfied, for much interest to inquire what effect the truncation of the series has (k=2,3,...). (Banks, 1990). The error incurred obviously decreases as one takes into account more and more terms when the function f(t) is con- (2.15) tinuous. However, in the neighbourhood of discontinuities, ripples occur, the magnitude of which remain roughly the same even as more and more terms are included in the Fourier expansion. These (2.16) ripples are referred to as Gibbs' oscillations. 2.2 Generalised Fourier expansion (2.17) As an example of using the generalised Fourier expansion, let us consider representing the function shown in Figure 2.1, in terms of (2.18) the Legendre Polynominals, which form an orthogonal set on the interval (-1,1). The corresponding normalised basis functions are Generalised Fourier expansion 2/5 given by Equations 2.24 to 2.27, where the Pn(t) are the Legendre In general, the coefficient Cn is zero when n is even, for this polynominals, which may be generated by the expression of Equa- particular example. Hence the function f(t) in Figure 2.1 can be tion 2.28, or by using the recurrence relation of Equation 2.29. represented in terms of the Legendre polynominals as in Equation 2.35. f(1)= (2.35) We have shown that an arbitrary function, f(t), can be expressed in terms of a superposition of orthogo