be represented in be as close as possible to f(t). Many measures of closeness exist, but terms of sums of sinusoidal components. a common measure is the mean squared error (MSE). We therefore Euler and d'Alembert showed that the wave equation (Equation require that the coefficients i=1,2, n are chosen to mini- 2.1.) could be satisfied by any well-behaved function of the variable mise its value in Equation 2.5. or x - ct, i.e. Equation 2.2. (2.5) (2.1) (x,t)=f(x+ct)+g(x-ct) (2.2) This expression is the mean squared error averaged over the interval (t1,t2). Defining the functions in Equations 2.6 and 2.7, then Boundary conditions, of course, impose restrictions on the func- the expression for the mean square error above can be written as in tions f and g, but apart from these restrictions the functions f and g Equation 2.8. are completely arbitrary. This result started a long controversy, since it had been shown by Daniel Bernoulli (1727), that the solution to (2.6) the wave equation could also be represented as a superposition of sinusoidal components. If the solution obtained by Bernoulli was as general as that of Euler and d'Alembert, it follows that an arbitrary (2.7) function can be represented as a superposition of sinusoidal compo- nents. Euler had difficulty accepting this, and it was Jean-Baptiste Fourier (1768-1830) whose work 'Memoire sur la Chaleur' ex- pounded the method now known as Fourier analysis and eventually showed the above equivalence. It is of fundamental importance in both linear systems theory, and signal processing in general, that the input and/or output signals of (2.8) interest can be represented as linear combinations of simple basis functions, and the Fourier series representation is one such method Using the identity of Equation 2.9 the expression for the mean that allows the signals of interest to be decomposed into harmonic squared error can be written as in Equation 2.10. components using a sinusoidal basis. This is clearly of interest if one is considering the spectral compositio