n of signals, and in this area, Fourier analysis and the Fourier transform are used widely. (2.9) It should be realised that the Fourier representation is just one of many possible representations, and the choice of the signal repre- sentation will be determined by the particular nature of the problem under investigation. Suppose that we have a function f(t) that we wish to represent on a finite interval (t1,t2), where can obviously represent time, space (2.10) or any other dimension relevant to our problem, in terms of a set of a basis functions 4,(1),42(1), 4'n (1). We will assume that these functions are orthogonal on the region of support (t1,t2), i.e. Equation 2.3. From the form of this expression, it is clear that the MSE is always greater than or equal to zero, since it is the sum of squared terms, and that it achieves its least value when the middle term above is (2.3) zero, i.e. as in Equation 2.11. <fl4.> (2.11) We will use the notation <4,14;> to denote the integral above. The idea of orthogonality expressed above, is the same as that applied to vectors and vector spaces, and our representation of f(t) in terms of the functions 4.(1),i=1,2, n is equivalent to Therefore, the best approximation of an arbitrary signal or func- tion f(t), in the mean squared error sense, over the region of support representing a vector f in terms of an orthogonal set of vectors that (t1,t), that can be represented as a linear superposition of orthogo- span the space containing f. nal basis functions, is achieved by choosing the coefficients ci We will assume that the representation of f(t) can be written as a according to the above expression, which is the normalised projec- linear combination of the basis functions 4 (1) as in Equation 2.4. tion of the function in the direction of the basis function 4- 2/4 Fourier analysis As is clear from above, when the coefficients C; are chosen using (2.19) the above criterion, the mean squared error is simply the averaged difference between two terms, and this difference can be made 