(solution) ( A generalization of the Fourier approach to approximate an energy signal ) Suppose that s ( t ) is

(solution) ( A generalization of the Fourier approach to approximate an energy signal ) Suppose that s ( t ) is

( A generalization of the Fourier approach to approximate an energy signal ) Suppose that s ( t ) is a deterministic, real-valued signal with finite energy E s = ( ∞ s 2 ( t )d t . Furthermore, suppose that there exists a set of orthonormal basis −∞ functions { φ n ( t ), n = 1, 2, … , N }, i.e., r ∞ φ n ( t ) φ m ( t )d t = r 0, m /= n . (P5.13) 1, m = n −∞ We want to approximate the signal s ( t ) by a weighted linear combination of these basis functions, i.e., N s ˆ( t ) = s k φ k ( t ), (P5.14) k =1 where { s k }, k = 1, 2, … , N , are the coefficients in the approximation of s ( t ). The approximation error incurred is e ( t ) = s ( t ) − s ˆ( t ). (P5.15) Find the coefficients { s k } that minimize the energy of the approximation error. What is the minimum mean square approximation error, i.e., ( ∞ e 2 ( t )d t ? −∞