( 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 ?

−∞