In this problem, we consider the discrete-time counterpart of the concepts introduced in

Chapter 3, Problem 3.69

(choose chapter or problem)

In this problem, we consider the discrete-time counterpart of the concepts introduced in 3.65 and 3.66. In analogy with the continuous-time case, two discretetime signals ;[n], where the cf>i[n] are orthogonal over the interval (N1, N 2 ), then N? M ::;2 lx[n]i2 = L lail 2 Ai. i= I (d) Let cf>i[n], i = 0, 1, ... , M, be a set of orthogonal functions over the interval (N1, N2), and let x[n] be a given signal. Suppose that we wish to approximate x[n] as a linear combination of the cf>i[n]; that is, M i[n] = L aicf>i[n], i=O where the ai are constant coefficients. Let e[n] = x[n] - i[n], and show that if we wish to minimize N, E = ::;2 le[n]i2, n=N1 then the ai are given by (P3.69-2) [Hint: As in 3.66, express E in terms of ai, cf>i[n], Ai, and x[n], write ai = bi + }ci, and show that the equations aE - = 0 and abi are satisfied by the ai given by eq. (P3.69-2). Note that applying this result when the cf>i[n] are as in part (b) yields eq. (3.95) for ak.] (e) Apply the result of part (d) when the cf>i[n] are as in part (a) to determine the coefficients ai in terms of x[n].