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

In this problem, we consider the discrete-time counterpart of the concepts introduced in Problems 3.65 and 3.66. In analogy with the continuous-time case, two discrete-time signals if

If the value of the constants are both 1, then the signals are said to be orthonormal.

(a) Consider the signals

Show that these signals are orthonormal over the interval (—N, N).

(b) Show that the signals

are orthogonal over any interval of length N.

(c) Show that if

Where the

are satisfied by the a_i given by eq. (P3.69-2). Note that applying this result when the are as in part (b) yields eq. (3.95) for a_k.]

(e) Apply the result of part (d) when the are as in part (a) to determine the coefficients a_i in terms of x[n].