1. Consider the passband basis functions defined in (8.56), where is f( t )  itself defined by (8.57). Demonstrate the validity of Properties 1, 2, and 3 of these passband basis functions

2. The water-filling solution for the loading problem is defined by (8.73) subject to the constraint of (8.70). Using this pair of relations, formulate a recursive algorithm for computing the allocation of the transmit power P among the N subchannels. The algorithm should start with an initial total or sum noise-to-signal ratio NSR(i) = 0 for iteration i = 0, and the subchannels sorted in terms of those with the smallest power allocation to the largest.