4. (40 pts, 8 pts for each part) Consider the following piecewise polynomial pulse: 0 t< -2 p(t) = a(t+2)²(t+1) −(t+1)(t

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4 40 Pts 8 Pts For Each Part Consider The Following Piecewise Polynomial Pulse 0 T 2 P T A T 2 T 1 T 1 T 1 (91.68 KiB) Viewed 34 times

4. (40 pts, 8 pts for each part) Consider the following piecewise polynomial pulse: 0 t< -2 p(t) = a(t+2)²(t+1) −(t+1)(t− 1) b(t-1) (t2)² -2<t<-1 −1≤t≤1 1<t<2 2 <t. (a) Find the constants a and b that makes the derivative p' continuous. Sketch p using these values for a and b. (b) Let u(t) be a linearly modulated signal for scalar sequence (: k € Z) using pulse p and rate 1: u(t) = Σrkp(t - k). k=-∞ Suppose that k € {-1,1} for all k. i. Let te Z. What are the possible values of u(t)? ii. Let 0 < t < 1. For which values of k does u(t) depend on Tk? iii. For those k, what are the values of r that maximize u(t)? iv. Find an upper bound on u(t) that is valid for all symbol sequences and all times t. X (c) Assume that the symbol sequence r has time average zero and is time-average uncorrelated. Find the power of u(t) (d) Suppose that we want to transmit u(t) as a conventional AM signal with carrier frequency 100. Write an expression for the resulting AM signal in terms of u. What restrictions are there on the amplitude factors appearing in your expression? (e) Find the maximum possible AM power efficiency for this scheme. (If you did not solve all of the previous parts, explain how the answer would depend on those answers.)