Answer Happy

1. Engineers have observed that when transferring a message I (an input signal through a physical medium some information is lost in the message O that is received (an output signal). Information may be lost due to dissipation as heat is transferred, or for other reasons, depending on physical context. For example, when we record music we do not record everything but a finite amount of information. If we represent the input signal by a function I(2), defined for xe [-7, 7], then one way to measure how much is lost is to compute the percentage loss: S*, \I(x) – O(x)/2dx A= S*. \I (2)|2dx E where 0(x) represents the output signal, also defined for x € (-77, 7]. For example, suppose we are given a transistor that sends an input I(x) to 1 O(x) = {"}{2}dt+ 3 (14) sin(e)dt) sin(x). a S" (tt = - -T = (a) What is the percentage loss if the input and output are equal? What is the percentage loss if the output is zero (i.e. O(x) = 0 for all x € (-2,7])? (b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the least loss. In Experiment 1, Mr. MacGyver computes A1 0.08. In Experiment 2, Mr. MacGyver computes A2 = 0.14. In which experiment is less information lost? (c) What is the output 01(x) determined by the transistor if the input is 11(x) = x?? Do not use approximations but compute the integrals exactly. = e2 -e-2 = Do not use approximations but compute the 2 (d) Now find the output O2(x) if the input is 12(x) integrals exactly Hint: Use integration by parts twice.