1. Engineers have observed that when transferring a message I [an input signal] through a physical medium some information is lost in the message 0 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 1(3). defined for .1: E [i'r, 1r]. then one way to measure how much is lost is to compute the percentage loss: 2 If. Inns) Gonzo 1\" cursive.- ' where 0(3) represents the output signal, also defined for r E [rr, 1T]. For example, suppose we are given a transistor that sends an input Km) to 0(3) = I(t)dt+ 1 (f Ift)si.u(i}dt) aim). _-- 7r -1T (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. 0(3) 2 0 for all m E [7r,71']}? (b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the least loss. In Experiment 1, Mr. MacGyver computes A1 = 9.08. In Experiment 2, Mr. Macnyer computes A2 = 0.14. In which experiment is less information lost? {c} What is the output 01(3) determined by the transistor if the input is Ids) = 32? Do not use approximations but compute the integrals exactly. (d) Now find the output 02(1'} if the input is Ir} = 8 EB . Do not use approximations but compute the integrals exactly. Hint: Use integration by parts twice. Some comments: :3 _ m e The function rs} = % has the name 'hyperbolic sine' and is denoted sinhfm]. Similarly, one can define the hyperbolic cosine coshfm}. The name 'hyperbolic' is due to the fact that the coordinates {3.3;} = (coshft),sinh{t}] satisfy the equation of a hyperbola 1:2 3:2 = 1 in a similar way that the coordinates (my) = [cos[3), 3111(9)) belong to the circle 32 + y2 = 1. e The expression of 0(3) given above is intimately related to the Fourier transform. which has many applications from solving differential equations to wavelet signal decomposition