1. Engineers have observed that when transferring a message I (an input signal) through a physical medium someinformation is lost in the message O that is received (an output signal). Information may be lost due to dissipationas heat is transferred, or for other reasons, depending on physical context. For example, when we record music wedo not record everything but a finite amount of information. If we represent the input signal by a function I (x),defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss:S", L[(2) – O(x)|PdxA =where O(x) represents the output signal, also defined for x E [-1, "].For example, suppose we are given a transistor that sends an input I(x) toO(2) = 1(t)dt +() sin(t)dt) sin(r).(a) What is the percentage loss if the input and output are equal? What is the percentage loss if the output iszero (i.e. O(x) = 0 for all x E [-1, 7])?(b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the leastloss. In Experiment 1, Mr. MacGyver computes A1 = 0.08. In Experiment 2, Mr. MacGyver computesA2 = 0.14. In which experiment is less information lost?(c) What is the output O1(x) determined by the transistor if the input is I1(x) = r?? Do not use approximationsbut compute the integrals exactly.et - e(d) Now find the output O2(x) if the input is I2(x) =integrals exactly.Hint: Use integration by parts twice.Do not use approximations but compute theSome comments:et - e-I• The function f(x) =has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can2define the hyperbolic cosine cosh(x). The name 'hyperbolic' is due to the fact that the coordinates (x, y)(cosh(t), sinh(t)) satisfy the equation of a hyperbola x? – y? = 1 in a similar way that the coordinates (x, y)(cos(0), sin(0)) belong to the circle r? + y? =• The expression of O(x) given above is intimately related to the Fourier transform, which has many applicationsfrom solving differential equations to wavelet signal decomposition.= 1.

Question

1. Engineers have observed that when transferring a message I (an input signal) through a physical medium someinformation is lost in the message O that is received (an output signal). Information may be lost due to dissipationas heat is transferred, or for other reasons, depending on physical context. For example, when we record music wedo not record everything but a finite amount of information. If we represent the input signal by a function I (x),defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss:S&#34;, L[(2) – O(x)|PdxA =where O(x) represents the output signal, also defined for x E [-1, &#34;].For example, suppose we are given a transistor that sends an input I(x) toO(2) = 1(t)dt +() sin(t)dt) sin(r).(a) What is the percentage loss if the input and output are equal? What is the percentage loss if the output iszero (i.e. O(x) = 0 for all x E [-1, 7])?(b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the leastloss. In Experiment 1, Mr. MacGyver computes A1 = 0.08. In Experiment 2, Mr. MacGyver computesA2 = 0.14. In which experiment is less information lost?(c) What is the output O1(x) determined by the transistor if the input is I1(x) = r?? Do not use approximationsbut compute the integrals exactly.et - e(d) Now find the output O2(x) if the input is I2(x) =integrals exactly.Hint: Use integration by parts twice.Do not use approximations but compute theSome comments:et - e-I• The function f(x) =has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can2define the hyperbolic cosine cosh(x). The name 'hyperbolic' is due to the fact that the coordinates (x, y)(cosh(t), sinh(t)) satisfy the equation of a hyperbola x? – y? = 1 in a similar way that the coordinates (x, y)(cos(0), sin(0)) belong to the circle r? + y? =• The expression of O(x) given above is intimately related to the Fourier transform, which has many applicationsfrom solving differential equations to wavelet signal decomposition.= 1.

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