Problems: 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(x), defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss: E, [I (x) – O(x)|*dx A = where O(x) represents the output signal, also defined for x E [-T, T]. For example, suppose we are given a transistor that sends an input I(x) to O(z) = 1(t)dt + = ( ( 1(t) sin(t)dt) sin(æ). (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 E [-1,"])? (b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the least loss. In Experiment 1, Mr. MacGyver computes A1 A2 = 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) = x²? Do not use approximations but compute the integrals exactly. 0.08. In Experiment 2, Mr. MacGyver computes et - e-I (d) Now find the output O2(x) if the input is I2(x) Do not use approximations but compute the 2 integrals exactly. Hint: Use integration by parts twice. Some comments: et – e-* • The function f(x) : define 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 x² – y? = 1 in a similar way that the coordinates (x, y) = (cos(0), sin(0)) belong to the circle x² + y² = 1. • The expression of O(x) given above is intimately related to the Fourier transform, which has many applications from solving differential equations to wavelet signal decomposition. has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can %3D
Problems: 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(x), defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss: E, [I (x) – O(x)|*dx A = where O(x) represents the output signal, also defined for x E [-T, T]. For example, suppose we are given a transistor that sends an input I(x) to O(z) = 1(t)dt + = ( ( 1(t) sin(t)dt) sin(æ). (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 E [-1,"])? (b) Suppose Mr. MacGyver tries several experiments to see which option transmits information with the least loss. In Experiment 1, Mr. MacGyver computes A1 A2 = 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) = x²? Do not use approximations but compute the integrals exactly. 0.08. In Experiment 2, Mr. MacGyver computes et - e-I (d) Now find the output O2(x) if the input is I2(x) Do not use approximations but compute the 2 integrals exactly. Hint: Use integration by parts twice. Some comments: et – e-* • The function f(x) : define 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 x² – y? = 1 in a similar way that the coordinates (x, y) = (cos(0), sin(0)) belong to the circle x² + y² = 1. • The expression of O(x) given above is intimately related to the Fourier transform, which has many applications from solving differential equations to wavelet signal decomposition. has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can %3D
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![Problems:
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(x),
defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss:
S", \I(x) – O(x)*dx
S", \I(x)|²dx
A =
where O(x) represents the output signal, also defined for x E [-n, 7].
For example, suppose we are given a transistor that sends an input I(x) to
1
O(2) = |
1(0)dt +(" () sin(t)dt) sin(2).
(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 E [–T,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 O1(x) determined by the transistor if the input is I1 (x) = x? Do not use approximations
but compute the integrals exactly.
ex
- e-x
(d) Now find the output O2(x) if the input is I2(x)
Do not use approximations but compute the
2
integrals exactly.
Hint: Use integration by parts twice.
Some comments:
et
- e-x
• The function f(x)
define 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 x2 – y? = 1 in a similar way that the coordinates (x, y)
(cos(0), sin(0)) belong to the circle x² + y? = 1.
• The expression of O(x) given above is intimately related to the Fourier transform, which has many applications
from solving differential equations to wavelet signal decomposition.
has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c324644-fca2-4afa-a450-d18d23424a82%2Fe1a6dd28-8dbc-441a-a122-a4f5a7e60ec0%2Fozgnmb8_processed.png&w=3840&q=75)
Transcribed Image Text:Problems:
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(x),
defined for x E [-1, 7], then one way to measure how much is lost is to compute the percentage loss:
S", \I(x) – O(x)*dx
S", \I(x)|²dx
A =
where O(x) represents the output signal, also defined for x E [-n, 7].
For example, suppose we are given a transistor that sends an input I(x) to
1
O(2) = |
1(0)dt +(" () sin(t)dt) sin(2).
(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 E [–T,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 O1(x) determined by the transistor if the input is I1 (x) = x? Do not use approximations
but compute the integrals exactly.
ex
- e-x
(d) Now find the output O2(x) if the input is I2(x)
Do not use approximations but compute the
2
integrals exactly.
Hint: Use integration by parts twice.
Some comments:
et
- e-x
• The function f(x)
define 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 x2 – y? = 1 in a similar way that the coordinates (x, y)
(cos(0), sin(0)) belong to the circle x² + y? = 1.
• The expression of O(x) given above is intimately related to the Fourier transform, which has many applications
from solving differential equations to wavelet signal decomposition.
has the name 'hyperbolic sine' and is denoted sinh(x). Similarly, one can
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