Let a linear system with input x(t) and output y(t) be described by the differential equation .
(a) Compute the simplest math function form of the impulse response h(t) for this system. HINT: Remember that with zero initial conditions, the following Laplace transform pairs hold:
Let the time-domain function p(t) be given by p(t) = g(3 − 0.5 t). (a) Compute the simplest piecewise math form for p(t).
(b) Plot p(t) over the range 0 ≤ t ≤ 10 .
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Let a linear system with input x(t) and output y(t) be described
by the differential equation...
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential...
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential equation given below: a causal system is dx(t) dse)+540+6y(t) = x(t) +T Assuming that the initial conditions are zero and using the Laplace transform determine [5 Marks] 15 Marks the following: a- Transfer function H(s) of the system. b- Impulse response h(t) of the system. Y (s) X(s)
solve all 22. The input-output relationship for a linear, time-invariant system is described by differential equation...
solve all
22. The input-output relationship for a linear, time-invariant system is described by differential equation y") +5y'()+6y(1)=2x'()+x(1) This system is excited from rest by a unit-strength impulse, i.e., X(t) = 8(t). Find the corresponding response y(t) using Fourier transform methods. 23. A signal x(1) = 2 + cos (215001)+cos (210001)+cos (2.15001). a) Sketch the Fourier transform X b) Signal x() is input to a filter with impulse response (1) given below. In each case, sketch the associated frequency response...
Consider a first-order system with input x(t) and output y(t). Let the time constant be the...
Consider a first-order system with input x(t) and output y(t). Let the time constant be the part of your birth date in the format of day, month (ddmm) in microseconds. Complete the following steps: 1. Write the differential equation representing the system. 2. Derive the transfer function H(s). A Note: Label all graphs appropriately. ddmm 3. Use H(s) with MATLAB to complete the following actions: • Find the poles are zeros. • Find the step response. • Find the impulse...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
Problem 3 A system is described by the following second-order linear differential equation d'y dz 5y(sin2t+...
Problem 3 A system is described by the following second-order linear differential equation d'y dz 5y(sin2t+ e-t)u(t) dt2 where y(0)y()0 Solve the differential equation using the Laplace Transform method.
Consider a CTLTI system described by the following ordinary differential equation with constant coefficients: N M...
Consider a CTLTI system described by the following ordinary differential equation with constant coefficients: N M dky(t) 2 ak ak dtk , dkx(t) Ok atk bk - 2 k=0 k=0 The system function H(s) is defined as the Laplace transform of the impulse response h(t) of the system. Write and prove the expression of H(s) as a function of the coefficients of the differential equation. Justify each single step of the proof from first principles (hypothesis, thesis, proof).
For a continuous time linear time-invariant system, the input-output relation is the following (x(t) the input, y(t) the...
For a continuous time linear time-invariant system, the
input-output relation is the following (x(t) the input, y(t)
the
output):
, where h(t) is the impulse response function of the
system.
Please explain why a signal like e/“* is always an eigenvector
of
this linear map for any w. Also, if ¥(w),X(w),and H(w) are
the
Fourier transforms of y(t),x(t),and h(t), respectively.
Please
derive in detail the relation between Y(w),X(w),and H(w),
which means to reproduce the proof of the basic convolution
property...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 +...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider a differential equation system for which the input x(t) and output y(t) are related by the differential equation d’y(t) dy(t) -6y(t) = 5x(t). dt dt Assume that the system is initially at rest. a) Determine the transfer function. b) Specify the ROC of H(s) and justify it. c) Determine the system impulse response h(t).
Problem 1. The input x(t) and output y(t) of an LTI system satisfy the differential equation...
Problem 1. The input x(t) and output y(t) of an LTI system satisfy the differential equation d’y(t) + wốy(t)=r(t), where wo is a fixed real number. A) Find the right-going impulse response of the system. B) Find the left-going impulse response of the system.
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential equation given below: a causal system is dx(t) dse)+540+6y(t) = x(t) +T Assuming that the initial conditions are zero and using the Laplace transform determine [5 Marks] 15 Marks the following: a- Transfer function H(s) of the system. b- Impulse response h(t) of the system. Y (s) X(s)
solve all
22. The input-output relationship for a linear, time-invariant system is described by differential equation y") +5y'()+6y(1)=2x'()+x(1) This system is excited from rest by a unit-strength impulse, i.e., X(t) = 8(t). Find the corresponding response y(t) using Fourier transform methods. 23. A signal x(1) = 2 + cos (215001)+cos (210001)+cos (2.15001). a) Sketch the Fourier transform X b) Signal x() is input to a filter with impulse response (1) given below. In each case, sketch the associated frequency response...
Consider a first-order system with input x(t) and output y(t). Let the time constant be the part of your birth date in the format of day, month (ddmm) in microseconds. Complete the following steps: 1. Write the differential equation representing the system. 2. Derive the transfer function H(s). A Note: Label all graphs appropriately. ddmm 3. Use H(s) with MATLAB to complete the following actions: • Find the poles are zeros. • Find the step response. • Find the impulse...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
Problem 3 A system is described by the following second-order linear differential equation d'y dz 5y(sin2t+ e-t)u(t) dt2 where y(0)y()0 Solve the differential equation using the Laplace Transform method.
Consider a CTLTI system described by the following ordinary differential equation with constant coefficients: N M dky(t) 2 ak ak dtk , dkx(t) Ok atk bk - 2 k=0 k=0 The system function H(s) is defined as the Laplace transform of the impulse response h(t) of the system. Write and prove the expression of H(s) as a function of the coefficients of the differential equation. Justify each single step of the proof from first principles (hypothesis, thesis, proof).
For a continuous time linear time-invariant system, the
input-output relation is the following (x(t) the input, y(t)
the
output):
, where h(t) is the impulse response function of the
system.
Please explain why a signal like e/“* is always an eigenvector
of
this linear map for any w. Also, if ¥(w),X(w),and H(w) are
the
Fourier transforms of y(t),x(t),and h(t), respectively.
Please
derive in detail the relation between Y(w),X(w),and H(w),
which means to reproduce the proof of the basic convolution
property...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider a differential equation system for which the input x(t) and output y(t) are related by the differential equation d’y(t) dy(t) -6y(t) = 5x(t). dt dt Assume that the system is initially at rest. a) Determine the transfer function. b) Specify the ROC of H(s) and justify it. c) Determine the system impulse response h(t).
Problem 1. The input x(t) and output y(t) of an LTI system satisfy the differential equation d’y(t) + wốy(t)=r(t), where wo is a fixed real number. A) Find the right-going impulse response of the system. B) Find the left-going impulse response of the system.
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