(a) Consider a stable LTIC system for which the input x(t) and output y(t) are related by the differential equation. d²y(t) dy(t) dt² dt 2y(t) = = x(t) Let X(s) and Y(s) denote Laplace transforms of x(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impulse response i) Determine the transfer function H(s) together with its ROC (region of convergence), and sketch the zero-pole plot of it ii) Determine h(t) if the system is stable iii) Sketch the magnitude response |H(w) of the system using the geometric evaluation, based on the zero-pole plot obtained in i)
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![(a) Consider a stable LTIC system for which the input x(t) and output y(t) are
related by the differential equation.
d²y(t) dy(t)
dt² dt
2y(t) =
= x(t)
Let X(s) and Y(s) denote Laplace transforms of x(t) and y(t), respectively,
and let H(s) denote the Laplace transform of h(t), the system impulse response.
i) Determine the transfer function H(s) together with its ROC (region of
convergence), and sketch the zero-pole plot of it
ii) Determine h(t) if the system is stable
iii) Sketch the magnitude response |H(w)] of the system using the geometric
evaluation, based on the zero-pole plot obtained in i)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd86a297b-9273-401b-8059-81b05b619264%2F1d856811-3c3e-4fdc-ae4c-9b670ac3a147%2Fu0bsgya_processed.png&w=3840&q=75)
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