3.6 In the following problems we use the inverse Laplace transform and the relation between input and output of LTI systems. (a) The Laplace transform of the output of a system is e-2s Y1(s)= (s +2)² +2 (s +2)³ s2 +1 find yı (t), assume it is causal. (b) The Laplace transform of the output y2(t) of a second-order system is -s2 – s+1 s(s2 + 3s + 2) Y2(s)=
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.
![3.6 In the following problems we use the inverse Laplace transform and the relation between input
and output of LTI systems.
(a) The Laplace transform of the output of a system is
(s +2)² +2
(s +2)³
-25
Y1(s)=
s2 +1
find yı(t), assume it is causal.
(b) The Laplace transform of the output y2(t) of a second-order system is
-s? – s+1
s(s² + 3s + 2)
Y2(s)=
239
232
CHAPTER 3 THE LAPLACE TRANSFORM
If the input of this system is x2(t) = u (t), find the ordinary differential equation that rep-
resents the system and the corresponding initial conditions y2(0) and dy2(0)/dt.
(c) The Laplace transform of the output y(t) of a system is
1
Y (s) =
s(s + 1)2 +4)
Assume y(t) to be causal. Find the steady-state response yss (t), and the transient y,(t).
Answer: (a) yı(t) = sin(t – 2)u(t – 2) + e-2ª u(1) +1?e-2'u(t);
dy2(0)/dt = 2; (c) transient y, (t) = [-(1/5)e¬ cos(2t) – (1/10)e¬' sin(2t)]u(t).
(b) y2(0) = –1
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