Homework 4: EE 3417, Linear Systems, Fall 2015
Handed 10/15/2015, Due 10/29/2015
Instructions:The purpose of this homework is to let you practice concepts in Laplace transforms, Inverse Laplace transforms, system response, transfer function. To receive full credit, please return handwritten or typed answers to me prior to the lecture on Oct 13. Be as detailed in your answers as possible, and show the steps you followed in arriving at your answers. In addition, MATLAB problems require you to print the code (*.m file), its output and associated plots. You are encouraged to consult with your classmates while you work on the homework, however, writing/coding and understanding final submissions must be your own work.
Problem 1: (Laplace transforms) (20 pts)
(a)Determine the Laplace transform of and r(t -T) (ramp function)
(b)Determine the Laplace transform of the causal sawtooth waveform shown in Fig below
(c)Determine Laplace transform of
(d)Obtain the Laplace transform of,
Problem 2: (Inverse Laplace Transform) (15pts)
(a)Apply the partial-fraction expansion method to determine x(t), given that its Laplace transform
(b)Determine the inverse Laplace transform of
(c)Determine the inverse Laplace transform of
Problem 3: (Transfer functions, System response, Stability) (25 pts)
(a)Consider the system transfer function
Compute the output for the input
(b)Obtain the impulse response of transfer function H(s) below, and indicate the Region of Convergence (ROC). Is the systemBIBO stable?
(c)Obtain the transfer function of the op-amp circuit shown in the Figure below:
Problem 4: (Block diagram realizations) (10 pts)
Using Direct FormII, determinehow many integrators are needed to realize the system with transfer function. Draw the realization Block diagrams for:
Problem 5: (Zero input response, impulse response, zero state response) (15 pts)
Consider the system with the I/O model
Compute the following
a)Impulse response of the system
b)Zero input response with initial conditions y(0-)=1;
c)Zero state response for the input
Problem 6: (Properties of Laplace Transform) (10 pts)
For an LTIC system with zero initial conditions (systems initially in zero state), if an input produces and output , then using Laplace transform show the following.
(a)The input produces and output
(b)The input produces an output . Hence show that the unit step response of a system is an integral of impulse response; i.e. .