ECE 381: Laboratory 4

Winter 2006

Analog system response by numerical integration

February 14

    Part II. Assignment

 

Part I.

  1. Refer to Section M2.4 in the course textbook and use the program MS2P4.m to show visually the process of convolution an impulse response with an input signal.  You must download MS2P4.m and modify the inline function definitions for x and h on lines 5 and 6.  Use the following signals.

·        x(t) = u(t+1)-u(t-1),    h(t) = u(t);  -2 < t < 2, 0<y<3

·        x(t) = u(t-1)-u(t-3),    h(t) = u(t-2)-u(t-3),  0<t<8,  0 <y<2

·        x(t) = 40u(t),  h(t) = (1/3).*exp(-2.*t)u(t),  0<t<4,  0<y<15

These are all examples that were used in class lectures.  Print the result the is given once the MATLAB program finishes.

Part II.

Given below are differential equations and input signals for various systems.

  1. Using the MATLAB roots function, find the characteristic roots for all those systems.
  2. Then find the zero-input response and the impulse response for all systems.  Depending on the results for your characteristic roots, you may do this either by hand or by using the MATLAB matrix functions to solve Ax=b by using x = inv(A)*b when appropriate.
  3. For systems (a), (b), (c), find the total response as you have in homework using convolution and combining the zero-state and zero-input signals together.
  4. For systems (a), (b), and (c), plot and print the zero-state response you found in step 3 and compare it with the result found using MS2P4.m (print out that also).
  5. For systems (d), (e), and (f), simply use MS2P4 to show the zero-state response.

 

Use MATLAB to numerically evaluate the zero-state, zero-input, and total response for each of the analog systems described by the following differential equations

 

        

System equation

Excitation

Initial Conditions

Solve for

(a)

y''(t) + 5y'(t) + 6y(t) = x(t);   

x(t) = 6u(t);   

y(0) = 0, y'(0) = 1;   

0 ≤ t ≤ 10.

(b)

y''(t) + 5y'(t) + 6y(t) = x(t);   

x(t) = 2e-tu(t);   

y(0) = 0, y'(0) = 1;   

0 ≤ t ≤ 8.

(c)

y''(t) + 4y'(t) + 3y(t) = x(t);   

x(t) = 36tu(t);   

y(0) = 0, y'(0) = 1;   

0 ≤ t ≤ 2.

(d)

y''(t) + 4y'(t) + 4y(t) = x(t);   

x(t) = 2e-2tu(t);   

y(0) = 0, y'(0) = 1;   

0 ≤ t ≤ 5.

(e)

y''(t) + 4y'(t) + 4y(t) = x(t);   

x(t) = 8cos(2t)u(t);   

y(0) = 0, y'(0) = 1;   

0 ≤ t ≤ 10.

(f)

y'''(t) + 4y''(t) + 5y'(t) + 2y(t) = x(t);   

x(t) = e-2tu(t);   

y(0) = 0, y'(0) = 1, y''(0) = 0;   

0 ≤ t ≤ 10.

In each case, obtain the three responses separately for the indicated interval of time, and plot them in one figure, on the same time axis. Comment on the results. In particular, can you verify the initial conditions from your plots? Do the zero-state and zero-input responses seem to add up to the total response? Can you identify the transient and steady-state responses in your plots?

Turn in all plots, MATLAB scripts and functions. Include your comments and answers to questions.

School of Computing and Engineering
University of Missouri - Kansas City

Last updated: February 10, 2006