ECE 381: Laboratory 2 Winter 2006
Complex exponentials and chirp signals January 31
    Part III. Assignment MATLAB notes | Note on chirp signals | Assignment
In this assignment, the MATLAB functions you will use include real, imag, abs, angle, cos, and sound as well as the built-in constants pi and j. You are supposed to generate the chirp signals in Exercises 1 and 2 using standard MATLAB routines and tools.

  1. Consider the analog chirp signal x(t) = cos(1000*2πt2).

    1. On one figure with a 3-by-1 subplot layout, plot x(t) over 0 ≤ t ≤ T for T = 2, 6, and 10. Use a small time step, say Δt = 25e-6 (sampling frequency of 40,000 Hz).
    2. What do the plots reveal as T is increased? Is this signal periodic? Should it be?
    3. The instantaneous frequency is the derivative of the argument for a sinusoid divided by 2π.  In this case, it would be the derivative of 1000*2πt2. How does its instantaneous frequency vary with time?
    4. Use the MATLAB sound function to listen to this chirp signal.  How does what you hear match your answer to (c)?

     

  2. Consider the analog chirp signal x(t) = cos[2πf0t + π(f1-f0)t2/T].  When this is used, the frequency of x(t) varies linearly from f0 to f1 over 0 ≤ t ≤ T. Use f0 and f1 as 0 and 4096 Hz, respectively, and T = 0.5 sec.. Generate the vector x corresponding to x(t), with Δt = 1/8192.  Play the following ten signals and listen:

    1. x 2. y = x(length(x):-1:1)     3. [x -x]     4. [y y]
    5. [x x x x]     6. [-y y -y y] 7. [x y] 8. [y x]
    9. [x y x y] 10. [[x -y]' [y -x]']

    Does what you hear make sense? How does the instantaneous frequency seem to vary for each signal?

     

  3. The general solution to a second order differential equation with complex roots is a sinusoid multiplied by an exponential of the form x(t) = eatcos(2πft+θ).  For all of the plots below, be careful to choose a range of time axis values that displays the signal well.
  1. Plot a signal whose amplitude envelope decays by 50% every 3 seconds, oscillates 2 times per second, and has a phase shift of 0.  What are the values for a, f, and θ?
  2. Plot a signal whose amplitude envelope never decays, oscillates 3 times per second, and has a phase shift of π/2. What are the values for a, f, and θ?
  3. Plot a signal whose amplitude envelope doubles in size every 3 seconds, oscillates 1 time per second, and has a phase shift of π. What are the values for a, f, and θ?
  4. Plot a signal whose amplitude envelope decreases to 75% of its previous value every 2 seconds, never oscillates (i.e., the sinusoidal term is a constant), and and has a phase shift of 0. What are the values for a, f, and θ?

For each of the above exercises, turn in all MATLAB scripts or diaries, plots, and explanations and comments.

School of Computing and Engineering
University of Missouri - Kansas City 		Last updated: January 28, 2006