EECS 210_________________________PROBLEM SET #1________________________Winter 2001

ASSIGNED: January 05, 2001. Read Sections 16.1 through 16.4 of textbook (yes that's Chapter 16!)
DUE DATE: January 12, 2001. Lab Book: Through Lab Lecture 1; also Chapter 1 of Appendix I.

    THIS WEEK: Basics of sinusoidal signals and Fourier series of periodic signals.
    This problem set requires 6 plots. Put them all on one page using subplot(3,2,*) (use help).
    Attach a printout of the Matlab computer code you used to generate this page of plots.
    1. The following result will be useful in interpreting Fourier series: Using it,
      we can use either sines and cosines separately, or phase-shifted cosines.
    2. Prove Acos(wt)+Bsin(wt)=Ccos(wt-D) for A,B>0 where C=(A2+B2)½ and tan(D)=B/A.
      HINT: cos(a-b)=cos(a)cos(b)+sin(a)sin(b) implies 2cos(a)cos(b)=cos(a-b)+cos(a+b).
    3. Plot 3cos(t)+4sin(t) for 0 < t < 20 using Matlab's linspace(0,20,1000) for the t-axis.
      Estimate the amplitude and phase from the plot, and compare to results using the formula.

  1. We wish to tune a piano to "AAA" (the leftmost piano key), which is a pure sinusoid at 27 Hz.
    We have a tuning fork that generates 27 Hz exactly. However, the piano is really tuned to 28 Hz.
    We strike the tuning fork and piano key simultaneously; each note has amplitude=½, phase=0.
    1. Describe in words what we hear. HINT: use the 2nd trigonometric identity above.
    2. Plot for 0 < t < 2 sec. (note this is a lot of cycles!) Try listening using sound.
      If you use sound, make sure you use headphones, so others aren't distracted!

  2. An odd periodic signal with period=1 is decomposed into sines at integer frequencies.
    The coefficient of the sine at k Hertz is (-1)k+1/k. Using Matlab, plot the sum of each of these:
    (a) the first term only; (b) the first 3 terms; (c) the first 8 terms. To what is it converging?
    1. A non-negative square wave with height=2, width=1, and period=2 is plotted below.
    2. Compute its Fourier series. Write out explicitly the first 5 nonzero terms.
    3. This signal is passed through an ideal low-pass filter which blocks all frequencies above
      3 Hz and has no effect on all frequencies below 3 Hz. Using Matlab, plot the output.
      Compare this to the input square wave. Discuss and explain the difference between them.

  3. Each key of the telephone tone dialer (see Pre-Lab for Lab #1) generates a sum of 2 sinusoids.
    For each key, this sum-of-2-sinusoids is periodic with period 1/n for some small integer n.
    Determine the 3 periods produced by keys 6,7 and 8. Answers are 3 from: 1/2,1/3...1/8,1/9.
    HINT: A sum-of-2-sinusoids at frequencies f1 and f2 Hertz repeats every T seconds, where
    T=n1/f1=n2/f2 for integers n1 and n2 such that n1/f1=n2/f2. Your job is to find the smallest T.
    "Never let your schooling interfere with your education"--Mark Twain