- The following result will be useful in interpreting Fourier series: Using it,
- Prove Acos(wt)+Bsin(wt)=Ccos(wt-D) for A,B>0 where
C=(A
^{2}+B^{2})^{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).

- 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.

we can use either sines and cosines separately, or phase-shifted cosines.

- Prove Acos(wt)+Bsin(wt)=Ccos(wt-D) for A,B>0 where
C=(A
- 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.

- Describe in words what we hear. HINT: use the 2
^{nd}trigonometric identity above. - 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!

- Describe in words what we hear. HINT: use the 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?

- A non-negative square wave with height=2, width=1, and period=2 is plotted below.
**Compute**its Fourier series. Write out*explicitly*the first 5 nonzero terms.- 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.

- 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

This problem set requires 6 plots. Put them all on one page using

Attach a printout of the Matlab computer code you used to generate this page of plots.