You should use one of the computer algebra systems below with this example. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
Nancy loved music and her stereo system. She could listen anytime she wanted to any music she wanted by slipping a small silvery disk into her CD player and pressing a few buttons. Then one day her wonderful 75 watts per channel amplifier died. Shopping for a replacement, she discovered prices had risen. But the salesman said not to worry -- quality had gone up. He told Nancy that she hadn't really used all 75 watts in her old amplifier and only needed that much power because her older amplifier didn't perform well near full power. He said that she could buy a 40 watt per channel amplifier and play it at the volume she usually used because the newer amplifiers performed very well even close to full power.
Nancy bought the recommended amplifier, went home, hooked it up to her system, put her favorite CD on, adjusted the volume to her usual listening level, sat back in her favorite chair ... and one of her speakers blew out. ... Why?
Ampifiers "clip" signals that are too loud. Mathematically, the result of "clipping" a signal f(t) can be described by a new function
g(t) = Max(-A, Min(A, f(t)))
where A is the level at which the signal is clipped.
A little knowledge about "clipping" makes Nancy's experience even more puzzling. The signal produced by her new amplifier was actually weaker than the signal produced by her old amplifier -- and yet this weaker signal blew out one of her speakers.
Use your CAS window to look at the result of clipping the signal
at the level 1.
If you have Mathematica you can also listen to the clipped signal and the original signal and compare the two. What do you hear? With the other computer algebra systems you can look at the clipped signal but you cannot yet listen to it.
If you are using Mathematica or have another way to hear a "clipped" signal you have a head start figuring out why one of Nancy's speakers blew out. Some additional knowledge about sound systems can also help. Sound is produced mechanically in a sound system by speakers and different sounds place different demands on the speakers. Low-pitched sounds require large speakers that can move a lot of air but because low-pitched sounds have low frequencies the speaker cone doesn't have to react very quickly. This combination is good because moving a lot of air requires a large speaker cone and a large speaker cone will react more slowly than a smaller speaker cone. High-pitched sounds, on the other hand, can be reproduced better using a smaller speaker cone because they require faster action and don't need to move as much air. Thus, most sound systems use a combination of speakers -- usually two or three speakers per channel with large speakers, called "woofers," for the bass and smaller speakers, called "tweeters," for the treble. Sometimes "midrange" speakers are used for the intermediate frequencies. A "cross-over" network separates the different components and routes them to the appropriate speakers.
If you have listened to a clipped signal you might be able to deduce from all this that Nancy blew out one of the tweeters in her sound system. Why?
The key to understanding Nancy's sound system and many other phenomenon involving complex signals is looking at the different components and particularly the different frequencies that make up a signal. In the preceding module we looked at a three-dimensional object from several different perspectives. Depending on our viewpoint we saw different features of the object. We will do exactly the same thing in this module. We will "look at" or, more precisely, "listen to" a complex signal from various different "viewpoints" -- using mathematical "ears" that are attuned to different frequencies.
By now an English major would be screaming "MIXED METAPHORS" at the top of his lungs. What are "eyes" and "ears" doing in the same metaphor? But that is the power of mathematics -- the same mathematics describes "looking" at" a three dimensional object from various different perspectives and "listening to" a sound signal with ears attuned to one particular frequency.