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Pay no attention to that man behind the curtain! Over time, modem technology advanced, enabling the local loop to provide higher and higher bandwidth This increasing bandwidth was made possible through clever signaling schemes that enabled a single signaling event to transport more than a single bit These modern modems, often called Shannon-busting modems because they defy the limits of signaling defined by Shannon, are commonplace today They enable baud levels to reach unheard-of extremes and permit the creation of very high bit-per-signal rates The analog local loop is used today for various voice and data applications in both business and residence markets The new lease on life that it enjoys, thanks to advanced modem technology as well as a focus by installation personnel on the need to build clean, reliable outside plant, has resulted in the development of faster access technologies designed to operate across the analog local loop.

Sound Power in Terms of Decibels The preceding formula can be worked inside out, so that you can determine the final sound power, given the initial sound power and the decibel change. To do this, you use the inverse of the logarithmic function, symbolized as log 1 or antilog. This function, like the logarithmic function, can be performed by any good scientific calculator, or by the calculator program in a personal computer when set to scientific mode. Suppose the initial sound power is P, and the change in decibels is dB. Let Q be the final sound power. Then:

Adding complex numbers is just a matter of adding the real parts and the complex parts separately. The sum of 4 j7 and 45 j83 is therefore (4 45) j(7 83) 49 j( 76) 49 j76.

This includes traditional high-speed modem access and such options as the Digital Subscriber Line (DSL)..

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Q = P antilog (dB/10) As an example, suppose the initial power, P, is 10 W, and the perceived volume change is 3 dB. Then the final power, Q, is equal to 10 antilog ( 3/10) = 10 0.5 = 5 W.

266 Impedance and admittance Subtracting complex numbers works similarly. The difference (4 j7) (45 j83) is found by multiplying the second complex number by 1 and then adding the result, getting (4 j7) ( 1(45 j83) (4 j7) ( 45 j83) 41 j90. The general formula for the sum of two complex numbers (a jb) and (c jd) is (a jb) (c jd) (a c) j(b d)

The plus and minus number signs get tricky when working with sums and differences of complex numbers. Just remember that any difference can be treated as a sum: multiply the second number by 1 and then add. You might want to do some exercises to get yourself acquainted with the way these numbers behave, but in working with engineers, you will not often be called upon to wrestle with complex numbers at the level of nitty-gritty. If you plan to become an engineer, you ll need to practice adding and subtracting complex numbers. But it s not difficult once you get used to it by doing a few sample problems.

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Decibels in the Real World Sound levels are sometimes specified in decibels relative to the threshold of hearing, defined as the faintest possible sound that a person can detect in a quiet room, assuming his or her hearing is normal. This threshold is assigned the value 0 dB. Other sound levels can then be quantified as figures such as 30 dB or 75 dB. If a certain noise has a loudness of 30 dB, that means it s 30 dB above the threshold of hearing, or 1000 times as loud as the quietest detectable noise. A noise at 60 dB is 1,000,000 (or 106) times as powerful as a sound at the threshold of hearing. Sound-level meters are used to determine the decibel levels of various noises and acoustic environments. A typical conversation occurs at a level of about 70 dB. This is 10,000,000 (or 107) times the threshold of hearing, in terms of actual sound power. The roar of the crowd at a rock concert might be 90 dB, or 1,000,000,000 (109) times the threshold of hearing. A sound at 100 dB, typical of the music at a large rock concert if you are sitting in the front row, is 10,000,000,000 (1010 ) times as loud, in terms of power, as a sound at the threshold of hearing.

Recall that in a circuit having reactance and resistance, the current and the voltage are not in phase. The extent to which they differ in phase is the phase angle. If there is no reactance, then the phase angle is 0 degrees. If there is a pure reactance, then the phase angle is either 90 degrees or 90 degrees. The power factor is equal to the cosine of the phase angle. You can use a calculator to find this easily.

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