s dealing with very large or very small numbers easier. Decibel Scale The decibel scale is a logarithmic scale which is used to represent and manipulate a wide range of numbers encountered initially in telecommunications (and nowadays a number of other fields). For example, sound level ratios may vary from 1 to 1012. Clearly it is cumbersome to write a number with 12 digits or more. A logarithmic representation is more compact and easier to manipulate: multiplications and divisions can be replaced by additions and subtractions. The conversion from a real to a decibel scale is simple: 'X' is the decibel representation of a real number 'x' if X = 10 * log (x) It follows that X =10(X/10) For example, if X= 100, then X=20 dB, or if X= 2521, then X=34.02dB. Also, if X=30 dB, then X = 1000, or if X = 13.2 dB, then = 20.89. As noted above for two real numbers X and y we have: x*y=X+Yandxly=X-Y Broadband Telecommunications Technologies and Management That is, multiplication in the real domain is equivalent to addition in the decibel domain, and division in the real domain is equivalent to subtraction in the decibel domain. In telecommunications signal power levels may also be represented using the decibel scale. Usually a signal power level is written with respect to a power unit such as 1 watt (W) or 1 milliwatt (mW). Using the decibel scale, the representation takes the form dBW or dBm to specify which unit has been used. For example, a power level of 4 W or 4000 mW can be written using the decibel scale as: 10*log,(4W)=6dBW Link budget calculations are very important in system design. They may be used to determine the distance over which communications can be made, or calculate the amount of necessary power that yields a desired receiver SNR. As Case Study 4.1 on the Voyager probe demon- strates, these calculations are common practice in any telecommunications system design, as they determine whether a channel can be established and the amount of information that may be transferred. Example 4.5 Consider a satellite th