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where B is the allocated channel bandwidth in Hz. The use of Nyquist filtering(or raised cosine filtering) helps to limit the spectral width without causingadjacent channel interference. If one assumes that the filter has a rolloff factora, then the required channel bandwidth can be expressed asB ¼ 1 þ aT sð3:6ÞBy substituting (3.2) in (3.6) and noting that r b ¼ 1=T b ; we can write themaximum bit rate asr b ¼ B log 2 M1 þ að3:7Þ3.2.6 Probability of Bit Error and Bit Error RateIn satellite communications it is desirable to minimize the average probabilityof bit error at the receiver subject to the constraints on received power andchannel bandwidth. The terms probability of bit error P e and bit error rateðBERÞ are used interchangeably in the literature although they differ slightly inmeaning in practice. P e is a mathematical expectation of the bit error rate for agiven system. BER is an empirical record of a system’s actual bit errorperformance. In essence, system performance can be quantified by firstmeasuring the bit error rate (BER) and then comparing the BER with theexpected probability of bit error.Probability of bit error is a function of the carrier-to-noise power ratio(C=N). This noise is often taken as thermal noise, which can be expressed asN ¼ kTB W ð3:8ÞThe carrier power is a function of bit energy and bit duration:whereC ¼ E bT bW ð3:9Þk ¼ Boltzmann’s constant ¼ 1:38 10 23 W=ðHz-KÞ:B ¼ noise bandwidth, Hz:T ¼ system noise temperature, K (note that 0 Celsius 273KÞ:E b ¼ energy of a single bit, J/bit:Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.