View File - University of Engineering and Technology, Taxila

Noise power density N 0bandwidth:N 0 ¼ N Bis the thermal power N normalized to a 1-Hzð3:10ÞBy substituting (3.10) in (3.8) and in view of (3.9), we expressE b¼ CT bN 0 N=B¼C BN r bð3:11Þwhere the bit rate r b ¼ 1=T b . If the bandwidth equals the bit rate, the energyper bit-to-noise power density ratio will equal the carrier-to-noise power ratio,that is, E b =N 0 ¼ C=N.The general expression for the bit error probability for an M-PSKsystem is written asP e ¼ 1 b erfcpffiffiffibs ffiffiffiffiffi!E bN 0hsin p i !2 bð3:12Þwhere erfcðÞ denotes the complimentary error function. The larger the erfc’sargument, the smaller the probability of error is.Error function, erf, is, by definition,erfðuÞ ¼p2 ffiffiffipð u0e t2 dtand the complimentary error function isð3:13ÞerfcðuÞ ¼1 erfðuÞ ð3:14ÞThe integrals for these functions require numerical evaluation. For a largeargument u, we can use a series expansion of the Laplace–Gauss integral of anasymptotic expansion:ð u0pffiffiffipe t2 dt ¼2e u22u112u 2 þ 1:31:3:5ð2u 2 Þ 2 ð2u 2 Þ 3 þ ...By substituting (3.15) in (3.13) and in turn in (3.14):erfcðuÞ ¼ e u2puffiffiffi 11p 2u 2 þ 1:31:3:5ð2u 2 Þ 2 ð2u 2 Þ 3 þ ...ð3:15Þð3:16ÞCopyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.