A tutorial on digital signalling (Part 2)
Keywords:bit error rate receiver filter sampler threshold comparator
Statisticians call these probabilities a priori probabilities, because they describe what is known about the received bit before the receiver observes r (t). We follow normal practice and assume the next step is to find and We will show the steps for the first of these in detail, since the calculation of the second term is similar.
Given that i = 0—that is, given that is transmitted—the receiver makes an error if we can then calculate by
Now and VT are constants, so the probability of error depends on the statistical behaviour of the noise term If we model the received noise n (t ) as Gaussian noise with zero mean and power spectrum then the noise at the output of the filter will also be Gaussian with zero mean and power spectrum where H (f ) is the frequency response of the filter. The variance of the noise is equal to its average power, so we have
With this characterisation of the noise, we can identify the decision statistic y (Ts ) = as a Gaussian random variable with mean and variance We can then write the conditional probability of error of Equation (5.6) as
where Q (x ) is the complementary normal distribution function defined in Appendix A.
With the assumption that i = 1, a very similar calculation gives us
Putting Equations (5.8) and (5.9) together into Equation (5.5) gives the overall probability of error as
The next step is to choose the threshold value VT to minimise Pe. This is a matter of differentiating Equation (5.10) with respect to VT and setting the derivative to zero. After some algebra, the optimum threshold value turns out to be
Substituting for VT in Equation (5.10) gives the probability of error
Equation (5.10) is illustrated in figure 3. The figure shows the two Gaussian probability density functions for y (Ts ), one assuming i = 0 and the other assuming i = 1. The figure is drawn for the case of polar keying, with and . The threshold is as given by Equation (5.11). The conditional probabilities of error and are the shaded regions under the density functions.
Figure 3: Probability density functions for the decision statistic give i = 0 and i = 1. |
Introduction to Wireless Systems By Bruce A. Black, Philip S. DiPiazza, Bruce A. Ferguson, David R. Voltmer, Frederick C. Berry, Published Jun 7, 2011 by Prentice Hall, is reprinted with permission by Pearson Publishing.
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