Digital Signaling Principles—Part II

Baseband Pulse Detection

In this section we address the problem of detecting symbol pulses in the presence of noise. We illustrate by assuming a binary line code received in additive white Gaussian noise.2  The role of the receiving elements is to reliably determine whether a 1 or a 0 was sent for each symbol in the incoming waveform. We assume that the receiver makes its decisions one symbol at a time and that each decision is made independently. The analysis that follows is classic; details can be found in many communication systems textbooks.³

The probability of error  is an important measure of digital receiver performance. This probability is calculated using an analytical model of a baseband system and is a basis for predicting the bit error rate (BER). The BER is an empirical performance measure given by the number of bits in error divided by the number of bits transferred during a test procedure. If the analytical model is accurate, and enough trials with enough samples per trial are performed during the test procedure, then we expect the probability of error to approximate the BER. We will use the two terms interchangeably, but keep in mind that they will be equal only if the analytical model is accurate.

A diagram of the system to be analyzed is shown in Figure 5.4. The diagram shows the signal and noise input to a baseband receiver. The signal is designated by si (t ), 0 ≤ t ≤ Ts , where i =  0 or 1 depending on whether the signal represents a 0 or a 1, respectively. The noise is written as n (t ), giving a received waveform of

The receiver consists of a filter, a sampler, and a threshold comparator. The filter has impulse response h (t ), to be discussed later. The sampler samples the filter output y (t ) at t = kTs , to obtain the decision statistic y (kTs ). The decision statistic is compared with a threshold VT . The receiver will decide that a 1 was transmitted if , and that a 0 was transmitted if In the following analysis we will set Since decisions are made independently for each received pulse, the results will remain general.

Since the filter is linear and time-invariant, its output can be found by convolution. We can write

where the limits on the last integral reflect the fact that the filter output is relevant only while a data pulse is being received. Substituting Equation (5.1) gives

Thus the filter output is the sum of a signal part and a noise part. Sampling at t = Ts  gives

The receiver’s performance is characterized by its probability of error Pe  . This is the probability that the receiver’s decision is incorrect. Since there are two ways of making an incorrect decision, we can write

where is the conditional probability of error given that a value k  was sent, and P [i = k ] is the probability that value k  was sent. The probabilities P [i =  0] and P [i =  1] depend only on the data source. 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 behavior 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 characterization 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 minimize 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 5.5. 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.

Next:  The Matched Filter

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.