In Chapter 9, Section 9.7.2, a radar-detection trade-off example was introduced. In this example, we had a signal plus additive Gaussian noise. The signal voltage was equal to V and the noise power (variance) equal to N, with threshold detection at a voltage value of T. Of interest were the calculations of the detection probability, P(d), and the false-alarm probability, P( fa). With the information in this chapter, we are now in a position to carry out a simple quantitative analysis.

Specifically, we calculate the detection and false-alarm probabilities when the noise and threshold are kept constant and the signal level is increased. Specific assumed values are:

Case 1: V = 4 volts, N = 4 watts, T = 1 volt Case 2: V = 8 volts, N = 4 watts, T = 1 volt Case 3: V = 10 volts, N = 4 watts, T = 1 volt

In all cases, the noise power remains the same, so that because ct2 = N, the noise standard deviation is equal to 2 volts.

For Case 1, the range from the mean value of 4 volts to the threshold of 1 volt is 3 volts. Therefore, the threshold is at 1.5 sigma from the mean. At a value equal to 1.5, from the normal probability table, the corresponding area under the right-hand portion of the normal is 0.4332. When added to 0.5, a detection probability of P(d) = 0.9332 is obtained. For noise alone (no signal present), we may calculate the false-alarm probability by recognizing that the mean value of zero is only 1 volt from the threshold. That is, the threshold is only 1 /2 = 0.5 sigma from the mean. The corresponding table lookup value is 0.1915. However, in this situation, this must be subtracted from 0.5 to yield the correct answer of 0.3085.

In Case 2, the signal is increased to 8 volts with the other parameters remaining the same. This means that the distance from the threshold is 8 -1 = 7 volts, which represents 3.5 sigmas from the mean. At a value of 3.5, the normal table lookup gives us a value of approximately 0.49975. When added to 0.5, the detection probability for this case becomes 0.99975. Because neither the threshold nor the noise variance was changed, the false-alarm probability remains the same as in Case 1.

For Case 3, the signal is further increased to 10 volts. The distance from the threshold is now 10 - 1 = 9 volts, which in this example is 4.5 sigmas. The table lookup results in a value of 0.499997, which when added to 0.5 yields a detection probability of 0.999997. As with Case 2, the false-alarm probability remains the same as in Case 1.

Although these values have been chosen so as to be amenable to lookup in the normal probability integral table, and are not necessarily realistic for a radar system, they illustrate the way in which one might make detection and false-alarm probability calculations, using the material presented in this chapter.

Example. In a pulse-signal-detection situation, (a) if the noise power is 9 watts, and the false-alarm probability is 0.0099, where is the threshold? In this case, ct = 3 and the area under the normal distribution is 0.5 -0.0099 = 0.4901. This yields a value from the normal table of 2.33. The threshold is therefore at 2.33 ct = (2.33)(3) = 6.99 volts. For part (b), what is the root-mean-square (rms) signal-to-noise ratio to achieve a detection probability of 0.9772? For 0.9772 - 0.5 = 0.4772, the normal table yields a value of 2, in which case 2ct = (2)(3) = 6. This value, when added to 6.99, results in an rms signal of approximately 13 volts. Thus, the rms signal-to-noise ratio is approximately 13/3 = 4.33.

These examples illustrate the way in which signals might be detected in the presence of noise. The signal pulse is corrupted by additive Gaussian noise during transmission, so that at the receiver, both signal and noise are present. The threshold detection scheme simply compares the signal-plus-noise value to the threshold and decides that a signal is present when the threshold is exceeded.

In the section in this chapter dealing with the binomial distribution, we calculated the probability of no errors in a sequence of eight pulses that represented an alphanumeric character. We now can see how this section and that previous section fit together in that we are now in a position to calculate the error probability for a single pulse. Although there is much more to be explored in this regard, such as errors of both types, we hope that the reader can see more concretely the value in the preceding Gaussian error model and, as well, the binomial computation from the previous section.