To illustrate the matter of trade-off analysis from a technical point of view, we briefly discuss here an example of detection of a radar pulse signal. We assume a simple threshold detection scheme whereby a target is declared to be present when, at the sampling time, the threshold (T) is exceeded. If the threshold is not exceeded, the decision is that no target was present.

Under the general assumption of a pulsed system with additive independent Gaussian noise:

a. When a target is present, the threshold detector ''sees'' a signal-plus-noise Gaussian distribution with a mean value of voltage equal to V and a noise (power) variance equal to N.

b. When a target is not present, the threshold detector ''sees'' a noise Gaussian distribution with a mean value of zero and a noise (power) variance equal to N.

In situation (a), we are interested in the probability of detection, P(d), that is, the probability that we will correctly detect a target when it is present. In (b), we wish to compute the false alarm probability, P( fa), that is, the probability that when no target is present, we may incorrectly conclude that there is a target. This can occur when, at the time of sampling, the noise alone is sufficiently large so as to exceed the set threshold.

Trade-offs between detection and false-alarm probabilities can occur when the threshold value (T), the pulse amplitude (V), and the noise power (N) are varied. The following three cases further explain this idea.

Case One: Increase Threshold; Pulse Amplitude and Noise Power Remain the Same

As we increase the detection threshold, less of the signal-plus-noise distribution remains to the right of the threshold value. Therefore, the detection probability decreases. This is an undesirable effect. However, less of the noise-alone distribution is to the right of the threshold, so the false-alarm probability also diminishes. This is a desirable consequence. Therefore, by the increase in threshold, we are ''trading'' to obtain better false-alarm performance [a lower P(fa)], but at the expense of detection performance [a lower P(d)]. A natural question is: Is there a threshold selection that allows us to meet both the detection and false-alarm proba bility requirements? By performing this trade-off analysis, that is, stepping the threshold through increasing and decreasing values, we determine the answer to this question.

Case Two: Increase Pulse Amplitude; Threshold and Noise Power Remain the Same

In general, the pulse amplitude is increased by increasing the power transmitted by the radar. This normally increases the cost of the radar system. The signal-to-noise ratio increases and the signal-plus-noise distribution has a larger mean value, but the same noise variance (N). In this case, the detection probability increases for a target at the same range. Another way to look at this case is to say that for the same P(d) as in Case One, we can see a target at a longer range. Pumping more power out of the transmitter results in an improved detection capability. But if the noise power remains the same, so will the false-alarm probability.

Case Three: Decrease Noise; Pulse Amplitude and Detector Threshold Remain the Same

Decreasing the noise may be achieved by designing a lower-noise frontend receiver. This increases cost and may also increase development time if we are pushing the state of the art. Less noise shows up as a decrease in the variance (N) of both the signal-plus-noise and the noise-alone distributions. This means that the detection probability increases and the false-alarm probability decreases! These are both desirable consequences. However, we must pay the price of the low-noise receiver and there are some natural limits as to how far the noise can be reduced.

We may also explore trade-offs that involve changing two of the preceding three key parameters at the same time. Such an exploration will reveal additional variations in the detection and false-alarm probabilities. In addition, other implementations and models may be considered (such as the integration of pulses), but the same basic notions of trade-offs remain. One is trying to find a balanced solution that satisfies user requirements. A more quantitative treatment of this particular example can be found in a variety of texts [9.9].