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In the process of detecting and estimating, simulation is used to provide a relationship between the probability of detection and the probability of false alarm. This relationship can be expressed as Pfa=1-Q(z) and Pd=1-Q(z-d), where Q(z) and Q(z-d) are the Gaussian Q-functions, representing the probability of a standard normal random variable being greater than z and z-d, respectively.
To better understand this relationship, it can be useful to visualize it by creating a graph that shows the relationship between Pfa and Pd for different values of d. This graph can help to identify the optimal values of d that maximize both Pfa and Pd. By examining various values of d, it is possible to find the best trade-off between these two values and optimize the performance of the detection and estimation system. Therefore, it is recommended to plot the relationship between Pfa and Pd for different values of d, in order to better understand the performance of the system.