Relationship Between Detection Probability and False Alarm Probability with Pfa=1-Q(z)

In detection and estimation processes, simulations are employed to establish the relationship between detection probability (Pd) and false alarm probability (Pfa). This relationship is mathematically expressed as Pfa = 1 - Q(z) and Pd = 1 - Q(z - d), where Q(z) and Q(z - d) represent the Gaussian Q-functions. These functions quantify the probability that a standard normal random variable exceeds the thresholds z and z - d, respectively. To enhance understanding of this relationship, visualization through plots showing Pfa versus Pd for various d values is recommended. In code implementation, this typically involves: 1. Defining a range of z values (e.g., using linspace in MATLAB or np.linspace in Python) 2. Calculating Q(z) using built-in functions like qfunc in MATLAB or scipy.stats.norm.sf in Python 3. Iterating over different d values to compute corresponding Pd curves 4. Plotting Pd against Pfa with distinct markers/colors for each d value Key algorithmic considerations include: - The Q-function can be computed via numerical integration or approximation formulas (e.g., Abramowitz-Stegun approximation) - Optimal d values maximize both Pd and Pfa, which can be identified through ROC (Receiver Operating Characteristic) curve analysis - Implementation should include axis labeling (Pfa vs. Pd) and legends indicating d values Such visualizations help identify optimal d values that balance detection capability and false alarm resistance, ultimately optimizing system performance in detection-estimation applications.