Consider the matched filter receiver. Develop expressions for the single pulse of known parameters probability of detection \(P_{D}\) and probability of false alarm \(P_{f a}\).

In the case of noise alone, the quadrature components of a radar return are independent Gaussian random variables with zero mean and variance \(\sigma^{2}\). Assume that the radar processing consists of envelope detection followed by threshold decision. (a) Write an expression for the \(p d f\) of the envelope. (b) Determine the threshold \(V_{T}\) as a function of \(\sigma\) that ensures a probability of false alarm \(P_{f a} \leq 10^{-8}\).