The purpose of this study is to investigate several artificial Neural Network (NN) architectures in order to design a cognitive radar system capable of optimally distinguishing linear Frequency-Modulated (FM) signals from bandlimited Additive White Gaussian Noise (AWGN). The goal is to create a theoretical framework to determine an optimal NN architecture to achieve a Probability of Detection (PD) of 95% or higher and a Probability of False Alarm (PFA) of 1.5% or lower at 5 dB Signal to Noise Ratio (SNR). Literature research reveals that the frequency-domain power spectral densities characterize a signal more efficiently than its time-domain counterparts. Therefore, the input data is preprocessed by calculating the magnitude square of the Discrete Fourier Transform of the digitally sampled bandlimited AWGN and linear FM signals to populate a matrix containing N number of samples and M number of spectra. This matrix is used as input for the NN, and the spectra are divided as follows: 70% for training, 15% for validation, and 15% for testing. The study begins by experimentally deducing the optimal number of hidden neurons (1-40 neurons), then the optimal number of hidden layers (1-5 layers), and lastly, the most efficient learning algorithm. The training algorithms examined are: Resilient Backpropagation, Scaled Conjugate Gradient, Conjugate Gradient with Powell/Beale Restarts, Polak-Ribiére Conjugate Gradient, and Variable Learning Rate Backpropagation. We determine that an architecture with ten hidden neurons (or higher), one hidden layer, and a Scaled Conjugate Gradient for training algorithm encapsulates an optimal architecture for our application.