In continuation of an earlier study, we explore a Neymann-Pearson hypothesis testing scenario where, under the null hypothesis (${\mathcal { H}}_{0}$), the received signal is a white noise process$N_{t}$, which is not Gaussian in general, and under the alternative hypothesis (${\mathcal { H}}_{1}$), the received signal comprises a deterministic transmitted signal$s_{t}$corrupted by additive white noise, the sum of$N_{t}$and another noise process originating from the transmitter, denoted as$Z_{t}$, which is not necessarily Gaussian either. Our approach focuses on detectors that are based on the correlation and energy of the received signal, which are motivated by implementation simplicity. We optimize the detector parameters to achieve the best trade-off between missed-detection and false-alarm error exponents. First, we optimize the detectors for a given signal, resulting in a non-linear relation between the signal and correlator weights to be optimized. Subsequently, we optimize the transmitted signal and the detector parameters jointly, revealing that the optimal signal is a balanced ternary signal and the correlator has at most three different coefficients, thus facilitating a computationally feasible solution.