  In continuation of an earlier study, we explore a Neymann-Pearson hypothesis
testing scenario where, under the null hypothesis ($\cal{H}_0$), the received
signal is a white noise process $N_t$, which is not Gaussian in general, and
under the alternative hypothesis ($\cal{H}_1$), 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.
