Suitable or optimal noise benefits in signal detection

• Six intervals of additive noises divided according to the two constraints.
• Derivation of the suitable additive noise to meet the two constraints.
• Formulation of the suitable noise for improvability or nonimprovability.
• Optimal noises to minimize PFA, maximize PD and maximize the overall improvement.

We present an effective way to generate the suitable or the optimal additive noises which can achieve the three goals of the noise enhanced detectability, i.e., the maximum detection probability (PD), the minimum false alarm probability (PFA) and the maximum overall improvement of PD and PFA, without increasing PFA and decreasing PD in a binary hypothesis testing problem. The mechanism of our method is that we divide the discrete vectors into six intervals and choose the useful or partial useful vectors from these intervals to form the additive noise according to different requirements. The form of the optimal noise is derived and proven as a randomization of no more than two discrete vectors in our way. Moreover, how to choose suitable and optimal noises from the six intervals are given. Finally, numerous examples are presented to illustrate the theoretical analysis, where the background noises are Gaussian, symmetric and asymmetric Gaussian mixture noise, respectively.