Further Results on the Detectability of Known Signals in Gaussian Noise

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Abstract
In a previous paper Mathews and David pointed out that the theoretical detectability of a known signal in noise, whose power spectrum is bandlimited, may be larger than 2E/N0. However, their results showed only a small improvement over 2E/N0, and nothing was said concerning noise with a rational power spectrum. For the case of a known signal in noise of rational power spectrum, a test statistic which maximizes (in a certain sense) the detectability has been derived. By using this test statistic the detectability can be calculated. For a signal which is a constant in noise whose power spectrum has only poles, the detectability increases as the number of poles is increased to make the noise more nearly bandlimited. The results for the rational power spectrum case suggested measuring derivatives as a test statistic for the bandlimited noise case. By using only derivatives for the statistic, the detectability of a known signal in bandlimited noise is shown to be proportional to the number of derivatives measured. Thus, examples are presented in which theoretical detectability increases without bound either as a rational noise approaches a bandlimited noise or as more derivatives of a bandlimited noise are measured. We feel these results have very little to do with subjective signal perception.

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