Abstract :
We consider the problem of detecting a Gaussian signal in nonGaussian noise using two sets of data, where one contains the signal (if present) and noise while the other contains noise only. The noise in the second data is essentially a delayed version of the one in the first which is Gaussian. What makes the noise in the second nonGaussian is the randomness of this delay. If the signal were strong, it would be more economical to ignore the second data, which contains noise only, and do the simple quadratic detection using the first data only since both the signal and the noise are Guassian in this case. However, if the signal is weak and the noises in two data are interdependent, it is worth attempting to cancel the noise in the first data by using the second data, thus effectively increasing the signal-to-noise ratio. Unlike the phase-matching of simple sinusoids, matching two broad-band Gaussian noises with random relative delay is greatly involved. Analytic derivation of the approximately optimum detection statistic is made feasibly by the weak signal approximation and the small parameter variation approximation. The random delay ? is decomposed into three parts in this detection structure: ?, ? and ?. ? tells which sampling interval ? falls in, and the effect of where in that interval is expressed by the correlation coefficient ?ei? between the sampled noise in the first data and the ?th later sampled noise in the second. The approximately optimum detector consists of three types of quadratic forms, which are averaged over ?. The first, q1, involves the first data only, the second, q2, the delayed version of the second data and the third, q12, both. In forming the quadratic forms with the first data the first n-? samples are more heavily weighted than the remaining since the noise in them are "matched" to the last n-? samples of the second data, thus having greater possibility of cancellation. With the second data, containing no signal at all, however, the firs- ? samples are simply ignored since they are independent of the noise in the first. Roughly speaking, q1 forms the usual quadratic form using the first data, q12 attempts to subtract from q1 the part of the noise which match the one in the second, and q2 acts as a correction term.
