, Nh is the number of hop frequency slots in the spread-spectrum bandwidth, and NT is the total number of hop time intervals in the observation. We note that the detector corresponding to (1.83) is equivalent to Figure 1.4 since in deriving (1.83) the log-likelihood ratio test was actually employed. Applying this same test in deriving Figure 1.4 would have resulted in the inclusion of a logarithm non-linearity between the summer and product blocks in this figure. However, since the logarithm of a product is iden- tical to the summation of the logarithms, we obtain the equivalence with (1.83). While the above results that have been reported in [9] are only for syn- chronous structures, they are not purely of academic interest since they can be used to derive upper bounds on performance for any practical asyn- chronous system. Furthermore, as the authors point out, the same maximum- likelihood approach can be used to derive asynchronous optimal and suboptimal detector structures. In [10], Krasner addresses the general problem of optimum detection of digitally modulated signals (a train of unknown symbols modulated onto a single or quadrature carriers) in the context of the LPI applica- tion, i.e., low symbol SNR, unknown symbol epoch, unknown carrier phase. By not restricting the symbol set to be binary and allowing vari- ous choices for the correlation properties of the symbol set, e.g., antipo- dal (balanced), orthogonal, or biorthogonal, the direct-sequence and frequency-hopping spread-spectrum modulations occur as special cases. Again the structures that are developed result from application of the maximum-likelihood theory to testing the hypothesis of signal plus noise against the null hypothesis of noise alone. As such, the presence of infor- mation modulation on the signal when present is of no consequence. Analogoous to [9], optimal detectors for the coherent and non-coherent Low Probability of Intercept Communications carrier cases are derived, here, however, in the presence of unknown sym- bol epoch (asyn