is criterion demands that the lowest possible hop rate be selected. However, since a lower limit is dic- tated by the second criterion, this limit becomes the optimum hop rate for all considerations. 1.4 FURTHER DISCUSSION The material presented in this chapter has focussed on providing a systems viewpoint of the tradeoffs required between the communicator/receiver and the interceptor for communication in an LPI environment. Emphasis has been placed on determining the system design parameters necessary for communicating in the presence of practical interceptor detectors, neverthe- less pointing out their relation to optimal structures. In the recent literature, several authors [9], [10], [11] have taken a more formal maximum-likelihood approach to the LPI communications problem, both from the standpoint of estimating the unknown synchronization para- meters associated with the communicator's modulation (e.g., carrier phase, chip or hop epoch) and determining the optimum interceptor detector structures given none, some, or all of this synchronization knowledge. In par- ticular, Polydoros and Weber [9] determine synchronous and asynchronous (with respect to the chip or hop offset of the spread-spectrum modulation), Low Probability of Intercept Communications coherent and non-coherent (with respect to the carrier phase) detectors for both direct sequence and frequency-hopped modulations. All of these struc- tures fall into the category of feature-extraction detectors, since in the absence of all knowledge except carrier frequency, the wideband radiome- ter is the obvious choice and is in fact an asymptotically optimum device as the predetection SNR decreases to zero. The following represents a summary of the results obtained in [9]. Let y(t) denote the signal received by the interceptor which, when the communicator is transmitting, is the sum of signal plus noise and otherwise consists only of noise. For a direct-sequence spread-spectrum modulation, application of the generalized likelihood ratio test leads to the f