ollowing decision rules. Assuming both chip timing epoch and carrier phase are known (a very idealistic assumption), then the generalized coherent syn- chronous radiometer is an implementation of the decision rule: decide signal present (1.77) decide signal absent where n is the detection threshold, (1.78) and N is the number of chips, each of duration Tc, in the observation time T used to define the likelihood ratio test. If the "In cosh" non-linearity is approximated by a square-law device as is appropriate at low predetection SNR, then the resulting structure is just a synchronous coherent energy detec- tor or radiometer. Relaxing the assumption of known carrier phase leads to the optimal non- coherent detector satisfying the decision rule: decide decide signal present absent (1.79) signal where (1.80) (1.81) j=1,2,...,N. (1.82) Here wo is the carrier radian frequency and Cij denotes the j-th binary element (+1or-1) of the i-th PN code pattern ci(t). In general, the index i Further Discussion ranges over the N-dimensional space corresponding to the 2N possible pat- terns (arrangements) of the N binary symbols within the T = NT--second observation interval. However, it can be shown that insofar as the decision rule is concerned, it is sufficient to divide the above space into two disjoint complementary subsets (G and G), each having 2N-1 elements, and sum only over those elements in either G or G. The rule for the subdivision is that, for each possible PN code vector Ci = (Ci1, Ci2, CiN), its negative (com- plement) belogns to the other subset. If as before the non-linearity "I0" is approximated by its low SNR equivalent Io(x) = 1 + x2/4, then a synchro- nous non-coherent energy detector results. For detection of frequency-hopped signals, it can be shown that the opti- mal synchronous non-coherent detector is an implementation of the decision decide signal present (1.83) decide signal absent where, analogous to (1.80), R is the envelope of the signal in the j-th hop frequency slot and the i-th hop time interval