at time k depends on dk and -1 because only the pulse xk-1(t;dx-1) interferes with the transmission during the k-th pulse XK k(t; ). By defining -1 as the state at time k we can model the L radio signals as a finite state process described by a trellis diagram where there are 2L states. The process of finding the data vector sequence {dk} that maximizes (2.22) can be implemented using the Viterbi algorithm [103]. This system is illustrated in Figure 2.16. Verdu [96] took examples and evaluted upper and lower bounds on their bit error probabilities for this optimum maximum-likelihood receiver. These bounds were derived as a function of the L signal para- meters {A1}, {T1}, and {01}. Then the results were averaged over uniform probability distributions for {01} and {T1} and for the worst case choice of the delays {T1}. Figure 2.17 shows Verdu's example of L = 2 equal energy (A1 = A2) radios transmitting to a single receiver. The chip sequence here was periodic with a three chip period and N = 3 chips per data bit as shown. Except at low values of Eb/No, the optimum ML receiver is able to demodulate both signals as well as if there was only one DS/BPSK radio transmitting. The conventional receiver here has the form of Figure 2.10 as compared with the optimum receiver shown in Figure 2.16. This result is similar to the comparison of conventional and maximum-likelihood This metric may be time varying because of the pseudorandom sequences used. Multiple Access MATCHED FILTER OPTIMUM MATCHED SIMULTANEOUS FILTER DETECTOR IS A 2L STATE VITERBI ALGORITHM MATCHED FILTER Figure 2.16. Optimum detector. WORST CASE CONVENTIONAL DETECTOR BEST CASE CONVENTIONAL DETECTOR UPPER BOUND OPTIMUM DETECTOR SINGLE USER USER I USER 2 Eg/No (dB) Figure 2.17. N = 3, L = 2 example. Spread-Spectrum Multiple Access with FH/MFSK Waveforms receivers for channels with intersymbol interference. Other examples considered by Verdu include L = 3 equal power radios using N = 31 chips per data bit and an L = 2 case with unequal signal powers. These exam