Thus the second transmitting radio achieves reli- able communication without any coding and sends data at rate R2 = 1.0 bits/symbol. (2.46) The rate 1/2 coded first radio sends data at rate R1 = .5 bits/symbol. (2.47) The above example is an illustration of how an optimum decoder simul- taneously decodes the signals from both radios. Such multiple access com- munication systems were first studied by Shannon [127] in 1961. In 1971, Ahlswede [128] determined the capacity regions for the two-user and three- user multiple access channels with independent sources, and van der Meulen [129] put forward a limiting expression and simple inner and outer bounds on the capacity region for the two-user multiple access channel. Liao [130] studied the general L-user multiple access channel with independent sources. Multiple Access DETECT/ QUANTIZER FREQUENCY OPTIMUM DEHOPPER DECODER DETECT/ QUANTIZER Figure 2.28. Optimum decoder. He formulated the capacity region for this channel and proved the funda- mental coding theorem. An extensive survey on the information-theoretic aspects of multiple access channels has been assembled by van der Meulen [131], [132]. As sketched in Figure 2.28, the optimum detector simultaneously decodes both radio signals. It regards the two transmitting radios shown as a single radio with a vector data symbol consisting of two binary data symbols. Thus there are four possible inputs and three possible outputs. The optimum decoder examines all possible sequences of vector data symbols and com- pares them with the channel output pattern of hard quantized energy detec- tor outputs. The resulting channel is shown in Figure 2.29. Since the two transmitting radios are spatially separated, the codes used in this channel are restricted. Liao [130], and then Slepian and Wolf [133], gave formulas for the capacity region for this discrete memoryless channel (fo,fd (1,0) (f,4,1) (1,1) (f,1,f1) (0,1) Figure 2.29. Channel for optimum decoder (M = 2, L = 2). Spread-Spectrum Multiple Access with FH/MFSK Waveforms Figu