re 2.30. Optimum decoder capacity region. with independent encoding of the two message sources. Their results are shown in Figure 2.30, which gives the data rates of the two transmitting radios that can be simultaneously achieved while maintaining arbitrarily small bit error probabilities. The rates R1 = .5 and R2 = 1.0 illustrated above represent one achievable point in this capacity region. It is interesting to compare the optimum case of the previous example with pure TDMA. Suppose in a TDMA mode the first radio transmits p fraction of the time while the second radio transmits at 1 - p fraction of the time. Since there would be no interference there are no errors and the average data rates are R1 = p bits/symbol (2.48) R2=1 - p bits/symbol 1 = R1 + R2. (2.49) This defines the capacity region for pure TDMA and is shown in Figure 2.31. Note that the simultaneously transmitting radios with optimum decoding can achieve higher data rates than pure TDMA. Construction of specific codes for the two-radio channel has been inves- tigated by several authors [134]-[138]. Chang and Weldon [139] and Chang and Wolf [140] extended these results to the general case of L-users and M frequencies. The use of trellis codes for these multiple access channels has also been investigated [141]-[147]. Figures 2.32 to 2.34 show the multiple access channels for L = 3, M = 2; L = 2, M = 3; and L = 2, M = 4. For arbitrary L and M there are ML inputs and the number of outputs is given by = min {L,M}. (2.50) Multiple Access Figure 2.31. Pure TDMA capacity region. (for for fo) (1,0) (fo,fo,f1) (fo,f1, fo) (fo,fl,f1) (1,1) (f1,fo,fo) (F1,FO,F1) (f1,f1,fo) (O, 1) (f1,f1,f1) Figure 2.32. Channel for optimum decoder (M = 2,L = 3). (f1,4) (f,,f2) (f2,f1) (100) (f1,f3) (110) (101) (f3,f1) (f2,f2) (010) (f2,f3) (011) (f3,f2) (001) (f3,f3) Figure 2.33. Channel for optimum decoder (M = 3, L = 2). Spread-Spectrum Multiple Access with FH/MFSK Waveforms (fo,or (1000) (fo,fi) (f1.fo) (1100) (fo,f2) (1010) (f2,fo) (F,F,) (0100) (f1,f2) (OI IO) (f2,f1) (f1,