e Po needed to produce a given overall (outer code) Pb is achieved by establishing a super channel (inner code) Pb = in (2.153) with K = 1 and L = Q: inner codePb=1-1-P)1 (2.156) We will consider three particular inner codes: the Odenwalder (7,1) code for K = 1 (we would use the rate 1/3 code except that the G(D) expression of (2.147) does not contain enough terms to be accurate at the higher Pb's defined by (2.156)), and the Trumpis (7,1) and (7, 1) codes for K = 2 and 3, respectively (this K refers to the MFSK data modulation rather than the parameter in (2.152) and (2.153); with the dual-K inner code, the two para- meters were synonymous). Analogous to (2.155), the overall code rate to be used in (2.133) is R = x i nner c code rate. (2.157) We should note that the inner Viterbi decoder makes hard binary subsym- bol decisions based on the soft decision/side information metric applied to the K-bit chips out of the FH/MFSK channel, SO D is still given by (2.132), (2.134), (2.138), or (2.139). The SNR's needed to achieve a system Pb of 10-5 with selected RS outer codes for worst case partial-band noise and multitone jamming are shown in Table 2.15. The concatenated systems employing the Trumpis inner codes are particularly attractive: in fact, the (511, 447) RS outer code combined with the Trumpis (7,1) convolutional inner code over the 4-ary channel achieves Pb = 10-5 in any non-adaptive jamming with EbIN1 = 8.31 dB, rep- resenting a 1 dB improvement over the best RS/Dual-K system in Table 2.14. Table 2.15 also adheres to the previously observed behavior that the worst case jammer is partial-band noise for K = 1 and multitone for K > 2. Also, whereas the performance improves with K in partial-band noise, it peaks at K = 2 in multitone jamming for small Pb's. 2.3.3 Coding with Diversity We demonstrated in the previous section that convolutional codes and con- catenated Reed-Solomon/convolutional codes can provide improved per- formance for FH/MFSK signals in worst case jamming over simple diversity. The next quest