GENERAL ANALYSIS OF ANTI-JAM COMMUNICATION SYSTEMS
4.10.3 Reed-Solomon Outer Codes
Figure 4.22 shows the most popular form of concatenation of codes. Basically, this approach is to create a super channel as in Figure 4.21, where the super channel begins at the input to a convolutional encoder and ends at the Viterbi decoder output. The convolutional encoder with the Viterbi decoder that forms part of the super channel is the “inner code.” The inner code reduces the error probabilities of the first coding channel (see Figure 4.1) which consists of the modulator, radio channel, and demodulator. This then forms a new coding channel for another “outer code” which can further reduce the error rate.
In general, it is difficult to analytically obtain the bit error statistics of the super channel for a specific convolutional code. In the following example, some simulation results assuming an ideal additive white Gaussian noise radio channel [14] are shown. These simulations show that burst lengths out of Viterbi decoders can be modeled by a geometric probability distribution.
Figures 4.23 and 4.24 show how decoded bit errors out of a Viterbi decoder tend to occur in bursts of various lengths with BPSK or QPSK modulation with three bit quantization. Generally, the decoded bits are error free for a while and then when a decoding bit error occurs the errors occur in a burst or string of length Lbwith a probability distribution that is geometric [14],
(4.125) Pr5Lb⫽m6⫽p11⫺p2m⫺1; m⫽1, 2,p
Rs⫽log2M⫺log231⫹ 1M⫺12Ds4.
Figure 4.22.Reed-Solomon outer code.
Figure 4.23.Histogram of burst lengths;Viterbi decoded constraint length 7,rate 1/2 convolutional code;Eb/N0⫽1.0 dB
Figure 4.24.Histogram of burst lengths;Viterbi decoded constraint length 10,rate 1/3 convolutional code;Eb/N0⫽0.75 dB
where
(4.126) and is the average burst length in data bits. The waiting time,W, between bursts has the empirical distribution
(4.127) where
(4.128) In (4.128), is the average waiting time and Kis again the convolutional code constraint length.
By choosing a Reed-Solomon (RS) outer code [15], we can take advan- tage of the bursty nature of the bit errors out of the inner code decoder. RS codes use higher order Q-ary symbols, where typically Q⫽2mfor some inte- germ. In most applications m⫽8 (Q⫽256). By taking m⫽8 bits or 1 byte to form a single Q-ary symbol, a burst of errors in the 8 bits results in only oneQ-ary symbol error which tends to reduce the impact of bit error bursts.
In Figure 4.22 the conversion from Q-ary symbols to bits and back again for the coding channel of the RS outer code is shown. In addition, to avoid bursts ofQ-ary symbol errors interleavers and deinterleavers may be used to pro- vide a memoryless (non-bursty) Q-ary coding channel for the RS code.
As mentioned above, the most commonly employed RS code has m⫽8 (Q⫽256) where each symbol is an 8-bit byte and a block length of N⫽Q
⫺1⫽255Q-ary symbols. To be able to correct up to t⫽16Q-ary symbols the number of data Q-ary symbols must be K⫽Q⫺1⫺ 2t⫽223 result- ing in a (255,223) block code using 256-ary symbols that can correct up to t⫽16 symbols. This is equivalent to a binary (2040, 1784) block code6that can correct up to t⫽128 bits in error as long as these bits are confined to at most 16 Q-ary symbols where Q⫽256.
Figures 4.25 and 4.26 show the performance of the concatenation system with no interleaving for two convolutional inner codes with Viterbi decod- ing. Ideal interleaving is assumed in Figure 4.27. These curves show the cur- rently most powerful (non-sequential) coding technique available.
Sequential decoding of convolutional codes with large constraint lengths can also achieve similar performance but with the possibility of losing data caused by buffer overflows [1], [2].
For the above examples where a convolutional code with the Viterbi decoder forms the inner code, it is difficult to determine conditional
W
q⫽ 1 W⫺K⫹2 .
Pr5W⫽n6⫽q11⫺q2n⫺K⫺1, n⫽K⫹1,K⫽2,p Lb
p⫽ 1 Lb
6Each 1784 data bits are encoded into a codeword of 2040 coded bits.
Figure 4.25. Non-interleaved performance statistics for concatenated coding scheme assuming no system losses; (7, 1/2) convolutional code (reprinted from [14]).
Figure 4.26. Non-interleaved performance statistics for concatenated coding scheme assuming no system losses; (10,1/3) convolutional code (reprinted from [14]).
Figure 4.27. Comparison of concatenated channel decoder bit error rates for sev- eral convolutional inner codes and a Reed-Solomon (255, 223) outer code with ideal interleaving assuming no system losses (reprinted from [14]).
probabilities of the super channel formed. A natural approximation is to assume a symmetric Q-ary super channel characterized by the single para- meterPs, the symbol error probability. For this model the super channel has conditional probabilities given by (4.79) with parameter Dsgiven by (4.81) andRsgiven by (4.124). This type of symmetric Q-ary super channel occurs for many of the spread-spectrum systems considered earlier. For example, in a symmetric M-ary channel of the type considered earlier, collect Lsym- bols to form Q-ary symbols for the outer channel where
(4.129) For the binary case with M⫽2, the choice of L⫽8 gives the Q⫽256 alpha- bet for the RS code. Similarly M⫽ 4 and L⫽4 also yields Q⫽ 256. For theseQ-ary symmetric channels the decoded bit error probability of the RS codePbis directly a function of the symbol error probability of the inner code denoted Ps. That is,
(4.130) In Appendix 4C, a table showing this relationship for various RS codes is presented.