b (3.57) To evaluate this, note that if k of the m coded symbols experience a pulse jammer then 1 Zini is a sum of k independent Gaussian random variables each of variance Nj/(2p). Using Hk to denote the condition of k pulse jammed symbols, we have Pr m > V mEb = 2m Eb (3.58) Unknown Jammer State Soft Decision ,0001 .0005 ,0001 p = 1.0 .51.3 Eb/N, (dB) Figure 3.13. Repeat code m = 9 with unknown jammer state/soft decision. Basic Concepts and System Models where k=0,1,...,m. (3.59) The bit error probability is, thus, (3.60) Figures 3.12 and 3.12 show Pb versus E/N, for various values of p where m = 5 in Figure 3.12 and m = 9 for Figure 3.13. Comparing these with the uncoded case shown in Figure 3.7 there is only slight improvement with the repeat code. It is still clear that pulse jamming can cause much more degra- dation compared to constant power jamming (p = 1). Our conclusion is that soft decision decoding with no jammer state knowledge is not very effective against pulse jammers. 3.5.2.2 Hard Decision Decoder The hard decision decoder makes a binary decision on each channel output symbol as follows: (3.61) i=1,2,...,m. The final decision is then given by, (3.62) For an odd integer m, the probability of error is the probability that (m + 1)/2 or more of the m symbol decisions are in error. The probability that a particular coded symbol decision is in error is given by Pr{Zm}>VE,/m} (3.63) The probability that more than half of the m coded symbol decisions are in Coded Direct-Sequence Spread Binary Phase-Shift-Keying error gives the overall bit error probability Pb = (n) E k(1 - E)m-k. (3.64) Figures 3.14 to 3.16 show Pb as given by (3.64) versus Eb/N, for various values of m. Here by increasing the value of m we can effectively combat pulse jamming. There is no loss in data rate nor any change in the spread- spectrum signal by increasing the number m of repeats of the data bit. These results show that against the worst pulse jammer the hard decision decoder does better than the soft decision decoder. At first,