) is valid, completing the desired proof. APPENDIX 2B. COMBINATORIAL COMPUTATION FOR n = 1 BAND MULTITONE JAMMING This is a combinatorial problem required to compute the exact performance for FH/MFSK signals with diversity in n = 1 band multitone jamming, assum- ing non-coherent linear combining and perfect side information. The prob- lem is deceptively simple to state, and difficult to resolve under certain conditions. Figure 2B.1. Frequency (row)-time (column) grid for computing probabilities asso- ciated with n = 1 band multitone jamming. Appendix 2B Consider a grid containing I rows and J columns, as shown in Figure 2B.1. One X is placed in each column: the row location is random, equally likely, and independent from column to column. The number of X's in the i-th row is denoted by li E [0,J], and lmax ; max1 sisilli). There are I distinct patterns of X's that can arise; we want to compute subsets of these defined by NL = number of patterns T /max = L. (2B.1) The difficulty here is that while the columns are statistically independent, the rows (specifically, the l's) are not. Clearly, 11max J, SO that N L = 0 for L E [1,J]. For L = 1, each X must lie in a different row, which can only occur if I J: (2B.2) For J/2 < L J, only one row can have L X's: there are I ways to choose that row, (L) ways to choose the columns that contain the X's in that row, and (I - 1) ways to place the remaining X's (all of these I - 1 other l's will be less than L since J - L < (J/2), SO that (2B.3) For smaller values of L, it is more difficult to derive a general, explicit expression for NL, but not impossible. For example, consider the next level of complexity,. J/3 < L < J/2. There are now patterns with two rows of L X's and I - 2 other rows with less than L X's; these patterns are counted twice in (2B.3). Furthermore, if 2L + 1 < J (which always occurs in the region J/3 , L < J/2 except when J is even and L = J/2 exactly), we must eliminate those patterns which contain L X's in one row and L + 1 or more X's in another row. The r