n-zero symbol b occurs in one period of an m-sequence {bn} exceeds the number No of times 0 occurs in one period, by 1. Nb = No + 1, for all b # 0. (5.60) This property, which states that symbols occur as equally often as possible within one period, is referred to as the balance property. Its proof follows directly from counting the number of times that a state L-tuple begins with a specific symbol in one cycle of the state sequence. Property R-2. Let {bn} be an m-sequence of elements from GF(q) with linear span L. The number Na of positions within one period of {bn} at which a J-tuple aja2 aj occurs, is given by fora=0,1JL (5.61) This generalization of the balance property also follows from state count- ing arguments. The probability of obtaining a particular J-tuple a in J independent trials, each trial having q possible equally likely outcomes, is given by q-1. If an Pseudonoise Generators Table 5.7 Comparison of binary m-sequence J-tuple statistics with Bernoulli sequence probabilities for L > J. Bernoulli m-sequence statistics No/N, N /N J-tuple Probability .467,.533 .492,.508 .498,.502 .200,.267 .238,.254 .247,.251 1.067,.133 .111,.127 .122,.125 m-sequence of GF(q) elements is to have the appearance of such a random process's sample function, it is necessary that the time-average statistic NaIN of the J-tuple event a be as close to q-1 as possible. For J , Property R-2 gives (5.62) for the all-zeros J-tuple (the worst case). Clearly No/N rapidly approaches q-1 as L increases (see Table 5.7). 5.4.2 The Shift-and-Add Property The next property is the foundation for several important characteristics of m-sequences. Property R-3. Let {bn} be an m-sequence over GF(q) with linear span L. Then for any T, T # 0 mod q1 - 1, the difference of the m-sequence {bn} and its T-shift {bn++} is another shift {bn++'(7)} of the same m-sequence. That is, (5.63) where T' (T) is defined for all T # 0 mod q1 1. Proof. The m-sequence and its T-shift both satisfy the linear recursion of degree L determined by its characteristic pol