We investigate a new possibility of frequency adaptation for spread spectrum communication systems. A special index is introduced to characterize the adaptation gain referring to the nonadaptive case. An existence of frequency band limit is developed for the allocated frequency band for possible frequency maneuver. The investigation method can be adapted both to direct sequence and frequency hopping spread spectrum techniques. In the paper only the narrowband interference case is treated
 Spread spectrum (SS) communications utilization is motivated by the need for reliable information transmission even under severe environmental conditions. A remarkable property of SS communications consists in its ability to successfully combat narrowband interference and jamming [Peterson et al., 1995; Torrieri, 1985]. Practically it is realized by suppression of the interference together with the part of the information bearing signal spectrum included in the output spreaded spectrum frequency band [Milstein, 1988; Sauliner, 1992; Fazal, 1994]. In doing so the remaining part of the desired signal energy could be scarce for required signal-to-noise ratio in respect to noise background and the performance will not be sufficient. Thus the only possibility is to change the frequency position for the other position with less number of interference. Frequency adaptation enables the choice of a position for the desired signal's spectrum and to change it timely. The purpose is to occupy at each moment of time the most favorable working frequency band [Goot and Girshov, 1977; Gott et al., 1979]. Nowadays this kind of signal adaptation is one of the most powerful techniques for mitigating interference. Especially the implementation of this kind of adaptation has been found advantageous for HF communications [Baker, 1993] and trophoscatter communications, where those are used for a long time [Betts and Ellington, 1970; Biggi and Courty, 1968].
 In this paper we investigate the possibilities of frequency adaptation for communication systems using signals with spectrum wider then the bands of interference and being able to suppress one or more interference sources within the signal bandwidth. Additional frequency adaptation can be applied to all cases of SS methods (Direct Sequence, Frequency Hopping, Time Hopping) as the protected signal.
2. General Relations for SS Improvement Index
 We define as b the frequency bandwidth of the SS signal. Suppose that technical tools enable us to put the band b at any position within a band B ≥ b (as shown in Figure 1) referred to as the adaptation band, and to replace the position of b at an arbitrary moment of time. The band B is affected by narrowband interference, but the SS communications technique allows the rejection of interference that has fallen into working band b.
 We assume that the performance is unsatisfactory if the number of such interference within bandwidth b is M or more. It is evident that frequency maneuvers would allow improvement of performance, since for an inadmissible number of interference the position of b may be shifted for a new, more suitable position.
 To characterize the investigated improvement quantitatively, we introduce the following index G:
where Q(B,b) represents the probability that the performance will be allowable for the signal of frequency band b using the adaptation within band B. Note, that if B = b we obtain the nonadaptive case.
 We describe here the model of narrowband interference by a random stream of impulses in the frequency domain. We assume that distances between the interference impulses are identically distributed random variables, as far as the interference is generated by independent sources.
 Let the interference within B located in such a way, that a performance cannot be sufficient. It means that sums of any M adjacent pauses between impulses of the stream are no more than b. Evidently the minimal number of impulses necessary for this event are as follows:
where [α] represents the symbol of integer part of number α.
 Let ϑl be the duration of the interval (frequency distance) between the (l-1)-th and l-th impulses and PL(B) is the probability that the total number of impulses within B is equal to L. Denote Θ and ξm:
is the conditional density function of variables ξ1, L, ξL+1 under the given Θ, fξ,Θ(ξ1, L, ξL+1, Θ) is the joint density function of these variables, and fΘ(Θ) is the density function of variable Θ.
 As seen from Figure 1, in a general case the first and the last values ϑ1 and ϑL+1 have other density functions in comparison to density fϑ(ϑ) of pauses ϑl. Due to statistical independence of the intervals and the identity of the distributions, using the well-known relation [Walrand, 1988], we can write the density functions for the first and last pauses as follows:
where is the average duration of the intervals.
 The identity of the distributions and the independence allow us to represent the joint density of variables ϑ1, L, ϑL+1 in the following form:
Supposing conditionally that ξ0 = 0, we convert relation (4) as follows:
The expression for index G is obtained by substitution of equation (12) into (1) with using in the nominator b instead of B.
 Thus, we obtained the expression for the improvement index G in a general form. For further clarity it is necessary to specify functions PL(B) and fΘ(Θ).
3. Specifications for the Case of Poisson Interference Stream
 We receive the assumption that the stream of interference is caused by many independent sources. Thus, we use the Poisson's model of the stream, since it is obtained as an asymptotic result in the case of action of a big number of independent sources [Lloyd, 1984], that is:
Now we use the well-known representation and substitute (13) into (12). Then, after some mathematical manipulations, we obtain the following expression:
Denote Smin = [Lmin/M], S = [L/M] so that L + 1 = MS + r, 1 ≤ r < M. Now, the last relation can be rewritten in the following form:
Evaluation of the expression for the value Θ(B,b), provided by formulas (15) and (16), is performed in Appendix A. From Appendix A results we obtain:
where al = B − lb.
 The integration in expression (17) may be fulfilled, but we would obtain a very complex and hard-to-visible expression, so that form (17) is more suitable for computations.
 The expression for Q(b,b) may be obtained from (16) and (17) with B = b, but it would be simpler to compute it directly as the probability that the sum of the first M intervals is more than b. Thus . Since the variables ϑm are distributed exponentially, this probability is equal [Lloyd, 1984]:
 By substitution of equations (17)–(19) into (1) we obtain an expression for the index G. The typical results of the computations are presented in Figure 2.
 The main conclusions of our investigations for improving adaptation efficiency of spread spectrum systems considering narrowband interference are as follows:
 1. We have developed a limit value for the index G characterizing the adaptation gain referring to the nonadaptive case. The limit depends on the desired signal bandwidth b and the allowable number of narrowband interference M within b. Our results disagree with the initial intuitive notion that the better performance is achieved for a wider adaptation band B.
 2. We obtained that the less value M the wider band B is required to achieve the limit. This may be explained by the fact, that if an original spread spectrum nonadaptive system can successfully combat the interference (large M), it would require less adaptation than a system with a small number of interference M within the band b.
 3. The wider the bandwidth b, the higher the limit is, but the wider bandwidth B is required to achieve the limit. The reason is that for wider bandwidth b it would be more difficult to find an allowable position for b within B.
 4. Our results provide a quantitative estimation for the adaptive gain of the frequency adaptive spread spectrum system compared to nonadaptive systems using the same b and M.