Asymptotic distributions of estimated cyclic autocorrelations of DSSS signals and the applications
 Dan Bao^{1}Email author,
 Riheng Wu^{2},
 Jingjing Cai^{1} and
 Peng Li^{1}
https://doi.org/10.1186/168714992013141
© Bao et al.; licensee Springer. 2013
Received: 1 February 2013
Accepted: 19 May 2013
Published: 28 May 2013
Abstract
Asymptotic distributions of estimated cyclic autocorrelations (CA) of direct sequence spread spectrum (DSSS) signals are derived in this paper. The estimation follows a zeromean complex normal distribution in which the variance exhibits a cyclic thumbtack form, and the cyclic period equals the symbol period. This property of the estimated CA can be used in the detection and recognition problem of DSSS signals. The asymptotic performances of detection and recognition are carried out, and the simulations also verify the theoretical analysis.
Keywords
Introduction
Direct sequence spread spectrum (DSSS) signals are widely used in commercial and military communications for their antijamming capabilities and low probability of interception. In DSSS systems, the information signal is modulated by a pseudonoise (PN) sequence before transmission. For cooperative communications, the PN spreading sequence is known to the receiver, which is used to carry out the despreading operation and recover the information data. Because of the processing gain in the matched filtering or correlation operations, the DSSS signals can be transmitted below the noise level. In the noncooperative communication scenario, however, the receiver may have no priori knowledge of the transmitter’s PN sequence. Hence, it is difficult for a noncooperative receiver to detect and despread the DSSS signals because the PN sequence used by the transmitter is unknown to the receiver. Besides, one more important problem is the recognition of the DSSS signals, since one may usually want to know whether an intercepted signal is a DSSS signal or an ordinary pulseamplitude modulated (PAM) signal in a noncooperative condition. Also, the detection and recognition problems of DSSS signals have special significance for cognitive radio systems.
To detect a DSSS communication hidden in the noise, a method that is based on the fluctuations of autocorrelation estimators is proposed by Burel in [1, 2]. In [3], the detection, symbol period, and chip width estimation of DSSS signals are carried out based on delaymultiply, correlation, and spectrum analysis, respectively. In [4], Deng presents an autocorrelation estimationbased detection method. It is suitable for the realtime detection of DSSS signals at low signaltonoise ratio (SNR) in a cognitive radio system. An approach is proposed in [5] to detect the baseband DSSS signal with narrowband interference based on blind source separation and fluctuations of the autocorrelation second moment. An algorithm for correlationbased detection of direct sequence spread spectrum signals with direction finding, including direction filtering and narrowband interference rejection, is implemented and evaluated in MATLAB in [6].
In [7], relying upon the asymptotic normality and consistency of k thorder cyclic statistics, asymptotically optimal χ^{2} tests are developed to detect the presence of cycles in the k thorder cyclic cumulants or polyspectra. The paper [8] deals with the analytical evaluation of the asymptotic detection and false alarm probabilities of multicycle and singlecycle detectors operating in additive white Gaussian noise, which are based on the cyclostationarity properties of the signal to be intercepted.
Selfrecovering receivers for DSSS signals in multipath with unknown spreading codes are discussed in [9], wherein a zeroforcing receiver/equalizer is proposed to recover the transmitted data. In [10], a method is proposed for estimating the pseudorandom sequence without any priori knowledge about the transmitter. Only the duration of the pseudorandom sequence is assumed to have been estimated. This approach is based on eigen analysis techniques.
Notice that the autocorrelation of DSSS signals is applied to resolve the detection problem in [1–6]. However, none of them deal with a theoretical interpretation for the fluctuation of the autocorrelation of DSSS signals and the asymptotic performances. Though the cyclic autocorrelation functions of the baseband DSSS PAM signal are derived in [11], the asymptotic distributions of the cyclic autocorrelations (CA) are absent. In this paper, we consider the asymptotic distributions of the estimated cyclic autocorrelations of DSSS signals, which may have the potential applications in noncooperative communication environments. Then, the theoretical properties of the CA are applied in detection and recognition problems of DSSS signals. The remainder of the paper is organized as follows: In Section 2, we evaluate the distributions of the estimation of the cyclic autocorrelation, from which some interesting properties are found. In Section 2, we apply the result of the former section to the problem of DSSS detection and recognition, where the optimal threshold is given. Section 2 provides some simulation examples to illustrate the performance of the proposed algorithm. Finally, Section 2 concludes the paper.
Asymptotic distributions of the estimation
is not identically zero when cycle frequency α≠0, where * denotes conjugation.
where A is a received amplitude, f_{ c } is a residual carrier, θ_{ c } is an unknown carrier phase, and n(t) is a complex additive white Gaussian noise with power spectrum density N_{0}.
where n is an integer, δ_{ n } is the Kronecker delta, δ(·) denotes the Dirac delta function, mod means modulo operation, and ε^{(T)}(α;n)δ(τ−n T_{ b }) represents the estimation error which vanishes asymptotically as T→∞. The estimation error ε^{(T)}(α;n)δ(τ−n T_{ b }) was usually ignored in former literatures just as noises, while it has interesting uses in some special cases, such as in the DSSS systems, because it has nonzero values only when τ=n T_{ b }. It is due to the fact that one just considered the first nonzero term of the CAF (21) before, while the second vanishing term was rarely utilized. In our opinion, by properly making use of the vanishing term, we will show in the following part that the CAF exhibits some vanishing but useful properties in testing the presence of the DSSS signals.
where $\sqrt{T}{\epsilon}^{(T)}(\alpha ;n)$ is defined as in (30), and $\frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau )$ is defined as in (17). The first term in (31) is well discussed in [7], so we just focus on the second term. Though it vanishes asymptotically as T→∞, the second term does exist, and due to the thumbtacklike AF $\frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau )$ of p(t), it makes some special peaks appear, in a stochastic manner, in the regions n T_{ b }−T_{ c }<τ<n T_{ b }+T_{ c } and −1/T_{ b }<α<1/T_{ b }.
which has also been discussed in [8].
It is well known that the DSSS signal has a low probability of intercept because of its low power spectrum density. The cooperative receiver can recover the information using the correlation operation or the matched filter. The advantage gained through the correlation receiver is called spreading gain, while the noncooperative receiver can rarely benefit from it. One may notice that the PN sequence c(n) does not appear in (35) and that the asymptotic covariance of ${\widehat{R}}_{\mathit{\text{xx}}\ast}(0;n{T}_{b})$ is proportional to spreading gain N_{ c }, which means that noncooperative receivers can also take advantage of the spectrum spreading gain using the CAF of the intercepted DSSS. In other words, the estimation of CAF resembles the correlation operation or the matched filter in some sense when the spreading code sequence c(n) is unknown.
Applications and asymptotic performance analyses

(A1) The intercepted DSSS signal r(t) is BPSKmodulated both for the information and the PN code sequence as in (10) and (11). The pulse shape q(t) is assumed to be rectangular without loss of generality.

(A2) The PN code sequence c(n) is unknown.

(A3) The bit period T_{ b } is known. However, in practice, T_{ b } can also be estimated using the CAF of r(t), while the estimation is not discussed here.

(A4) The signal r(t) is oversampled. The sample period T_{ s } is a fraction of T_{ c }, and the sample clock is not necessarily synchronized to the PN code in practice.
and x(t) is defined as in Equation 10. Equation 36 is a typical formulation of a classification problem, but when y(t)=0, Equation 36 turns into a special case as a detection problem.
For setting a threshold for the hypothesis testing problem, the asymptotic distribution of λ will be derived next.
The notation N_{0} in (41) shows the power spectrum density of the complex additive white Gaussian noise n(t) as in (9). From (41), it is easy to find the difference between the simple PAM signal and the DSSS signal. The greater the spectrum spreading gain N_{ c } is, the more different the test statistic λ becomes.
which means that H_{1} holds if λ ≥ η, and vice versa. Since the threshold η has been set, the probability of detection can be evaluated using the distribution of λ under H_{0}, which is defined as ${P}_{D}\stackrel{\Delta}{=}Pr\{\lambda \ge \eta {H}_{1}\}$.

Step 1. Using (39), estimate the CAF of received signal r(t).

Step 2. Calculate the test statistic λ as in (38).

Step 3. Set an expected value of false alarm rate P_{ F }. Using P_{ F }, find a threshold η by checking the Gaussian distribution tables, such that P_{ F }= Pr{λ ≥ η}.

Step 4. If λ ≥ η, declare that H_{1} holds, which means that a DSSS is present in the received signal or the type of the signal is DSSS, and vice versa.
Simulations
The chip period T_{ c }=8T_{ s } and the carrier frequency f_{ c }=0.022/T_{ s }, where T_{ s } is the sample period. Set the observation interval T be 10^{4}T_{ s }. The theoretical distributions of λ getting through asymptotic analysis are represented by lines, and the experimental ones getting through simulations are represented by markers.
Conclusion
The asymptotic distributions of the estimated CA of DSSS signals are derived. The variance of the estimation resembles a series of thumbtack forms, and the cyclic period equals the symbol period. Though it vanishes as the observation time becomes infinite, the properties of the estimated CA can be used in the detection and recognition problem of DSSS signals. A good agreement is obtained between theoretical and simulation results. The simulations also show that noncooperative receivers can also take advantage of the spectrum spreading gain using the CAF of the intercepted DSSS just as the cooperative receivers do.
Appendix 1
because n(t) is a complex additive white Gaussian noise.
By using (32), substituting (52) into (51) and (50) into (49), the asymptotic covariance and relation under H_{ i }, i=0,1 can be respectively expressed as in (41).
Appendix 2
Declarations
Acknowledgements
We want to thank the helpful comments and suggestions from the anonymous reviewers. This research was supported partially by the Fundamental Research Funds for the Central Universities (grant no. K5051202002).
Authors’ Affiliations
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