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Asymptotic distributions of estimated cyclic autocorrelations of DSSS signals and the applications
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 141 (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.
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
Consider a complexvalued continuous time series x(t) with a zero mean. It exhibits widesense cyclostationarity if the cyclic autocorrelation function (CAF)
is not identically zero when cycle frequency α≠0, where * denotes conjugation.
A linear timevariant system with inputoutput relation [12]
where x(t) is the input, y(t) is the corresponding output, and h(t,u) is the impulse response function, is called a linear almostperiodically timevariant (LAPTV) system if the timevariant impulse response function can be expressed as the Fourier series expansion [12]
where G is a countable set. The CAF of the output for the LAPTV system can be expressed as the convolution of the input CAF R_{x x∗} and the ambiguity function of h(t,u) [12]
where \underset{\tau}{\otimes} denotes convolution with respect to τ, and r_{12}(α;τ) denotes the ambiguity function of h(t,u)
Equation 4 can be specialized to the case of the linear timeinvariant systems for which the coefficient of the impulse response function is given by
Hence, from (3) one has [13]
where δ_{ σ } is the Kronecker delta, which δ_{ σ }=1 for σ=0 and δ_{ σ }=0 otherwise. It results in
Consider a DSSS system, where the equivalent complex baseband received signal r(t) contaminated by Gaussian noise can be expressed as
where x(t) is a DSSS PAM signal
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}.
In Equation 10, {a(k)}_{k∈Z} is an information code sequence of independent and identically distributed binary symbols taking on values ±1 with equal probability, T_{ b } is the bit period, and
is a spreading waveform, where {c(n),n=0,⋯,N_{ c }−1} is the N_{ c }length spread spectrum code sequence or called PN code sequence with c(n)∈{+1,−1}, T_{ c } is the chip period such that T_{ b }=N_{ c }T_{ c }, and q(t) is the chip pulse shape. Specifically, q(t) is the rectangular pulse, in which q(t)=rect(t/T_{ c }), where
For the convenience of analysis, Equation 10 can be rewritten as
where ⊗ denotes convolution, and x_{ δ }(t) is the ideal sampled signal
with δ(t) denoting Dirac’s delta. Now the CAF of x(t) can be expressed as by using Equation 8
where {R}_{{x}_{\delta}{x}_{\delta}^{\ast}}(\alpha ;\tau ) is the CAF of x_{ δ }(t),
and \frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau ) is the ambiguity function (AF) of p(t). The AF \frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau ) in (15) can be expressed as [11]
where
and
The AF is a useful tool to analyze the response of a matched receiver, which is often used to represent the magnitude of the matched receiver output for a coherent signal. Due to the white noiselike pseudonoise sequence of the spreading waveform, the AF of p(t) has a property of superb timefrequency concentration, which means that the energy of p(t) is concentrated in a very small timefrequency region −T_{ c }<τ<T_{ c } and −1/T_{ b }<α<1/T_{ b }. An example of the AF of the spreading waveform with N_{ c }=31 and T_{ c }=1 is plotted in Figure 1, which shows that the magnitude of \frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau ) looks like a thumbtack.
Take a closer look at the CAF of x_{ δ }(t). Consider the following estimator of {R}_{{x}_{\delta}{x}_{\delta}^{\ast}}(\alpha ;\tau ):
For the whiteness of the sequence {a(k)}_{k∈Z}, one has the following expression for {\widehat{R}}_{{x}_{\delta}{x}_{\delta}^{\ast}}^{(T)}(\alpha ;\tau ):
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.
Next, the asymptotical distribution of ε^{(T)}(α;n) will be evaluated as follows. The estimator {\widehat{R}}_{{x}_{\delta}{x}_{\delta}^{\ast}}^{(T)}(\alpha ;\tau ) defined in (20) is meansquare sense consistent [7], i.e.,
Additionally, \sqrt{T}[{\widehat{R}}_{{x}_{\delta}{x}_{\delta}^{\ast}}^{(T)}(\alpha ;\tau ){R}_{{x}_{\delta}{x}_{\delta}^{\ast}}(\alpha ;\tau )] is asymptotically complex normal [7]. From the asymptotic normality of CAF estimators that
where \stackrel{D}{=} denotes convergence in distribution, C N(μ,Γ,C) stands for a complex normal density where μ is the location parameter, Γ is the covariance matrix, and C is the relation matrix [14]. From Theorem 1 in [7], the parameters in (23) can be given as
and
where {S}_{\mathit{\text{abcd}}}^{\gamma}{(\omega )}_{\tau ,\rho}^{(\ast )} are the (conjugated) cyclic cross spectrum which is defined as [7]
The notation cum(.) in (26) represents the cumulant which defined in [7]. As in [8], the cyclic spectrum of x_{ δ }(t) can be written as
By substituting (14) into (26), using (27), the parameters of the cyclic cross spectrum can be simplified, accounting for the whiteness of the sequence {a(k)}_{k∈Z} when τ=n T_{ b } and τ≠0
and
From what that has been discussed above, we may draw a conclusion that if ε^{(T)}(α;n)δ(τ−n T_{ b }), the estimation error of {\widehat{R}}_{{x}_{\delta}{x}_{\delta}^{\ast}}^{(T)}(\alpha ;\tau ) is not identically zero and behaves in a random manner when τ=n T_{ b }, and \sqrt{T}{\epsilon}^{(T)}(\alpha ;n) follows asymptotic normal distribution that
By substituting (21) into (15), the estimation of the CAF of x(t) can finally be expressed as
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 }.
When T→∞, the CAF of x(t) can be simplified to a theoretical expression as
which has also been discussed in [8].
From (31), {\widehat{R}}_{\mathit{\text{xx}}\ast}(\alpha ;\tau ) follows an asymptotic complex normal distribution that
Since the support of \frac{}{}{r}_{p{p}^{\ast}}(\alpha ;\tau ) is on −N_{ c }T_{ c }<τ<N_{ c }T_{ c } and ε^{(T)}(α;n) is zeromean complex normal distributed, the parameters in (33) can be defined as follows:
where τ % T=τ−round(τ/T)·T. More specifically, the asymptotic distributions of {\widehat{R}}_{\mathit{\text{xx}}\ast}(\alpha ;\tau ) at some special points can be derived from (34). For instance, \underset{T\to \infty}{\text{lim}}\sqrt{T}{\widehat{R}}_{\mathit{\text{xx}}\ast}(0;n{T}_{b}) follows asymptotic normal distributions with the parameters defined as
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.
An example of the {\widehat{R}}_{\mathit{\text{xx}}\ast}(\alpha ;\tau ) is plotted in Figure 2. We notice that peaks appear in the regions n T_{ b }−T_{ c }<τ<n T_{ b }+T_{ c }, n=1,2 and −1/T_{ b }<α<1/T_{ b } just as discussed above.
Applications and asymptotic performance analyses
From the properties of the CAF estimation mentioned in the above section, it can be seen that the CAF can be used to check the presence of a DSSS, and it also has the potential of recognizing a DSSS. The following assumptions are considered:

(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.
For both the detection and recognition, a unified binary hypothesis testing problem is formulated as follows:
where y(t) is a simple PAM signal defined as
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.
From the discussion in the former section, it is seen that peaks appear in particular regions for the magnitude of the CAF of r(t). We find that the hypothesis testing problem is actually equivalent to checking whether the covariance of the CAF at τ=n T_{ b },n≠0, and α=0 is over a threshold or not. Thus, we present the test statistic as follows:
where N is the number of the lags. In (38), {\widehat{R}}_{\mathit{\text{rr}}\ast}(0;n{T}_{b}) represents the estimated CAF from the received DSSS r(t)
For setting a threshold for the hypothesis testing problem, the asymptotic distribution of λ will be derived next.
To get the asymptotic distribution of λ, it is necessary to get the asymptotic distribution of {\widehat{R}}_{\mathit{\text{rr}}\ast}(0;n{T}_{b}) first. The asymptotic distribution of {\widehat{R}}_{\mathit{\text{rr}}\ast}(0;n{T}_{b}) under H_{ i }, i=0,1 is as follows:
where {\Gamma}_{r}^{i}(0;n{T}_{b}) and {C}_{r}^{i}(0;n{T}_{b}) are the asymptotic covariance and relation under H_{ i } defined similar to (24) and (25) which can be expressed for T that is large enough as follows (see Appendix Appendix 1 for more details):
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.
Suppose that {\widehat{R}}_{\mathit{\text{rr}}\ast}(0;n{T}_{b}){\widehat{R}}_{\mathit{\text{rr}}\ast}^{\ast}(0;n{T}_{b})n=1,2,\dots ,N is statistically independent and identically distributed. Therefore, based on the central limit theorem, when N is considerably large, the distribution of λ approaches Gaussian distribution, i.e.,
with
and
where the derivation of the variants uses the fourth moment of {\widehat{R}}_{\mathit{\text{rr}}\ast}(0;n{T}_{b}) (see Appendix Appendix 2 for more details). Particularly, when y(t)=0, Equation 36 is reduced to a simple detection problem, which can be used to decide whether a DSSS signal exists in the intercepted signal or not. In this case, the parameters of (42) can be reduced to a simple detection version as
and
By following the conclusion of (42), we present a DSSS detector based on a constant false alarm rate approach. A threshold η should be worked out according to a probability of false alarm which is defined as {P}_{F}\stackrel{\Delta}{=}Pr\{\lambda >\eta {H}_{0}\}. Because of the distribution of λ as in (42), η can be found by searching for the Gaussian distribution tables, such that P_{ F }= Pr{λ ≥ η}. Then, the detector is given as
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}\}.
Therefore, the detection algorithm for the presence of the DSSS is implemented using the following steps:

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
To verify the accuracy of asymptotic Gaussian distributions for the test statistics of the recognition problem where y(t) is given by (37), histograms of estimated λ for simulated signals are plotted in Figure 3, where they are compared with the corresponding theoretical values. The DSSS signals are generated using (10) with N_{ c }=31, A=1, N=100 and for the SNR being fixed at −10 dB, where the SNR is defined, in decibels, as
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.
The detectability of the presented test statistics is investigated in terms of the receiver operation characteristic. The probabilities of recognition P_{ R } are plotted versus the probability of false alarm P_{ F } for the SNRs from −12 to −15 dB in Figure 4 where the theoretical curves are represented by lines, while the experimental results are represented by markers. The lowest false alarm probability in the simulation in Figure 4 is 10^{−3}, so from the experiments, one can see that the recognition probability is above 0.9 when P_{ F }=10^{−3} and the SNR=−13 dB.
When the PAM signal y(t)=0 in the hypothesis testing problem of Equation 36, the problem becomes a detection problem. The probabilities of detection P_{ R } versus the probability of false alarm P_{ F } are plotted in Figure 5 for the SNRs from −12 to −15 dB. The condition in Figure 5 is the same with that in Figure 4, while the only discrimination is the absence of y(t) in Figure 5. As we can see from the comparison of Figures 4 and 5, the detection probabilities are slightly greater than the recognition probabilities for the absence of y(t) in the detection problem.
Numerical experiments are carried out to evaluate the influence of the summation number of the lags N on the performance of the DSSS recognition. The results plotted in Figure 6 demonstrate that the probability of recognition P_{ R } increases rapidly as N becomes larger, and it becomes almost 1 at a moderate number of N for which the computational load will not be dramatically unacceptable. When SNR becomes lower, the situation becomes worse in which increasing N will not make the P_{ R } increase rapidly. Thus, it is necessary for one to choose a moderate N to achieve an optimal tradeoff between the performance and the computational load. The computational load of the algorithm is the order of [N T(1+ log2T)] when the FFT is used.
A simulation is also performed to evaluate the effect of the carrier frequency f_{ c } on the recognition problem of DSSS. In Figure 7, the square, asterisk, and circle markers represent P_{ R } when f_{ c }=0,0.022 and 0.053/T_{ s }, respectively. The simulation results in Figure 7 show that f_{ c } cannot affect P_{ R }, which verify the theoretical analysis of (42).
Next, the simulation is concerned with the asymptotic performance of the proposed test statistics. Figure 8 shows experimental probabilities of recognition when T=256, 1,024, 4,096, and 8,192 with respect to theoretical values. Note that when the observation interval T increases, the experimental probabilities rapidly approach the theoretical ones. The simulation demonstrates that it does not need a very large T to get a good performance in applications.
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
Using the superimposition of the two independent zeromean complex time series (r(t)=x(t)+n(t)), the cyclic cross spectrum can be expressed as [8]
Each quantity in (49) can be evaluated and expressed as
because n(t) is a complex additive white Gaussian noise.
The conjugate cyclic cross spectrum can be expressed as
where each quantity can be evaluated and expressed as
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
The characteristic function of a zeromean complex Gaussian variable z\stackrel{D}{=}\mathit{\text{CN}}\left(0,\Gamma ,C\right) is given by [14]
The relation between the m th moment of a complex random variable and the characteristic function is given in [15]. The fourthorder moment of z can be achieved by
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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).
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Bao, D., Wu, R., Cai, J. et al. Asymptotic distributions of estimated cyclic autocorrelations of DSSS signals and the applications. J Wireless Com Network 2013, 141 (2013). https://doi.org/10.1186/168714992013141
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DOI: https://doi.org/10.1186/168714992013141
Keywords
 Asymptotic Distribution
 Power Spectrum Density
 Impulse Response Function
 Ambiguity Function
 Cognitive Radio System