IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 6, JUNE 2004
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Anti-Jamming Filtering in the Autocorrelation Domain Rueywen Liu, Life Fellow, IEEE, and Rendong Ying
Abstract—An anti-jamming filtering technique is presented that it fully eliminates the jamming (or noncooperative) signal and its performance is noise-independent. This technique uses multiple receivers; a filtering process explores the nonoverlapping properties of the signals in the autocorrelation domain; and a matching of the input and the output statistics. A limited simulation study supports the theory. Index Terms—Anti-jamming filter design, autocorrelation matching, blind channel equalization and blocking, wireless communication.
I. INTRODUCTION
T
HE main objective of filtering is among received signals, to accept those that are desired and to reject those that are undesired. Most filtering techniques explore the nonoverlapping properties of these signals, mainly in the frequency domain facilitated by Fourier transform and analog/digital filters [1], or in the time-frequency domain facilitated by the wavelet transform and filter banks [2]. The performance of a filtering technique depends heavily on the domain that is chosen. For example, the filtering of two sinusoidals with different frequencies can be difficult in the time domain, but simple in the frequency domain, because they are overlapping in the time-domain, but nonoverlapping in the frequency domain. In the case of jamming signals, it is usually heavily overlapping with the desired signal in both time and frequency domain. Therefore, a filtering process in these domains can be difficult. Even the frequency-hopping technique, which creates the nonoverlapping in the time-frequency domain, cannot avoid the partial-band interference due to the jamming signal [3]. The jamming signal cannot be fully eliminated by these techniques. Recently, some filtering techniques explore the nonoverlapping properties in the “statistics domain.” For example, one method explores the situation when the desired signal is cyclostationary while the jamming signal is not [4]. Another method explores the situation when the jamming signal is temporally correlated while the desired signal is i.i.d. [5]. In both cases, the jamming signal can be fully eliminated if the noise power can be correctly estimated. In general, these methods are noise-dependent.
Manuscript received August 21, 2003; revised October 29, 2003. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Richard C. Kavanagh. R. Liu is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail:
[email protected]). R. Ying is with the Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/LSP.2004.827950
In this letter, we present a filtering technique by which the jamming (or noncooperative) signal can be fully eliminated and its performance is noise-independent, which is fully supported by our simulation. This technique explores the nonoverlapping property of these signals in the domain of autocorrelation functions. It is based on the Principle of Autocorrelation Matching (see [6] and references therein). We have extended the principle to the case when the signals are mutually noncooperative, and they are allowed to be correlated. A closed-form solution is also obtained. Like [4] and [5], this method is also blind and uses second-order statistics, but unlike them it is a direct method, i.e., the jamming signal is fully eliminated without first identifying the channel. Hence, the computation complexity is reduced. II. THE MODEL Consider a simple model with two receivers given by (1) where desired signal; jamming signal; additive noises; received signals; coefficients of the channel matrix H. The jamming signal can be intentional or unintentional such as the interfering signal from neighboring cells. It is usually noncooperative with the desired signal and, hence, training signals are not accessible. Throughout this letter, we assume that , , and are zero-mean and stationary, and that the noises and the desired and jamming signals are uncorrelated. However, we allow the correlation between the two noises and between the signals and . For simplicity, we consider the real case. Its extension to the complex case is straightforward. Consider a anti-jamming filter of the form (2) Then, the composite equation is given by (3) where
and
. When the channel matrix is known, methods such as optimal combining [7] can be used to achieve optimal solutions.
1070-9908/04$20.00 © 2004 IEEE
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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 6, JUNE 2004
However, since the part of the channel matrix associated to the jamming signal is not known nor can be estimated due to the fact that the jamming signal is noncooperative, alternative criteria needed to be considered. In this letter, the anti-jamming performance is characterized by the criterion
the noises are white. When the noises are not white, then we . It can be computed if its distribution is known, have or estimated if there is a duration of time when the signals are absent. Under the above assumption, (5) becomes
(4)
(7)
with which (3) becomes
As such, the jamming signal , in theory, is fully eliminated, and the desired signal is retained up to a sign factor. We will show that the deterministic criterion (4) can be fully achieved under any noisy environment, i.e., the anti-jamming performance is noise-independent. Although the existence of the anti-jamming filter is assured when the channel matrix is nonsingular, the design of such a filter is nontrivial because the channel coefficients are usually not known. Training signal techniques cannot be used to identify the channel because it is not accessible, due to the presence of the (noncooperative) jamming signal. We propose a blind technique under which such a filter can be obtained directly without first identifying the channel matrix. This technique extends the Principle of Autocorrelation Matching [6] to the case when the jamming signal is noncooperative with the desired signal and to the case when the signals are allowed to be correlated. A closed-form solution for the filter is also obtained. III. FILTERING IN THE AUTOCORRELATION DOMAIN The autocorrelation of
The half interval is like a passband, with which the noise is completely “filtered out.” We will show the following theorem. Theorem 1: Let the channel matrix H be nonsingular, and for the noises exist. If there exists a set of let the cutoff lag in the sampling points autocorrelation domain for which the correlation matrix
.. .
.. .
(8)
.. .
satisfies one of the following two rank-conditions. i) If and are uncorrelated1 (9) ii) If
and
are correlated (10)
Then, the Anti-Jamming Criterion (4) is satisfied if and only if the autocorrelation matching condition is satisfied, i.e., the autocorrelation of the output matches that of the desired signal , in the set
can be obtained from (3)
(11) (5)
where , and similarly for , , and ; and . It consists of two parts; one that related only to the signals and as shown in the first three terms, and one that related to the noises as shown in the denote the length of , i.e., last term. Let
where . Proof: Substituting Anti-Jamming Criterion (4) to (7) yields (11), i.e., the necessity condition is established. For sufficiency, imposing (11) to (7) yields
(12) or, in a matrix form
and similarly for others. Definition 1: The lag noises if
.. .
is said to be a cutoff lag for the (6)
We assume the existence of for which and . When the noises are white, i.e., , this assumption is equivalent to the case when both the jamming signal and the desired signal are nonwhite. Since the jamming signal is required to have high energy in a limited bandwidth, it is usually nonwhite. The desired signal can be designed to be nonwhite. Therefore, this assumption can be easily fulfilled when
.. .
.. .
(13) Either rank-condition (9) or (10) establishes the Anti-Jamming Criterion (4). The theorem is proved. When the signals are randomly chosen, Condition (9) or (10) is likely to be satisfied if the number of nonzero rows of is large enough, which can be fulfilled by the design of alone; with . for example, 1This
condition can be replaced by a weaker one: [r ( ) +
r ( ), for some .
r ( )]
=
LIU AND YING: ANTI-JAMMING FILTERING IN THE AUTOCORRELATION DOMAIN
In summary, we have shown that under some minor conditions, when the autocorrelation matching condition (11) is fulfilled, the jamming signal can be fully eliminated, and the anti-jamming performance is noise independent. This theory is supported by the simulation to be shown later. IV. FILTER DESIGN IN THE AUTOCORRELATION DOMAIN A closed-form solution for the Autocorrelation Matching Condition (11) is presented here. For simplicity, only the outline of the process is presented here and its detail can be easily worked out and is omitted. Equation (11) is a set of simultaneous quadratic equations of the form: for (14) An objective function can be used to find the solution
(15) The minimal point can be found from the derivative of (15), i.e., (see (16) at the bottom of the page) where the coefficients and are homogeneous polynomials of , and of degree 2. Let . Then, (16) becomes
Since according to Criteron (4), after eliminating (17) becomes
(17) ,
which can be rearranged into a polynomial form of degree 4 (18) Hence, a closed form solution for can be found. Among the four solutions, the one that minimizes the objective function (15) is the correct one, and from which the solution can be found. V. SIMULATION The purpose of the simulation has two parts: one is to verify Theorem 1 with long data length, where is correctly estimated, and the other is to show its applicability to
527
. The Anti-Jamming Index (AJI) short data length with is used to test the filtering efficiency (19) It is one when the jamming is completely blocked and thus, Theorem 1 is verified. Since AJI does not take into account of the noise power, we also show the BER-plots for comparison. In this simulation, the channel matrix is chosen as for simplicity. The noises are Gaussian, white, uncorrelated, and with same power . As such, the power of the composite noise is shown to be by (3). The signals and are generated by the system given in Fig. 1. They are heavily correlated, and also heavily overlapping in both time and frequency domain. . The power ratio between and is show to be In view of , the signal-to-noise ratio is given by . In the following simulations, varies from 1 to 9 and SNR varies from 0 to 10 db. The matching in (11) of the . Monte Carlo simulaautocorrelations are at tion is run 100 times. are given in Fig. 2. The case for data length The AJI-plots shown in Fig. 2(a), as Theorem 1 predicted, are for all bunched together at one; specifically, SNR. The BER-plots are given in Fig. 2 (b). The case for zero jamming power is also plotted as shown by the dashed line. It is interesting to note that all the plots are bunched together and they all are close to the plot for zero jamming power. It shows that the BER-plots are essentially unaltered with or without jamming signals. This is interesting because the coding–decoding system embracing the equalized system would perform the same with or without the jamming signal. Next, we study the effect of short data length. When , the autocorrelations of signals and noises are no longer the same as the ideal ones. As such, we set our AM filter matching the autocorrelation of the actual desired signal. Similar plots are given in Fig. 3(a) and (b), and they show similar results. The AJI-plots are still bunched together around 1, except for the case when the rank condition of the correlation matrix (8) for the set is nearly violated. VI. CONCLUSION We have presented an anti-jamming filtering technique in the autocorrelation domain. By exploring their nonoverlapping properties, it is proved that the jamming (or noncooperative) signal can be fully eliminated and the anti-jamming performance is noise independent. The limited simulation and fully supports the theory within the range . This method seems to be jamming-proof for overlaps in the time and/or frequency domain. It may be
(16)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 6, JUNE 2004
Fig. 1. Pre-filters that generate signal
s
and jamming
j
.
When this signal processing technique is applied to communication systems, the desired signal can play the rule of training signals, as shown in WCDMA [8]. Furthermore, because our technique is frequency independent, it retains the advantages of frequency reuse [9]. This anti-jamming filtering technique has also the potential to be extended for multiple-access communication systems with multiple paths, as shown in [6] and [8]. ACKNOWLEDGMENT The authors wish to acknowledge constructive discussions with T. Kailath and J. A. Nossek. REFERENCES
Fig. 2. (a) AJI-plots and (b) BER-plots for
N = 10 000.
Fig. 3.
N = 500.
(a) AJI-Plots and (b) BER-plots for
vulnerable, according to Theorem 1, to the violation of the rank conditions of the correlation matrix (8). Such violation may be mitigated by selecting a larger set , and hence taller the matrix (8).
[1] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989. [2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. [3] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [4] A. Chevreuil, F. Desbouvries, A. Gorokhov, P. Loubaton, and C. Vignat, “Blind equalization in the presence of Jammer and unknown noise: Solutions based on second-order cyclostationary statistics,” IEEE Trans. Signal Processing, vol. 46, pp. 259–263, Jan. 1998. [5] A. Belouchrani and M. G. Amin, “A two-sensor array blind beamformer for direct sequence spread spectrum communications,” IEEE Trans. Signal Processing, vol. 47, pp. 2191–2199, Aug. 1999. [6] R. Liu, H. Luo, L. Song, B. Hu, and X. T. Ling, “Autocorrelation-a new differentiation factor for random multiple access wireless communication,” in Proc. IEEE Int. Symp. Circuits and Systems , vol. III, Phoenix, AZ, May 2002, pp. 643–646. [7] J. H. Winters, “Optimal combining in digital mobile radio with cochannel interference,” IEEE J. Select. Areas Commun., vol. SAC-2, pp. 528–539, July 1984. [8] L. Song, B. Hu, R. Liu, and X. T. Ling, “A new multiplex-access scheme based on the diversity of autocorrelation,” in Proc. IEEE Int. Conf. Communications, vol. 1, 2003, pp. 720–724. [9] A. J. Viterbi, CDMA: Principle of Spread Spectrum Communication. Reading, MA: Addison-Wesley, 1995.
