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1、精品论文Blind Beamforming for Noncircular SignalsYougen Xu, and Zhiwen LiuDepartment of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, ChinaE-mail: yougenxuAbstractBlind beamforming for extracting noncircular signals without a prior knowledge on the desired steering vector is c
2、onsidered in this paper. Three second-order statistic based blind schemes, termed as subspace decomposition, self-reference, and extraction power maximization methods, respectively, are proposed to extract one desired non- circular signal from a number of statistically independent circular interfere
3、nces in the presence of circular Gaus- sian noise with arbitrary and unknown correlation structure. Two joint second- and fourth-order cumulant based methods are also developed for the case of multiple noncircular interferences. One is ESPRIT (estimation of signal parameters via rotational invarianc
4、e techniques) type method. The other represents a real-valued extension of the WL-MVDR (widely linear minimum variance distortionless response). Numerical examples are shown to illustrate the performance of the proposed methods. The latest version of this online-only showing has also been submitted
5、to IET Signal Processing.Keywords: Array signal processing, adaptive arrays, noncircular signals, blind beamforming1.IntroductionBlind beamforming has found numerous important applications in radar, sonar, wireless communi- cations, and biomedical imaging, mainly because it can restore one or more d
6、esired signals from multi- ple cochannel interfering signals and noise without a prior knowledge of the array manifold and the need for training signal 1-2. In contrast, the non-blind beamforming techniques, such as MVDR (minimum variance distortionless response 3) and LCMV (linear constrained minim
7、um variance 3), require the information about the steering vector of the signal-of-interest (SOI), thereby necessitating an array calibration procedure. However, the advance calibration of the array characteristics is usually expensive and may become uncertain especially in the presence of multipath
8、 propagation and environ- mental changes.The very earlier attempt for blind beamforming seems to be achieved by assuming structural proper- ties of the array manifold. In such methods, the direction-of-arrival (DOA) of the desired signal is first estimated by using direction finding techniques such
9、as MUSIC (multiple signal classification 4) and ESPRIT (estimation of signal parameters via rotational invariance techniques 5). The DOA estimate then can be used to construct the desired steering vector, after which a certain beamformer can be de- signed to extract the signal from that direction. S
10、trictly speaking, this two-step method is not a truly “blind” method since it requires also array calibration. More importantly, DOA estimation may be complex and inadequate when the array is sensitive to both DOA and signal polarization, or, very fre- quently, suffers from severe element mutual cou
11、pling. Later, new types of blind beamformers were proposed that are not based on the receiving channel structure, but instead exploit the structural char- acters of the signal waveform. A promising example is the constant modulus algorithm (CMA) that can extract signals with constant amplitude (such
12、 as phase modulated signals) 6-7. However, CMA is very sensitive to the initialization condition, and generally poorly convergent. An analytical CMA variant provided later in 8 was observed to be able to avoid such convergence problems.The cyclostationarity property of many man-made communication si
13、gnals can also be exploited for blind beamforming by processing the incoming signals at the carrier frequency (i.e., without demodula- tion). A popular cyclostationarity-exploiting blind method labeled as SCORE (spectral self-coherence restoral 9) can extract the signal at a known (or can be estimat
14、ed to alleviate the effect of cycle fre- quency perturbation caused by channel uncertainty such as Doppler shift 10) cycle frequency auto- matically. Some other interesting cyclostationarity resorted blind beamformers can be found in 11-13 (and references therein). In the literature, there are still
15、 some efforts made on the exploitation of the temporal correlation structure of the colored or nonstationary signals, see, for example, 14-15 and references therein. A fundamental and necessary requirement of the algorithms there is that the incom-This work was supported by the National Natural Scie
16、nce Foundation of China under Grant 60602037.- 29 -ing signals have distinct power spectral densities (PSDs).Note that all the aforementioned property-restoral type blind beamformers can be accomplished by using second-order statistics (SOS) alone (based on the fact that the desired signal and the i
17、nterferences are temporally separable). For the more general case of arbitrary non-Gaussian signals, a variety of blind methods based on the higher-order statistics (HOS) have been proposed 16-22 (and references therein). A striking example is JADE (joint approximate diagonalization of eigen-matrice
18、s 16), wherein second-order statistics are used to whiten the signal contribution of the received noisy data prior to cumulant based diagonalization (iterative). A major drawback of JADE is the requirement of the knowledge about noise covariance matrix. Still, it is assumed by JADE that all the inci
19、dent signals are non-Gaussian and have nonzero and unequal kurtoses. A natural cure for these shortcomings is to whiten the signal via the HOS instead, as has done in 19-20. Also, a closed-form cumulant based blind beamformer was proposed in 17 to recover a single non-Gaussian signal from multiple G
20、aus- sian interferences, while an ESPRIT-type method was suggested in 21 to avoid whitening step. Some other work on cumulant based blind beamforming can be found in 22-23 and references therein.More recently, a few efforts have been made on the exploitation of noncircularity of signals such as ampl
21、itude modulated (AM) signal and binary phase-shift keying (BPSK) signal. For example, the Ta- kagi factorizing technique was used in 24-25, based on SOS only. However, algorithms there require that the signals to be separated have distinct noncircularity rates a condition violated by many non- circu
22、lar signals encountered in communications (such as rectilinear signals). Also, JADE and ICA (in- dependent component analysis) have been extended for noncircular signal blind beamforming in 23-24 and 26-27, respectively. Still, some interesting work on DOA estimation of noncircular signals can be fo
23、und in 28-30 (and references therein). In this paper, we limit ourselves also to blind beamforming for noncircular signals. We propose three SOS based blind methods for the case of one noncircular SOI corrupted by a number of circular interferences. We further provide a new mixed-order blind algorit
24、hm for the case where both the desired signal and interfering signals are noncircular. We still extend a re- cently developed noncircularity-exploiting MVDR beamformer 31 to a real-valued form in a blind situation.The paper is organized as follows. In Section II, we formulate the problem. In Section
25、 III, we present three methods for blind beamforming in the presence of circular interferences. In Section IV, we pro- pose two mixed-order blind methods for the case of noncircular interferences. We then provide several numerical examples in Section V and finally conclude the paper in Section VI. T
26、hroughout the paper, we use uppercase and lowercase boldface letters to denote vectors and matrices, respectively. Symbols“ ”, “T ”, and “ H ” represent complex conjugate, transpose, and complex conjugate transpose, re-spectively. Furthermore, “ E ” and “ cum ” signify statistical expectation and th
27、e fourth-order cumulant,respectively.2.Problem FormulationA complex signals(t) is said to be noncircular (at order 2) ifE s 2 (t) 0 . In other words, anoncircular signal has nonvanishing conjugate correlation as well as nonzero correlation (that is,ss2 = E | s(t) |2 ). Generally,E s 2 (t) = =e j 2 ,
28、 where = is referred to as the noncircularity rate.Note that 0 = 129. For example, a BPSK signal has a noncircularity rata of 1 (completely non-circular signal). Typical examples of noncircular signal include AM, BPSK, amplitude phase-shift keying (ASK), minimum shift keying (MSK), etc 28, 29.Consid
29、er an array of N elements, with arbitrary unknown response patterns and locations. Assumejthat there are Q interfering signalss i (t), j = 1, .,Q , and a noncircular desired signal,sd (t ) , im-pinging upon the array. Further, the additive noise (either white or colored) present is assumed to be cir
30、cular with unknown covariance. With these basic assumptions, the received baseband signal at the k-th sensor can be modeled asd dxk (t) = ak ( )s (t) +Qj =1i iak (j )s j (t) + nk (t )(1)where djandiare the parameter vector (contains DOA and polarization etc.) of the desired signaland the j-th interf
31、ering signal, respectively;ak ()is the response of the k-th sensor to the signalwave-front with parameter vector ;nk (t) is the additive noise at the k-th sensor. It is assumed thatthe desired signal and the interfering signals are statistically independent, and all the signals are statis- tically i
32、ndependent of the noise. Note that the model (1) applies also to the case of multipath propaga- tion and smart jamming. Furthermore, it can be rewritten in matrix notation, as1x(t ) = x (t) Tx 2 (t) xN (t)sd (t) n1(t) i s1 (t) n2 (t)a da i a i = ( ) ( 1 ) ( Q ) + # # (2)def= As i (t) Q def= s (t ) n
33、 (t) N n (t ) = a(d )sd (t) + Aisi (t) + n(t )= As(t) + n(t )Ti i iwherea() = a1(), .,aN ()is the generalized response vector,A = a(1 ), .,a(Q ) , A =a(d ), Ai ,si (t) = si (t), ., s i (t)T . In what follows, we assume that A has full column rank, and1 Qthe noise is circular and Gaussian, that is,R=
34、 E n(t)n H (t) OnNR = E n(t)nT (t) = Owhere “ON ” denotes an N Nn zero matrix, andRn ,Rn Nare called the noise covariance matrixand the noise conjugate covariance matrix, respectively.The optimum beamforming weight vector for a so-called “informed 16” beamformer is given by1 dwopt = Rx a( ) , where
35、is a scalar for maintaining a specified response for the desired signal,and Rxis the array covariance matrix, defined asR = E x(t)x H (t) = AR AH + R(3)wheresR = E s(t)sH (t)x s nis the source covariance matrix.We address here the problem of optimum beamforming with an array of arbitrary sensors who
36、se re- sponses and locations are completely unknown (without any a prior knowledge about the steering vec- tor(s) of the desired signal(s), aiming at extracting the noncircular desired signal(s) from the circular noise plus a number of circular or noncircular interferences.3.Circular Interference Ca
37、seIn this section, we present three approaches for estimating the steering vector (instead of DOA) of the desired signal by exploiting the conjugate cumulant redundancy. These approaches make it possible for the signal-selection applications where a few noncircular SOIs must be separated from very d
38、ense interference environments.A. Subspace decomposition (SD) methodThe second-order conjugate covariance matrix (also termed as “improper- 32”, “elliptic- 28” or“pseudo- 27” covariance matrix), is defined asRx = E x(t)xT (t)= Ai E si (t)si (t)T (Ai )T + E n(t)n(t)T +d a(d )aT (d )s= d a(d )aT (d )
39、= AR AT(4)whered = E sd (t)2 0 ,Rs= E s(t)sT (t) = diag(d , 0, ., 0) . The conjugate covariance ma-trixRx has the following singular value decomposition (SVD):d d d Hwhere udRx = u ,U n is the principal left singular vector ofO v ,Vn Rx according to the largest singular value(5)d ,U n , Vncontain th
40、e left and right subordinate singular vectors, respectively. It can be verified that11ud = a(d ) , where is a nonzero scalar. Then the optimal weight of the beamformer can be de-termined according to the following criteria:min wH R w + | P w |2subject to wH ud = 1(6)w x nwhere P= U U H , is the regu
41、larization parameter, | |denotes Euclid norm. This permits an n nsoft smoothing of the eigenvalues ofmization asRx . Using Lagrange multiplier scheme, we can obtain the opti-SD xnKw = (R + P )1ud (7)kkIn practical applications, we generally do not have access to the true covariance matrix and the tr
42、ue conjugate covariance matrix. They can be estimated from the received data in a batch manner, as Rx =1 x(t )x H (t )(8) Rx K k =1K 1 K=k =1kkx(t )xT (t )(9)where K is the length of the available data samples (snapshot number). Note that RxandRx arealso known as the sample covariance matrix and the
43、 s ample conjugate covariance matrix, respectively.Furthe r, the estimates of Pnandud , denoted byP nand d, can be obtained by eigendecompos- u1 dingRx . The optimization solution then iswSD = (Rx + Pn ) u .We note that the desired steering vector can also be estimated without singular value decompo
44、sition.The above method has an element counterpart. Indeed, we have from (4)1N2 Rx (:, m) = a(d )(10)where 2N m =1is a scalar. Using (6) we may also obtainwSD .B. Self-reference (SR) methodLetr(t) = c H x (t) = c H a (d )sd (t) + cH Ai si (t ) + c H n (t )(11)where c is a control vector obeyingcH a (d ) = (cd ) 0 . For example,c = e(N ,m ) , wheree(N ,m )is anN 1vector, ande(N ,m ) (j ) = j ,m = (j m) , with() being the Kronecker delta function.Note further thatE r(t)sd (t) = E cH a (d )sd (t) sd (t ) + E c H Ai si (t) sd
