ieee transactions on signal processing, vol. 52, no. 6

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 6, JUNE 2004 1549

Super-Exponential Blind Adaptive BeamformingKehu Yang, Member, IEEE, Takashi Ohira, Fellow, IEEE, Yimin Zhang, Senior Member, IEEE, and

Chong-Yung Chi, Senior Member, IEEE

Abstract—The objective of the beamforming with the exploita-tion of a sensor array is to enhance the signals of the sourcesfrom desired directions, suppress the noises and the interferingsignals from other directions, and/or simultaneously provide thelocalization of the associated sources. In this paper, we present ahigher order cumulant-based beamforming algorithm, namely,the super-exponential blind adaptive beamforming algorithm,which is extended from the super-exponential algorithm (SEA)and the inverse filter criteria (IFC). While both SEA and IFCassume noise-free conditions, this requirement is no longerneeded, and all the noise components are taken into account inthe proposed algorithm. Two special conditions are derived underwhich the proposed blind beamforming algorithm achieves theperformance of the corresponding optimal nonblind beamformerin the sense of minimum mean square error (MMSE). Simulationresults show that the proposed algorithm is effective and robust todiverse initial weight vectors; its performance with the use of thefourth-order cumulants is close to that of the nonblind optimalMMSE beamformer.

Index Terms—Adaptive array processing, blind beamforming,high-order cumulants, inverse filter criteria (IFC), super-exponen-tial algorithm (SEA).

I. INTRODUCTION

THE concept of the adaptive antenna [1] was first proposedmore than three decades ago. Since then, the adaptive array

or, in general, adaptive beamforming techniques, have beenwidely investigated for various purposes [2], [3], [20], [26]. Ina variety of applications, the objective of the beamforming isto enhance the signals of the sources from desired directions,suppress the noises and the interfering signals from otherdirections, and/or simultaneously provide the localization ofthe associated source [20]. The optimal beamformer (or spatialfilter), in the sense of minimum mean square error (MMSE)criteria, is specified by the Wiener–Hopf equation [2], [3],provided that the reference signal waveform or, alternatively, the

Manuscript received October 28, 2002; revised June 24, 2003. This work wassupported in part by the Telecommunications Advancement Organization (TAO)of Japan. The associate editor coordinating the review of this manuscript andapproving it for publication was Dr. Fulvio Gini.

K. Yang was with ATR Adaptive Communications Research Laboratories,Kyoto 619-0288, Japan. He is currently with the National Key Laboratory ofRadar Signal Processing, School of Electronic Engineering, Xidian University,Xi’an 710071, China (e-mail: [email protected]).

T. Ohira is with ATR Adaptive Communications Research Laboratories,Kyoto 619-0288, Japan (e-mail: [email protected]).

Y. Zhang is with Center for Advanced Communications, Villanova University,Villanova, PA 19085, USA (e-mail: [email protected]).

C.-Y. Chi is with Department of Electrical Engineering, National Tsing HuaUniversity, Hsinchu, Taiwan 30013 R.O.C. (e-mail: [email protected],[email protected], [email protected]).

Digital Object Identifier 10.1109/TSP.2004.827194

spatialsignature(spatial responseoftheassociatedantennaarray)of thedesiredsource isavailable.However, it isoftenencounteredin practical situations that the duration of the reference signal isnot sufficiently long or that the reference signal is not availableat all. The former situations happen, for example, in manymobile communication applications where the length of pilotsignals is limited to maintain a high communication capacity.The later situations arise in communication countermeasureand various passive radar and sonar applications. In suchsituations, adaptive algorithms based on the MMSE criterioncannot work properly. As a result, it becomes increasinglyimportant to develop blind adaptive beamforming algorithmsin the absence of a reference signal.

The constant modulus algorithm (CMA) is a well-knownalgorithm originally developed for blind equalization [15], [16]and has recently been used for blind beamforming [17]. Insteadof using training signals, the CMA uses the constant amplitudeproperty of the desired signals. Analyses reported in [18] and[19] have shown that, in certain cases where, for instance,the source signals are of high signal-to-noise ratio (SNR)and are not closely spaced, the steady-state and convergenceperformances of the CMA-based blind beamforming algorithmare very close to those obtained from the correspondingnonblind ones. However, the CMA-based blind beamformersusually incur considerable performance degradation whenthese conditions are not satisfied.

Applying higher order statistics, which are usually describedin terms of cumulants [7], or cyclostationary properties ofsignals to blind beamforming is considered to be a new trendto effectively make use of the inherent properties of thesignals [11]–[14]. However, the methods reported in the openliterature so far are only applicable to certain specific signalenvironments. For example, the kurtosis (fourth-order cumu-lant) maximization algorithm [11] globally converges only innoise-free situations. The fourth-order cumulant-based blindbeamforming approach [12], [13] assumes that the interferingsignals are Gaussian distributed, that incoming signals arecoherent, or that the incoming signals are independent andwith the identical fourth-order cumulants. On the other hand,to apply cyclostationarity in blind adaptive beamforming, apriori knowledge of the exact cyclic frequency of the desiredsignal is required [14]. Adaptive beamforming algorithmsbased on cyclic correlation do not work properly when thecyclic frequency of the desired source signal is not accuratelyestimated [14].

The recent development of the cumulants-based blind decon-volution algorithms, namely, the super-exponential algorithm(SEA) [4], [6] and the inverse filter criteria (IFC) [5], [8], [9],opened a new avenue for blind deconvolution and equalization

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1550 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 6, JUNE 2004

[4]. The extensions of SEA [6] to multi-input multi-output(MIMO) blind deconvolution problems proved to be effectivefor blind equalization in multiuser wireless communications[10], [22]–[24]. All the formulations of these single-inputsingle-output (SISO) or MIMO blind deconvolution problemsmake the following assumptions:

1) infinite length of the overall channel (convolution of thechannel and the equalizer);

2) finite length of the FIR filters;3) the absence of noise.

The noise-free assumption is basically justified by using a high-input SNR. According to [6], [10], and [22]–[24], an implicitcondition required for formulating the SEA is that the numberof independent sources (which also include independent inter-symbol interference components, if there are any) must exceedthe degrees of freedom (DOFs) of the system [3]. Violation ofthis condition will result in failure to apply SEA because thecorresponding correlation matrix is rank deficient and, there-fore, noninvertible [6], [22], [23]. This is a strict constraint andis in conflict with the well-known requirement that the systemDOFs must exceed the number of independent sources for effec-tive interference suppression. Situations with the number of theindependent sources being less than either the number of arraysensors or the number of the total taps of the space-time FIR fil-ters are often encountered in various communications and radarapplications. In spite of such importance, applications of SEAto such practical scenarios have not been well investigated thusfar.

Another very important factor to be considered is the systemnoise, which highly limits the system performance. In practice,noise is always present, and the input SNR could be moderate oreven low (examples of such situations include countermeasurein communications, radar, and sonar, etc.). This fact underscoresthe importance of developing new formulations of the SEA withthe noises taken into account.

The purpose of this paper is to formulate an SEA-basedblind beamforming algorithm with the consideration of arraynoise and interference signals. The proposed algorithm worksrobustly, independent of whether the number of independentsources is more or less than the DOFs of the array system.Without loss of generality, we confine our formulation to theclassical adaptive beamforming for independent non-Gaussiansource signals.

The main contributions of this paper are three fold. First,this paper presents a super-exponential blind beamforming al-gorithm with the consideration of both noise and interferencesignals. Second, two special conditions are derived for the algo-rithm to converge to the optimal beamformer specified by theWiener–Hopf solution. Third, this paper proves that the inversefilter criteria [8], [9] can be used as the objective function for theproposed beamforming algorithm, and a complete blind beam-forming algorithm is subsequently obtained.

The paper is organized as follows. In Section II, we formulatethe super-exponential blind beamforming algorithm and provethe two special conditions. Section III reveals the relationshipbetween the proposed algorithm and the existing IFC, and thecomplete blind beamforming algorithm is then derived. Sec-

tion IV presents simulation results of the conventional applica-tions. Section V concludes this paper.

II. FORMULATION OF BLIND ADAPTIVE BEAMFORMING

A. Formulation of Nonblind Adaptive Beamforming

Let us consider an antenna array of elements withincident non-Gaussian sources. The signal vector

at the antenna array is expressed as

(1)

where and (the su-perscript “ ” denotes transpose) are the array steeringvector and the array noise vector, respectively, and

is a diagonal matrixdenoting array channels’ complex gains (gains and phases).The steering vector, for example, of a half-wavelength spacedlinear array can be expressed as

with the phase reference point taken at the center of the array.To simplify the notation without loss of generality, hereafter, letus use instead of .

The following assumptions are made for the sources’ signalsand the employed antenna array.

A1) The source signals , are indepen-dent, stationary, and non-Gaussian processes with zeromean and variance .

A2) The noise vector is zero-mean, spatially white(but not limited) with and

, where the superscript“ ” denotes conjugate transpose, expresses thenoise power at each element, and is theidentity matrix. is uncorrelated with all thesource signals.

A3) The steering vectors of all the associated sources arelinearly independent when .

For the purpose of later use, we denote ,, , ,

, and ,and we define

(2)

and

(3)

Then

(4)

YANG et al.: SUPER-EXPONENTIAL BLIND ADAPTIVE BEAMFORMING 1551

With straightforward manipulations, the signal vector of (1) canbe expressed as the following linear vector model:

...

...

(5)

In order to avoid confusion between the subsequent SEA ex-tension and that presented in [6], [10], and [22]–[24], we high-light the following two cases for ease of understanding.

Case 1) . This represents the kind of typicalcases in communications, radar, and sonar applica-tions.

Case 2) . This is the case where the associated systemdoes not have enough DOFs.

According to [6], [10], and [22]–[24], an implicit condition nec-essary to formulate the exsiting SEA is that the number of in-dependent sources (including independent intersymbol compo-nents, if there are any) is greater than the DOFs of the system [3].It is obvious that Case 2 satisfies this requirement. The above-mentioned extension [6], [10], [22]–[24] can be directly appliedto Case 2, regardless of whether the system noises are consid-ered or not.

Remark 1: Note that in the absence of noises and for(i.e., the assumption A2 in [23]), the signal model expressed by(1) and (5) is a special case of [23, (1) and (2)] by setting .It is clear that in (5) is of full column rank if asthe noises are absent.

In this paper, we concentrate on Case 1 with the presence ofnoise. In (5), the dimension of the combined signal vectoris 1, which, regardless of the value of and , isalways greater than .

Using assumptions A1 and A2, the correlation matrix of thearray signal vector of can be expressed as

(6)

The output of adaptive antenna is formulated as

(7)

where is the complex weight vector. De-fine

(8)

as the gain vector of all the signals and the noises. Then, theoutput of the adaptive antenna can be rewritten as

(9)

Note that there are only weights that are actually controllable,and the gains corresponding to all the signals and the noisesdepend on the controllable weights.

Assume that , is the source signal ofinterest. Using assumptions A1 and A2, the mean square error(MSE) between and the output of the adaptive antennacan be expressed as

MSE

(10)

Minimizing the MSE with respect to results in the optimalweight vector, represented by the following Wiener–Hopf equa-tion, which is called the optimal adaptive beamformer or MMSEbeamformer [2], [3], [26]:

(11)

where

(12)

where the superscript “ ” denotes complex conjugate.

B. Super-Exponential Iteration Principle

Without loss of generality, we present the super-exponentialiteration principle in the same manner as that of [6] with the useof a finite dimensional complex gain vector . Define

(13a)

(13b)

where , and is

the norm of . Substituting and into(13a) and (13b), we obtain the following iteration equation:

(14)In the case that , ,

, for , we have

(15)

When and , if , , then,as , goes to

(16)


From (15), it is seen that is forced to super-exponentially con-verge to the desired response, i.e., the magnitude of the leadingcomponent approaches 1, and the others approach 0. For moregeneral cases, let us reconsider the iteration procedure with a fi-nite-modulus complex weighting sequence, and then, (13a) and(13b) become

(17a)

(17b)

where , , and . For, the equation corresponding to (14) becomes

(18)Similar to (15), in the case where ,

, for , we have

(19)

When , (19) can be rewritten as

(20)

The convergent behavior of the above iteration sequence is de-scribed by the following theorem.

Theorem 1: For an arbitrary weighting sequence with ele-ments of finite modulus, i.e., , ,

, the conditions for the iterative updated vector(20) to super-exponentially converge to a vector that the magni-tude of the leading component equal to 1 and the other compo-nents equal to 0, i.e.,

(21)

are , and

(22)

The proof of Theorem 1 is given in Appendix A.

C. Formulation of Blind Adaptive Beamforming

It is seen that for , the associated gain vector is updatedby (8) with a given weight vector , i.e.,

(23)

On the other hand, the gain can be directly updated

based on , i.e., ,

in accordance with (17a) without .This implies that there are evident differences between of(23) and of (17a). However, due to the attractive propertyof the super-exponential convergence shown in Theorem 1, itis desired that the gains , be updatedunder (17a) and (17b). The best way to solve the problem is tocombine the iteration procedures of (17a) and (17b) with that of(23) and then to minimize all the differences. This leads to thefollowing constrained minimum least square (LS) problem, i.e.,

subject to (24)

Define

(25)

Then, substituting (23) into (24) yields

subject to

(26)With the use of the method of Lagrange multipliers, the solutionfor can be derived straightforwardly as

(27)

Define

(28)

and with the definitions of (6) and (27), (28) can also be ex-pressed as

(29)

where can be estimated from the samples (or array snapshots)of , and will be estimated subsequently through ahigh-order cumulant. Regarding (26), (27), and (29), we havethe following remark.

Remark 2: As mentioned in Remark 1, in the absence ofnoise, is of full column rank when . In this case,the correlation matrix is rank-deficient. This implies that insuch a case, it is difficult to obtain an analytical solution forthe blind beamformer from (26). As a result, the iterationprocedure given by [23, (39) and (40)] becomes inapplicable.


Therefore, it is necessary to take the noises into account, andtreating the noise components as independent sources is a con-venient and effective way to solve the problem for Case 1.

Next, we present the method of estimating . First of all,let cum denote the joint cumulant of randomvariables , and define

cum

cum (30)

where is the output of the beamformer at the thiteration, i.e.,

(31)

Using the linearity property of cumulants [7], i.e.,cum cum , and substituting(31) and (5) into (30), (30) can be rewritten as (32), shown atthe bottom of the page. Define

cum (33)

as the cumulants of order of all sources. From (25),(28), and (32), we conclude that

cum (34)

According to (31), can be estimated by and thedata samples .

Equations (29), (31), and (34), rewritten as

(35a)

cum (35b)

(35c)

define the super-exponential blind beamfoming algorithm. A re-mark on it is given as follows.

Remark 3: It is noted that the above blind super-exponentialbeamfoming algorithm is valid for and can be appliedregardless of the array configurations and the number of non-Gaussian (i.e., nonzero cumulants) sources, which, therefore,is very important to diverse applications, e.g., communications,radar, sonar, etc.

When the noises are Gaussian distributed, then forand , equals zero [7]. In this case, thebeamforming weight vector given by (35a) can be written as

(36)It is evident from (36) that the iterative weight vector is a linearcombination of the optimal weight vectors corresponding to allthe source signals. Furthermore, substituting (36) into (8) yields

(37)

where

(38)

is the gain vector of the optimal adaptive antenna with respectto the th source, and

(39)

It is obvious that the optimum is also a linear combinationof .

From (36) and (37), it is seen that the convergent beamformerdepends on the initial vector , , input SNRs, the

spatial responses of the sources, and spatial correlation betweenthe sources.

We know that due to the constraint of (24), alwayshold for and . If the initial weightvector is chosen such that

(40)

cum cum

cum

cum

cum

...cum

(32)


then, from (18) or (37), it is obvious that for arbitrary,

(41)

where denotes the phase of . Note that (40) and (41)indicate that the super-exponential blind beamforming algorithmconvergesfasterfora largervalueof .However,a largervalueof usually gives rise to a larger variance of the estimated

[see(35b)] thus leadingtoalargervarianceof thedesignedbeamfomer. Nevertheless, for an appropriate , as thegainofall theundesiredsignalsare smaller than thatof thedesiredsignal under , the super-exponential beamforming algorithmwill converge fast and closely to the optimal beamformer. In otherwords, if is chosen such that

(42)

then under an appropriate , we have

(43)

where is an error vector with much smaller than .

In order to give more insight into the behavior and the per-formance of the proposed algorithm, we present two theoremsin the following for two special cases, respectively. In the firstcase, the SNRs of all source signals approach infinity, whereasin the second, the desired source is spatially orthogonal to theother sources.

Theorem 2: When , , and the noise com-ponents are Gaussian distributed, the algorithm specified by(35a)–(35c) performs perfect suppression of all the interferingsignals if , , and

where is defined by (33).Theorem 2 is proved in Appendix B.Remark 4: Theorem 2 implies that the higher the SNRs of all

the source signals, the closer the blind beamformer approachesthe nonblind optimal beamformer (the MMSE beamformer). Inaddition, the blind beamformer approaches the nonblind onewhen the SNRs of all the sources approach infinity.

Theorem 3: When , , and the noise com-ponents are Gaussian distributed, the algorithm specified by(35a)–(35c) converges to the nonblind optimal (MMSE) beam-former if the steering vector of the th ( ) sourceis spatially orthogonal to those of all the other sources, i.e.,

(44)

and

(45)

where

cum

(46)

(47)

(48)

(49)

The proof is in Appendix C.Remark 5: Note that under (45), the algorithm converges to

the conventional beamformer shown by (C8) in Appendix C.This provides some insights of the performance behavior of theproposed algorithm.

III. OBJECTIVE FUNCTION OF THE BLIND

ADAPTIVE BEAMFORMING

In many adaptive processing problems, we know that appro-priate objective functions, or cost functions, are very importantin developing adaptive algorithms and evaluating their perfor-mance. It has been shown in [5], [8], and [9] that the followinginverse filter criteria

(50)are suitable objective functions for blind deconvolution, where

represents the output of the associated blind equalizer,, and the cumulant of is defined as

cum

cum (51)

In the absence of noises, the objective function is bounded bythe maximum modulus of the cumulants of all the signal com-ponents [5], [8], [9]. In this section, our purpose is to show thatthe inverse filter criteria can also be an objective function for theproposed blind beamforming algorithm.

Replacing in (50) with the in (7)–(9), one can obtain

(52)where

cum (53)


represents the th-order cumulant of the th source signal.According to (4), we have

cum (54)

Defining

(55)

which is the root mean square (RMS) of the total output power,(52) can then be expressed as

(56)

Since most of the digital signal waveforms employed in com-munications and radar applications are of symmetric distribu-tions, it can be shown that all their odd-order moments and cu-mulants are equal to zero. In the following, the situation where

is considered. In this case, (56) becomes

(57)

From the above equation, we see that for a given ,, the necessary and sufficient condition

for is that and ,, , which corresponds to the perfect

suppression of all the interfering signals described in Theorem2. In practice, when the blind beamformer makes ,

, , , then (57) can beexpressed as

(58)where the ratio will decrease as increases.This indicates that the larger the , i.e., the higher the em-ployed order of the cumulant, the more the reduced contribu-tions of , , to . The term

denotes the suppression ratio of the beamformer tothe th source. The higher the suppression ratio, the smaller thecontribution of the th source to the objective function.

The gradient of the objective function with respect to theweight vector is derived as

sgn cum

(59)

where sgn expresses the sign of the cumulant. Because is a nonlinear function with re-

spect to , the stationary points [5], [8], [11] of (58) can be ob-tained from

(60)

We have the following theorem to show the relationship betweenthe stationary points and the algorithm.

Theorem 4: The super-exponential iterative algorithm spec-ified by (35a)–(35c) converges to the stationary points of theobjective function (58).

Theorem 4 is proved in Appendix D.The associated cumulant terms in (35a)–(35c) and (50) with

order (2,2), (3,3), and (4,4) are derived in Appendix E.From Appendix D, we see that the stationary points of the

objective function cannot be expressed by an analytical solu-tion, and the algorithm described by (35a) and (35b) satisfies(D2), which specifies the stationary points. Because the initial

could be different, the algorithm may converge to the dif-ferent stationary points. In essence, Theorem 4 shows that theinverse filter criteria can be an objective function of the blindbeamforming. Based on Theorem 4, the complete super-expo-nential blind adaptive beamforming algorithm can be cast intothe following three steps.

Step 1) Let , given the accuracy , andthe order of the cumulant.

Step 2) Let and , perform (35a)–(35c), andobtain .

Step 3) If , then stop; Elselet , go to Step 1.

IV. COMPUTER SIMULATIONS

In order to confirm the effectiveness of the proposed algo-rithm, computer simulations are performed. In the simulations,a six-element uniform linear array with half-wavelengthspacing is employed. The array is not calibrated, i.e., thecomplex gains of the array elements are perturbed from theirnominal value. Herein, we assume that the perturbation gainsare time-invariant, whereas their exact values are unknown.The respective nominal values of the amplitude and phase ofthe complex gains are set to 1 (0 dB) and 0 rad. We choose theperturbation gain from the samples of uniformly distributedrandom numbers, where the standard deviation of the amplitudeis assumed as 0.1, and that of the phase is assumed as 0.1 rad.As a result, the perturbation gain vector used in the simulationsis randomly generated, and the set of the samples

is selected.We consider the scenario where three signal sources illuminate

the antenna array from the far field. All the source signals arequadrature phase shift keying signals. The strongest source isassumed to be the desired one, which is located at 12.3 from thebroadside of the antenna array (the broadside here is consideredas the reference direction-of-arrival signals). The power of thesecond source is assumed to be 0.5 dB lower than that of thedesired signal, and its incident angle is 16.5 from the broadside.The power of the third source is 2 dB lower than that of the desiredsignal, and its incident angle is from the broadside.

As mentioned in Sections II and III, different initial weightvectors may lead to different convergent results. For compar-ison, the following two initial weight vectors are considered.


Fig. 1. Output SINR of the desired source.

I1)

I2) .

It is obvious that under I1, a beam toward the desired sourceis formed, whereas the array response (pattern) under I2 is om-nidirectional. Although better beamformer performance is ex-pected when I1 is used, nevertheless, I2 is often used in practicebecause the a priori information of the spatial signature of thedesired source is usually not available.

In Fig. 1(a) and (b), we depict the output SINR versus theinput SNR under the initial weight vectors I1 and I2, respec-tively. The number of the samples employed is 5000, and

is used. In these figures, “BB” stands for the proposedblind algorithm and “MMSE” stands for the nonblind MMSEoptimal beamformer. It is evident from these figures that whenthe input SNR is not too low (input SNR 5 dB), the higherthe order of the cumulants used, the closer the performance ofthe blind algorithm approaches that of the MMSE beamformer.It is also seen that the results under I1 are slightly better than

Fig. 2. Comparison of the convergence performance.

that under I2. When the input SNR is high, on the other hand,all the results of the blind algorithm with both initial weight vec-tors I1 and I2 and different cumulant orders of (2,2), (3,3), and(4,4) approach that of the MMSE beamformer with negligibledifference.

Fig. 2(a) and (b) show the convergence performance of theblind algorithm under I1 and I2, respectively, where the inputSNR is 10 dB, and 5000 data samples are employed. From thesefigures, we see that the convergence rates under both the twoinitial weight vectors are basically the same.

To investigate the effects of the number of the array signalsamples on the performance of the proposed algorithm, Fig. 3shows the output SINR versus the number of the array data sam-ples, where the input SNR is fixed to 10 dB. It is seen that thelonger the samples, the more closely the blind algorithm underboth I1 and I2 will converge to the optimum result. However,because the power of the second source signal is close to thatof the desired source signal, it would be possible for the algo-rithm to mistakenly converge to the result corresponding to thesecond source when I2 is employed. Fig. 3(c) shows the phe-nomenon for the case of the sample size equal to 1000. This isnatural because all blind algorithms are blind to the order of allthe corresponding signals, provided that the a priori informa-tion is available to distinguish them.

Finally, we show the beam patterns in Fig. 4 with both initialconditions, where 5000 samples are used, and the input SNR


Fig. 3. Effects of number of samples on performance.

is 10 dB. It is seen that the convergent beam patterns corre-sponding to both initial conditions are very close to that of theoptimal nonblind MMSE beamformer.

As a conclusion of the simulation results depicted inFigs. 1–4, we see that the proposed beamforming algorithm isrobust and effective. The simulation results demonstrated thatthe cumulant orders, i.e., (2, 2), (3, 3), and (4, 4) are applicable,and using the order (2, 2) leads to lower computations. Al-though using higher order comulants is helpful to improve theconvergence rate, the algorithm will be more sensitive to the

Fig. 4. Convergent patterns.

accuracy of cumulant estimation when a higher order is used.Regarding the initial array patterns, the simulation results showthat the algorithm under the omnidirectional pattern by I2 isrobust in most situations, even when there are small differencesin power between the desired source signal and the undesiredsource signals.

V. CONCLUSIONS

In this paper, we have presented the super-exponential blindadaptive beamforming algorithm, which is an extension fromthe super-exponential blind deconvolution theory and the in-verse filter criteria. This extension theoretically considers thepresence of noise such that the proposed beamforming algo-rithm is applicable, regardless of whether the number of inde-pendent sources exceeds the system degrees of freedom or not.We have also proved two special conditions under which theperformance of the proposed blind algorithm approaches thatof the optimal nonblind MMSE beamformer. Simulation resultshave shown that the proposed algorithm is effective and robust,even when the initial weight vector corresponding to the om-nidirectional pattern is used. It should be noted that using thehigher order cumulants increases the computational complexityand the sensitivity to the estimation accuracy of the cumulantsand decreases the robustness to the model assumptions.


APPENDIX APROOF OF THEOREM 1

A. Proof

When , (20) can be expressed as

(A1)

Under the condition (22), that is

(A2)

where , , and the

nominator of super-exponentially ( ) converges tozero as , i.e., as in (A3) and (A4), shown at the bottomof the page. Therefore, as

(A5)

(A6)

Thus, the theorem has been proved.

APPENDIX BPROOF OF THEOREM 2

A. Proof

By substituting (27) and (25) into (23) and by (2), the gainvector at the iteration can be simplified as

...

... (B1)

where

(B2)

Because the cumulants of Gaussian noises are equal to zeroswhen and , (B1) thus becomes

... (B3)

With the use of Sherman–Morrison–Woodbury formula [25],under assumption A3, the inversion of can be written as

(B4)

Therefore

(B5)

Similarly

(B6)

(A3)

(A4)


When , , (B5) and (B6) approachthe following results:

(B7)

(B8)

and substituting (B7) and (B8) into (B3), we have

... (B9)

where

(B10)

Because (B9) actually corresponds to a special case of (18) con-sidered in Theorem 1, applying the conclusion of Theorem 1proves Theorem 2.

APPENDIX CPROOF OF THEOREM 3

A. Proof

Similar to the proof of Theorem 2, when and the asso-ciated noises are Gaussian distributed, we rewrite the iterativegain vector of (B3) as (C1), shown at the bottom of the page.Define

(C2)

(C3)

The condition that , ,implies

(C4)

Because of (6), can be expressed as

(C5)

According to the Sherman–Morrison–Woodbury formula [25],the inversion of can be denoted as

(C6)Similarly, by employing the Sherman–Morrison–Woodbury for-mula to and with (C4), we obtain

(C7)

Therefore

(C8)

Substituting (C8) into (C1) and by (C4), we have (C9), shownat the bottom of the next page, where

(C10)

(C11)

According to (C9), the following inequalities stand:

(C12a)

(C12b)

(C12c)

It can be seen that , depends on ,, whereas we are only concerned with the

iterative behavior of , . Let

, i.e., for

(C13)

.... . .

... ...

(C1)


By (C13), (C12a) and (C12b), the following relations can beeasily inferred:

(C14a)

(C14b)

(C15)Define

(C16a)

(C16b)

(C16c)

so we have

(C17)

(C18a)

(C18b)

Repeatedly, we have the iterative equations

(C19a)

(C19b)

(C19c)

...

...

(C9)


Therefore

(C20)

When , as , thefollowing term will super-exponentially converges to zero:

(C21)

Because of (C8), (C21) implies that

(C22a)

(C22b)

Therefore, from (C9), we have

(C23)

It can be easily seen from (C23) that as increases, ap-proaches the gain vector of the optimal nonblind MMSE beam-former up to a complex rotating phase factor. Thus, Theorem 3has been proved.

APPENDIX DPROOF OF THEOREM 4

A. Proof

When the super-exponential iterative algorithm converges,(35a)–(35c) can be written as

(D1a)

cum (D1b)

(D1c)

Substituting (D1a) into (59), we have

sgn

cum (D2)

where is used. Based on the linearity property ofthe cumulants [7] and with (D1b) and (D1c)

cum

(D3)

which means

sgn (D4)

Therefore

(D5)

and Theorem 4 has been proved.

APPENDIX ECUMULANTS cum FOR 2, 3, 4

Because analytical signals (complex envelope) in communi-cations, radar, etc., are often of symmetrical distributions withodd order cumulants equal to zero, the following are formulaefor computing th-order cumulants of for 2, 3, and4 as defined in [7]:

cum

(E1a)

cum

(E1b)

cum

(E2a)

cum

(E2b)

cum

(E3a)

cum

(E3b)

ACKNOWLEDGMENT

The authors would like to give thanks to Dr. B. Komiyama,ATR Adaptive Communications Research Laboratories, for hisencouragement and helpful discussions.


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Kehu Yang (M’99) received the B.E., M.S., andPh.D. degrees from Xidian University (formerlythe Northwest Telecommunications EngineeringInstitute), Xi’an, China, in 1982, 1984, and 1995,respectively.

He joined Xidian University in 1985, where hebecame an Associate Professor in May 1996. FromDecember 1998 to May 2002, he was a visitingresearcher at ATR Adaptive Communications Re-search Laboratories, Kyoto, Japan. From June 2002to October 2002, he was a research fellow at Xi’an

Research Institute of ZTE Corporation. From November 2002 to December2002, he was with Xi’an Haitian Antenna Technologies Co., Ltd., as the leaderof the R&D Department 3. In January 2003, he re-joined Xidian University asan Associate Professor. His main research interests are array signal processingfor radar and smart antenna for mobile communications.

Takashi Ohira (S’79–M’83–SM’99–F’04) was bornin Osaka, Japan, in April 1955. He received B.E. andD.E. degrees in communication engineering fromOsaka University, in 1978 and 1983, respectively.

In 1983, he joined NTT Electrical CommunicationLaboratories, Yokosuka, Japan, where he wasengaged in research on monolithic integration ofmicrowave semiconductor devices and circuits.He developed GaAs MMIC transponder modulesand microwave beamforming networks aboard theJapanese domestic multibeam communication satel-

lites Engineering Test Satellite VI (ETS-VI) and ETS-VIII at NTT WirelessSystems Laboratories, Yokosuka. Since 1999, he has been engaged in researchon wireless ad hoc networks and microwave analog adaptive antennas aboardconsumer electronic devices at ATR Adaptive Communications ResearchLaboratories, Kyoto, Japan. Concurrently, he was a Consulting Engineer forthe National Astronautical Space Development Agency (NASDA) ETS-VIIIProject in 1999 and was an Invited Lecturer at Osaka University from 2000to 2001. He co-authored Monolithic Microwave Integrated Circuits (Tokyo,Japan: IEICE, 1997).

Dr. Ohira serves as IEEE MTT-S Japan Chapter Vice-Chairperson. He re-ceived the 1986 IEICE Shinohara Prize and the 1998 Japan Microwave Prize.

Yimin Zhang (SM’01) received the M.S. and Ph.D.degrees from the University of Tsukuba, Tsukuba,Japan, in 1985 and 1988, respectively.

He joined the faculty of the Department of RadioEngineering, Southeast University, Nanjing, China,in 1988. He served as a Technical Manager at theCommunication Laboratory Japan, Kawasaki, Japan,from 1995 to 1997 and was a Visiting Researcher atATR Adaptive Communications Research Laborato-ries, Kyoto, Japan, from 1997 to 1998. Since 1998,he has been with the Villanova University, Villanova,

PA, where he is currently a Research Associate Professor with the Center for Ad-vanced Communications. His research interests are in the areas of array signalprocessing, space-time adaptive processing, multiuser detection, blind signalprocessing, digital mobile communications, and time-frequency analysis.


Chong-Yung Chi (S’83–M’83–SM’89) was born inTaiwan, R.O.C., on August 7, 1952. He received theB.S. degree from the Tatung Institute of Technology,Taipei, Taiwan, in 1975, the M.S. degree from theNational Taiwan University, Taipei, in 1977, and thePh.D. degree from the University of Southern Cali-fornia, Los Angeles, in 1983, all in electrical engi-neering.

From July 1983 to September 1988, he was withthe Jet Propulsion Laboratory, Pasadena, CA, wherehe worked on the design of various spaceborne radar

remote sensing systems including radar scatterometers, synthetic apertureradars, altimeters, and rain mapping radars. From October 1988 to July 1989,he was a visiting specialist at the Department of Electrical Engineering,National Taiwan University. He has been a Professor with the Departmentof Electrical Engineering since August 1989 and Chairman of Instituteof Communications Engineering since August 2002, National Tsing HuaUniversity, Hsinchu, Taiwan. He was a visiting researcher with the AdvancedTelecommunications Research (ATR) Institute International, Kyoto, Japan, inMay and June 2001. He has published more than 110 technical papers in radarremote sensing, system identification and estimation theory, deconvolutionand channel equalization, digital filter design, spectral estimation, and higherorder statistics (HOS)-based signal processing. His research interests includesignal processing for wireless communications, statistical signal processing,and digital signal processing and their applications.

Dr. Chi is an active member of Society of Exploration Geophysicists, amember of European Association for Signal Processing, and an active memberof the Chinese Institute of Electrical Engineering. He was a technical committeemember of both the 1997 and 1999 IEEE Signal Processing Workshops onHigher Order Statistics (HOS) and the 2001 IEEE Workshop on StatisticalSignal Processing (SSP). He was also a member of International AdvisoryCommittee of TENCON 2001 and a member of International ProgramCommittee of the Fourth International Symposium on Independent ComponentAnalysis and Blind Source Separation (ICA-2003). He was a co-organizerand a general co-chairman of 2001 IEEE SP Workshop on Signal ProcessingAdvances in Wireless Communications (SPAWC-2001), the InternationalLiaison of SPAWC-2003, and Technical Committee Member and InternationalLiaison (Asia and Australia) of SPAWC-2004. He was a Program CommitteeMember for the International Conference on Signal Processing (ICSP-2003)and a Technical Committee Member of the Third IEEE International Sympo-sium on Signal Processing and Information Technology (ISSPIT’03). He wasthe Chair of the Information Theory Chapter of the IEEE Taipei Section fromJuly 2001 to June 2003. Currently, he is also an Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING and an Editorial Board Member of theEURASIP Journal on Applied Signal Processing.

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