Adaptive Array Beamforming with Robust Capabilities Under Random Phase Perturbations
Ju-Hong Lee and Kuang-Peng Cheng
Abstract—The problem of adaptive array beamforming with multiple-beam constraints in the presence of steering error caused by random phase perturbations is considered. We first construct a cost function consisting of terms that utilize a posteriori information due to the received array data and a priori information due to the probabilistic distribution of phase per-turbation, respectively. Then, an appropriate estimate of the actual phase angle vector associated with each of the desired signals can be obtained by performing a nonlinear optimization problem based on the cost function. An implementation algorithm is further presented to iteratively solve the problem. Theoretical analysis regarding the convergence property of the it-erative procedure is also investigated. Finally, several computer simulation examples are provided for illustration.
Index Terms—Adaptive array beamforming, random phase perturba-tions, steering error.
I. INTRODUCTION
An adaptive array beamformer is designed for automatically pre-serving the desired signals while canceling the interference and noise. The only a priori knowledge for a main-beam or a multiple-beam con-strained beamformer is the actual direction vectors of the desired sig-nals. A direction vector of a desired signal can be obtained from the information of the array sensor locations, signal impinging directions, and the propagation characteristics. However, the information may not be perfectly known in practice. This results in a mismatch between the presumed steering vectors and the actual direction vectors. The perfor-mance of a steered beam adaptive array beamformer is very sensitive to the mismatch [1]–[5].
To cure the problem due to the above mismatch, most robust tech-niques propose to impose additional constraints such as multiple linear constraints, derivative constraints, and norm constraints on the array weight vector [5]–[15]. However, imposing additional constraints de-teriorates the array capability in suppressing interference and noise. In contrast, Shahbazpanahi et al. [16] present a robust approach based on the worst-case performance optimization for curing the problem of array performance degradation due to the signal covariance matrix with some fixed error. Li et al. [17] propose a diagonal loading ap-proach for the beamforming problem of the desired signal with non-random steering vector error. Recently, based on the assumption that the steering vector error is an additive Gaussian random vector, two methods have been presented in [18] to find two appropriate closed-form solutions for estimating the optimal steering constraint vector. All of the above-mentioned techniques [5]–[18] are developed under the situation of adaptive beamforming with main-beam constraint. In many applications, such as satellite communications [19], an antenna array must possess beamforming capability to receive more than one signal with specified gain requirements while suppressing all jammers. This purpose can be effectively achieved by using an antenna array
Manuscript received June 13, 2003; revised January 21, 2004. This work was supported by the National Science Council under Grant NSC90-2213-E002-043. The associate editor coordinating the review of this paper and approving it for publication was Dr. Martin Haardt.
The authors are with the Department of Electrical Engineering, Na-tional Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: juhong@ cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TSP.2004.838961
with multiple-beam pattern [19], [20]. Recently, a technique for adap-tive beamforming with the capability of providing multiple-beam con-straints (MBCs) has been presented in [21].
In this correspondence, we consider the problem of adaptive beam-forming with MBC in the presence of random phase perturbations. A robust method in conjunction with an iterative procedure is presented for solving the considered problem. To find the optimal phase angle vector, we construct acost function consisting of the squared norm of the projection of the steering vector on the noise subspace and a constraint related to a likelihood function associated with the random phase error vector. Minimizing the squared norm of the projection of the steering vector on the noise subspace is equivalent to maximizing the squared norm of the projection of the steering vector on the signal plus interference subspace. The constraint related to a likelihood func-tion associated with the random phase error vector is utilized to prevent the obtained optimal phase angle vector for each desired signal from be-coming one of the interference phase angle vectors. Since the resulting minimization problem is highly nonlinear, we use a gradient method to iteratively search for the solution. It is shown that using the con-straint related to a likelihood function of the random phase error vector provides the advantage of properly adjusting the step size during the gradient search procedure. The analysis of the convergence property of the proposed method is also presented. Several computer simulation examples show the effectiveness of the proposed method.
II. PROBLEMFORMULATION
Consider a uniform linear array (ULA) with M sensors and in-terelement spacing equal to half of the smallest wavelength of the signals with specified gain/null arrangements. Let K narrowband and far-field signals be impinging on the array from direction angles i; i = 1; 2; . . . ; K, off broadside. The signal received at the m th array sensor can be expressed as
xm(t) = K i=1
si(t)am(i) + nm(t); m = 1; 2; . . . M (1) wheream(i) = exp(j(2dmsin i)=i); iis the wavelength of the i th signal, and dmis the distance between themth and the first array sensors,si(t) is the complex waveform of the ith signal, and nm(t) is the spatially white noise with mean zero and variance2nreceived at themth array sensor. In matrix form, we can write the data vector received by the ULA as follows:
xxx(t) = BBsss(t) + nnn(t)B (2) where the matrix BBB = [aaa(1) aaa(2) . . . aaa(K)] with the direction vector of the ith signal given by aaa(i) = [a1(i) a2(i) . . . aM(i)]T, the signal source vector issss(t) = [s1(t) s2(t) . . . sK(t)]T, and the noise vector isnnn(t) = [n1(t) n2(t) . . . nM(t)]T. The superscript T denotes transpose operation. Under the assumption that sss(t) and nnn(t) are uncorrelated, the M 2 M ensemble correlation matrix of xxx(t) is Toeplitz–Hermitian and given by
RR
Rx= [Rkl] = [R(k 0 l)] = Efxxx(t)xxx(t)Hg = BBRBRsBR BBH+ 2nIII (3) where the superscript H denotes the complex conjugate transpose. RR
Rs= Efsss(t)sss(t)Hg has rank K if the K signals are uncorrelated. Let the ULA use aweight vectorww = [w1w w2 . . . wM] for processing the received data vector xxx(t) to produce the array output signal 1053-587X/$20.00 © 2005 IEEE
y(t) = wwwHxxx(t). Assume that the selective gain/null requirements are specified by assigning a gaincpat the direction vectoraaa(p) for p = 1; 2; . . . ; P , where P denotes the number of signals with gain/null constraint. Then, the optimal weight vector for the adaptive array can be found from the following constrained optimization problem [19]:
MinimizeEfjy(t)j2g = wwwHRxwRR ww Subject to GGGHwww = ccc (4) where the matrixGG = [aaa(1) aaa(2) . . . aaa(PG )] denotes the constraint matrix, andccc = [c1 c2 . . . cP] denotes the gain vector. Accordingly, the optimal weight vector is given by
wwwo= RRR01
x GGG GGGHRRRx01GGG 01
ccc: (5)
Substituting (5) intoEfjy(t)j2g = wwwHRRRxww yields the correspondingw array output power equal to
Efjy(t)j2g = wwwHRRRxww = cccw H GGGHRRR01 x GGG
01
ccc: (6)
In the presence of random phase perturbation, we consider that the phase angle error vector for the signal with gaincpdue to an additive perturbation is given by
2ep= 2p0 2dp (7)
where2pand2dpdenote the phase angle vectors associated with ac-tual direction vectoraaa(p) and the presumed direction vector aaad(p), respectively. Without loss of generality, let the mth entry of the direction vector aaa(p) be expressed as am(p) = exp(j pm) and the corresponding phase angle vector be constructed as 2p= [ p1 p2 . . . pM]T. Similarly, let themth entry of the direc-tion vectoraaad(p) be expressed as adm(p) = exp(j dpm) and the phase angle vector be constructed as2dp= [ dp1 dp2 . . . dpM]T. According to the discussion of [5], the phase angle error vector2ep shown by (7) can be assumed to be a real Gaussian random vector with mean zero and covariance matrixDDDp. Based on this discussion, we consider the case of Gaussian random phase perturbation. Hence, the probability density function (PDF) for2epis given by
PDF(2ep) = [(2)Mdet(DDDp)]01=2exp 0 2epDTDD01p 2ep =2 : (8) Accordingly, the likelihood function (LF) regarding the phase angle error vector can be defined as
LF= exp 0 2TepDDD01p 2ep =2 = exp 0 (2p0 2dp)TDDD01
p (2p0 2dp) =2 : (9) Next, consider the case that the P signals with gain/null con-straint are uncorrelated and the P phase angle error vectors 2ep; p = 1; 2; . . . ; P are also uncorrelated. We construct an M 2 P phase angle error matrix 9 and its vectorized version 0 with size PM 2 1 for the P uncorrelated signals as follows. 9 = [2e1 2e2 . . . 2eP] and 0 = [2T
e1 2Te2 . . . 2TeP]T. The joint PDF for9 can be expressed as
PDF(9) = P p=1
[(2)Mdet(DDDp)]01=2 expf0(0T010)=2g (10)
where is aPM2PM block diagonal matrix with the pth M 2M diag-onal block matrix given byDDDp. The corresponding likelihood function regarding the phase angle error vector0 is given by
LFP = expf0(0T010)=2g = exp 0 P
p=1
(2p0 2dp)TDDD01
p (2p0 2dp) 2 :(11) To deal with the problem of array beamforming with MBC in the pres-ence of phase angle errors, as described above, we present a robust method based on the likelihood function given by (11).
III. ROBUSTMETHOD
From the property of a gain constrained array beamformer, it is well known that the output power of the beamformer will achieve its maximum when each presumed direction vectoraaad(p) of the constraint matrix GGG coincides with the actual direction vector aaa(p); p = 1; 2; . . . ; P . Moreover, from the eigendecom-position of RRxR [22], we can expressRRxR = Mi=1ieeeieeeHi, where 1 2 1 1 1 J+P J+P +1 = . . . = M = 2
nare the eigenvalues ofRRRxin the descending order, andeeeiare the corresponding eigenvectors.J is the number of interferers. The eigenvectors associ-ated with the minimum eigenvaluen2 are orthogonal to the direction vectors of the signals with specified gain/null constraints and inter-ferers. Therefore, the subspaces spanned byEEEn= feeeJ+P +1; . . . ; eeeMg (called the noise subspace) andEEsE = feee1; eee2; . . . ; eeeJ+Pg (called the signal plus interference subspace) are orthogonal. We can rewriteRRxR as follows: R R Rx= M i=1 ieeeieeeH i = EEEs3sEEEHs + EEEn3nEEEHn (12) where3s = diagf1; 2; . . . ; J+Pg, a nd 3n= n2III, where III de-notes the identity matrix with appropriate size. Based on (6), (11), and (12), an appropriate cost function regarding the phase angle errors is defined as J(8) = P p=1 (sssp)HEEEnEEEH n(sssp) 0 2 exp 0 P p=1 (2sp0 2dp)TDDD01 p (2sp0 2dp) 2 (13)
where sssp = [sp1 sp2 . . . spM]T = [exp(jp1) exp(jp2) . . . exp(jpM)]T; 2sp = [p1 p2 . . . pM]T, a nd 8 = [2T
s1 2Ts2 . . . 2TsP]T. The first term of (13) represents the squared norm of the projection of the constraint vectorssssp; p = 1; 2; . . . ; P on the noise subspace spanned byEEEn. The second term is the likelihood function related constraint. denotes a positive weighting parameter providing the relative weight between these terms. As a result, the optimal solution8oof minimizing (13) can then be used as an appro-priate estimate of8P = [2T1 2T2 . . . 2TP]Tformed by the actual phase angle vectors 2p; p = 1; 2; . . . ; P for array beamforming. However, the cost function of (13) is ahighly nonlinear function of the phase angle vectors2sp; p = 1; 2; . . . ; P . Thus, aclosed-form solution for the optimal solution cannot exist. We resort to an iterative procedure to solve this problem, as follows. First, we rewrite (13) as follows: J(8) = (S)HW(S) 0 2 exp 0 P p=1 (2sp0 2dp)TDDD01p (2sp0 2dp) 2 (14)
where the MP21 vector S = [sssT1 sssT2 . . . sssTP]T, a ndW is aPM2PM block diagonal matrix with thepth M 2M diagonal block matrix given byEEnEE EEHn. Then, the gradient vector ofJ(8) is computed according to 8J(8) = 02Re j EEnEE EEH nsss1 sss31 0 2Re j EEEnEEEHnsss2 sss32 . . . 0 2Re j EEEnEEEH nsssP sss3P T + exp 0 P p=1 (2sp0 2dp)TDDD01 p 2 (2sp0 2dp) 2 01(8 0 8d) (15) where Refxg denotes the real part of x, the superscript3the complex conjugate, and the Hadamard (or elementwise) product [23]. 8d= [2T
d1 2d2T . . . 2TdP]T. Then, we update the phase angle vector8 and the corresponding steering constraint vectorssspas follows:
8k+1 = 2k+1 s1 T 2k+1 s2 T . . . 2k+1 sP T T = 8k0 " 8J(8k) (16) sk+1 pm = exp jpmk+1 ; p = 1; 2; . . . ; P m = 1; 2; . . . ; M (17) where the superscriptk denotes the kth iteration and " the preset pos-itive step size. From (16), we note that the second term includes the factor of likelihood function related to each of the phase angle vector errors 2sp 0 2dp; p = 1; 2; . . . ; P at the kth iteration. Hence, it would be expected that the resulting gradient approach for finding the optimal solution of8 can provide a more appropriate estimate of 8 since the resulting step size becomes variable according to the expo-nential term, as shown in (15). In the literature, Smith [24] uses a con-jugate gradient and Newton’s method to maximize the array output signal-to-interference plus noise ratio (SINR) by adjusting the phase terms of the so-called phase-only weights for achieving the optimal phase-only adaptive nulling.
Next, we present an appropriate scheme for choosing the initial guess for the MP2 1 vector S = [sssT1 sssT2 . . . sssTP]T in order to initiate the iterative process of the proposed robust method. According to the optimal weight vector given by (5) under the assumption ofP = 1 and the desired signal with direction vectoraaa(p), the output of the adaptive array is approximately given by
yp(t) = wwwHopxxx(t) sp(t)gp+ wwwHopnnn(t) (18) based on the assumptions thatM > K and the interference signals are suppressed enough, wheregp wwwHopaaa(p) denotes the array gain for the specified signal, where subscriptp represents the results obtained by using the desired signal with direction vectoraaa(p); p = 1; 2; . . . ; P . Equation (18) reveals that the output of the adaptive array can be used as a reference signal to find the actual phase angle vector2p. Consider the cross-correlation betweenxxx(t) and yp(t). We ha ve
Efxxx(t)yp(t)3g = Efxxx(t)xxx(t)Hgwwopw
= RRRxwwopw pgpaaa(p) + 3 nw2wwop (19) wherepdenotes the power associated with the specified signal. In practice, the noise power2nis unknown. However, it can be obtained by setting the smallest eigenvalue of the autocorrelation matrix as the
estimate ofn2. From (19), we can therefore adopt the following vector as the initial guess for each ofsssp:
vvvp= RRxwR wwop0 2
nwwwop: (20)
Based on (20), it is clear that the direction vectoraaa(p) is approxi-mately proportional tovvvpwith a proportional constantpg3p. Hence, an appropriate initial guesssssopforssspcan be constructed as follows:
uuup= [up1; up2; . . . ; upM]T= (vp1)01vvvp
s0pm= jupmj01upm; m = 1; 2; . . . ; M (21) ssso
p= [s0p1; s0p2; . . . ; s0pM]T (22) forp = 1; 2; . . . ; P , where vp1denotes the first entry ofvvvpandvvvpthe result given by (20). The superscript “o” represents the initial guess. In other words, we keep only the phase portion ofvvvpand then take the phase referencing to the first element ofvvvpto form the initial guesssssop. Finally, we construct an initial guessSo= [(ssso1)T(sss2o)T . . . (sssoP)T]T ofS for carrying out the proposed iterative process.
For practical implementation, we compute the sample data correla-tion matrixRRRx(i) using i data snapshots as follows: RRRx(i) = (1 0 1=i)RRRx(i 0 1) + (1=i)xxx(i)xxx(i)H, which is used as the estimate of the requiredRRRx, wherexxx(i) denotes the ith data snapshot sampled fromxxx(t). The required computational complexity is O(M2). Using the approach of [22] to perform the eigendecomposition ofRRRx(i) pro-vides the basis matrixEEEn(i) and needs O(M2). To construct the matrix EE
En(i)EEEn(i)HrequiresO(M3). Therefore, the required computational complexity is aboutO(M3) + O(M2) + PM2+ PM in order to obtain the first term of the right-hand side of (15). Moreover, the computa-tional complexity for obtaining the second term of the right-hand side of (15) is about3MP + M. As a result, the computational complexity required for computing the gradient of J(8) when receiving i data snapshots is aboutO(M3) + O(M2) + PM2+ 4MP + M.
IV. CONVERGENCE OF THEPROPOSEDMETHOD
To ensure the convergence of the proposed method, we have to show that the cost function to be minimized as given by (14) possesses the property of J(8k+1) < J(8k). Let the vector AA
Akrepresent the second term of (16), i.e., AAAk = 0" 8J(8k) = [(AAAk
1)T (AAAk2)T . . . (AAAkP)T]TandAAAkp= [Akp1 Akp2 . . . AkpM]T; p = 1; 2; . . . ; P . Assume that AAAk is anonzero real vector with anorm small enough. Then,(AAAk)TAAAk> 0, i.e.,
(AAAk)Tf2"Refj[W(Sk)] (Sk)3g 0 " 2 expf0[(8k0 8d)T01(8k0 8d)]=2g 2 01(8k0 8d)g > 0: Hence (AAAk)Tf2Refj[W(Sk)] (Sk)3gg > 2 expf0[(8k0 8d)T01(8k0 8d)]=2g 2 (AAAk)T01(8k0 8d) (23)
and the objective function after the(k + 1)th iteration is given by J(8k+1) = (Sk+1)HW(Sk+1) 0
2 expf0[(8k+10 8d)T01(8k+10 8d)]=2g: (24) According to the above definition, (16), and (17), we have
sk+1 pm = exp jpmk+1 = exp j pmk + Akpm 1 + jAk pm exp jpmk p = 1; 2; . . . ; P; m = 1; 2; . . . ; M hence; Sk+1 Sk+ (jAAAk) Sk: (25)
(a) (b)
(c) (d)
Fig. 1(a). Array beampattern with respect to the desired signal of (a) 17 , (b)051 , (c) output SINR versus the number of snapshots, and (d) array output SINR versus the SOIs’ power, for Example 1.
Substituting (25) into (24) and performing the necessary algebraic ma-nipulations yields J(8k+1) [Sk+ (jAAAk) Sk]HW[Sk+ (jAAAk) Sk] 0 2 expf0[(8k+ AAAk0 8d)T01 2 (8k+ AAAk0 8d)]=2g (Sk)HW(Sk) 0 2 expf0[(8k0 8d)T01(8k0 8d)]=2g + (Sk)H [0jAAAk]W(Sk) + (Sk)HW[jAAAk] (Sk) + (Sk)H [0jAAAk]W[jAAAk] (Sk) + f(8k0 8d)T01AAAk+ (AAAk)T01AAAk=2g 2 expf0[(8k0 8d)T01(8k0 8d)]=2g J(8k) + 2Ref(Sk)HW[jAAAk] (Sk)g + (8k0 8d)T01AAAk 2 expf0[(8k0 8d)T01(8k0 8d)]=2g (26)
since the norm of AAAk is small enough, and we neglect the terms (AAAk)T01AAAkand(Sk)H [0jAAAk]W[jAAAk] (Sk).
From (26), it is clear that we have to show
2Ref(Sk)HW[jAAAk] (Sk)g + [(8k0 8d)T01AAAk]
expf0[(8k0 8d)T01(8k0 8d)]=2g 0 (27) for anyk to ensure the convergence. Based on (23), the condition of (27) can be reformulated as follows:
2Ref(Sk)HW[jAAAk] (Sk)g
+(AAAk)T2Refj[WSk] (Sk)3g 0 (28) which can be repressed as the following condition:
P p=1 2Re Sk p H E E EnEEEH n jAAAkp Skp + P p=1 AA Akp T 2Re jEEEnEEEnHSkp Skp 3 0: (29)
(a) (b)
(c)
(d) (e)
Fig. 2. Array beampattern with respect to the desired signal of (a) 25 , (b)025 , (c) 50 , (d) output SINR versus the number of snapshots, and (e) array output SINR versus the SOIs’ power for Example 2.
Next, the left-hand side of (29) is equal to P p=1 2 Re Skp H EEEnEEEHn jAAAkp Skp + AAAk p T j EEnEE EEH nSkp Skp 3 = P p=1 2Re Skp H EEEnEEEHn jAAAkp Skp + EEEnEEEH nSkp T jAAAk p Skp 3 = P p=1 2Re Skp H EEEnEEEHn jAAAkp Skp 0 Sk p H EEEnEEEH n H jAAAk p Skp 3 = P p=1 2Re 2j Im Sk p H E E EnEEEH n jAAAkp Skp = 0: (30) Hence, the result given by the left-hand side of (28) is always equal to zero, i.e., 2Ref(Sk)HW[jAAAk] (Sk)g +(AAAk)T2Refj[WSk] (Sk)3g = 0: (31) Consequently, we obtain 2Ref(Sk)HW[jAAAk] (Sk)g + (8k0 8d)T01AAAk 2 expf0[(8k0 8d)T01(8k0 8d)]=2g < 0: (32) It follows from (26) and (32) thatJ(8k+1) < J(8k). This ensures the convergence property under suitably small selections of".
V. COMPUTERSIMULATIONRESULTS
A ULA with interelement spacing equal to half of the minimum wavelength of the signals with specified gain/null requirements is used. All results presented are obtained by averagingL independent runs with independent noise samples for each run. The noise power level for each of the simulation example is set to 1. The array beampattern with respect to the desired signal ofrepresents a beampattern ob-tained by using the searching vectorsss with phase angle equal to the presumed phase angle plus the phase angle error associated with the desired signal offor finding the power gainjwwwHsssj2.
Example 1: Three signal sources with signal-to-noise (SNR) equal to 5, 4, and 2 dB, respectively, are impinging on the array with size M = 8 from direction angels 17; 051, and 67, respectively. The specified signals are the first two signals withc1 = c2 = 1, and the third one is the jammer. Let the random phase angle error vector for each of the specified signals have zero mean and covariance matrix DDDp = 0:3III; p = 1; 2, where III is the 8 2 8 identity matrix. Fig. 1 plots the simulation results in terms of the array beampatterns using 15 000 data snapshots and the corresponding array output SINR with and without utilizing the proposed method. The values ofL; ", a nd used are 50, 0.05, and 0.001, respectively. The results without random phase angle errors and of using the method of [17] with loading value equal to 2000 are also shown. The output SINRs obtained by utilizing 15 000 data snapshots for the result of using the proposed method, the result without random phase angle errors, and the result of using the di-agonal loading [17] are 13.06 dB, 13.58 dB, and 10.92 dB, respectively. We observe from this figure that the proposed method can effectively cope with the performance degradation due to the random phase angle errors.
Fig. 3. Output SINR and number of iterations versus the relative weight for Example 1.
Fig. 4. Output SINR and number of iterations versus the step size for Example 1.
Example 2: Four signals with SNR equal to 5, 6, 7, and 5 dB, respectively, are impinging on the array with size M = 15 from direction angels 25; 025, 50, and 0, respectively. The specified signals are the first three signals with c1 = c2 = c3= 1, and the fourth one is the jammer. Let the random phase angle error vector for each of the specified signals have zero mean and covariance matrix DDDp = 0:5III; p = 1; 2; 3, where III is the 15 2 15 identity matrix. Fig. 2 depicts the simulation results in terms of the array beampatterns using 15 000 data snapshots and the corresponding array output SINR with and without utilizing the proposed method. The values ofL; ", a nd used are 50, 0.05, and 0.001, respectively. The results without random phase angle errors and of using the method of [17] with loading value equal to 2000 are also shown. The output SINRs obtained by utilizing 15 000 data snapshots for the result of using the proposed method, the result without random phase angle errors, and the result of using the diagonal loading [17]
are 17.48, 17.75, and 14.11 dB, respectively. Again, we observe from this figure that the proposed method can effectively cure the performance degradation due to the random phase angle errors.
Finally, Figs. 3 and 4 show the array output SINRs and the numbers of iterations versus and ", respectively, for Example 1, after using 15 000 data snapshots and taking the average of 20 independent runs. The proposed method provides array performance with robust capabil-ities not very sensitive to the choice of and " in the ranges shown by the figures. However, the number of iterations for obtaining the con-vergent results is sensitive to the choice of and " and decreases in general as the value of" increases.
VI. CONCLUSION
We have illustrated that the performance degradation of an adaptive beamformer with multiple-beam constraints due to random phase angle errors is significant and presented an efficient method for solving the problem. A new cost function for curing the problem has been developed. The resulting minimization problem is highly nonlinear but can be solved through the use of an iterative procedure. In conjunction with a steepest-descent algorithm, the estimates for all of the signals with specified gain constraints can be obtained simultaneously. The convergence property regarding the proposed method has been investigated. Simulation examples have shown the effectiveness of the proposed method.
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Online Bayesian Activity Detection in DS/CDMA Networks Thanh Ngoc Bui, Vikram Krishnamurthy, and H. Vincent Poor
Abstract—An on-line Bayesian based multiple hypotheses Shiryayev Sequential Probability Ratio Test (SSPRT) for the detection/isolation of new active users in a multiuser code division multiple access (CDMA) environment is presented. This SSPRT algorithm makes use of a priori knowledge of the user activity parameter. Comparison by simulation between this SSPRT algorithm and the non-Bayesian Matrix cumulative sum (CUSUM) shows that when such information is available, the SSPRT algorithm that uses this information can achieve better performance than the non-Bayesian approach.
Index Terms—Activity detection, CDMA, multiuser detection, sequential probability ratio test.
I. INTRODUCTION
Multiuser detection (MUD) has been shown to be an important demodulation technique for use in direct sequence code division multiple access (DS/CDMA) systems. Though many MUD schemes have been proposed, their performance depends significantly on the assumptions made about the interference parameters (for example,
Manuscript received March 16, 2003; revised December 15, 2003. This work was supported by NSERC Canada, the British Columbia Advanced Systems Institute, the U.S. National Science Foundation under Grant ANI-03-38807, and the New Jersey Center for Pervasive Information Techonology. The associate editor coordinating the review of this paper and approving it for publication was Dr. Joseph Tabrikian.
T. N. Bui is with the Department of Electrical and Electronic Engineering, University of Melbourne, Victoria 3010 Australia.
V. Krishnamurthy is with the Department of Electrical and Computer En-gineering, University of British Columbia, Vancouver, BC, V6T 1Z4 Canada (e-mail vikramk@ece.ubc.ca).
H. V. Poor is with the Department of Electrical Engineering, Princeton Uni-versity, Princeton, NJ 08544 USA (e-mail poor@princeton.edu).
Digital Object Identifier 10.1109/TSP.2004.838944 1053-587X/$20.00 © 2005 IEEE
