Barroso, V.A.M. & Moura, J.M.F. “Beamforming with Correlated Arrivals in Mobile Communications” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 69 Beamforming with Correlated Arrivals in Mobile Communications 1 Victor A.N. Barroso Instituto Superior T ´ ecnico, Instituto de Sistemas e Rob ´ otica Jos ´ e M.F. Moura Carnegie Mellon University 69.1 Introduction 69.2 Beamforming Minimum Output Noise Power Beamforming (MNP) 69.3 MMSE Beamformer: Correlated Arrivals 69.4 MMSE Beamformer for Mobile Communications Modelof the ArrayOutput • MaximumLikelihoodEstimation of H 69.5 Experiments 69.6 Conclusions Acknowledgments References 69.1 Introduction The classical definition of a beamformer basically specifies its goal: to estimate the signal waveform arriving at the array from a given direction. Beamformers are spatial processors that combine the signals impinging on an array of captors. Combining the outputs of the captors forms a narrow beam pointing towards the direction of the source (look direction). This narrow beam can discriminate between sources spatially located at distinct sites. This important property of beamformers is used to design techniques that localize active or passive sources particularly in RADAR/ SONAR systems. In the last two decades, beamforming methods have had significant theoretical and practical advances. This, together with other technological advances, has broadened the application of so- phisticated beamforming techniques to a diversity of areas, including imaging, geophysical and oceanographic exploration, astrophysical exploration, and biomedical. See [19, 20] for an excellent overview of modern beamforming techniques and applications. Communications is another attractive application area for beamforming. In fact, beamforming has been widely used for directional transmission and reception as well as for sector broadcasting in satellite communications systems. More recently, due to the drastic increase of users in cellular 1 Initial date of submission of this article September 28, 1995. c  1999 by CRC Press LLC radio systems [10, 15, 18], including indoors and outdoors mobile systems, it is increasingly being recognized that the design of base station and mobile antennas based on beamforming methods improves significantly the system’s spectrum efficiency [1, 11]. In turn, this enables accommodating larger numbers of users [3, 16]. The most striking argument in favor of using advanced beamforming techniques suchasadaptive or blind beamforming for mobile communicationsisbasedonthe idea of Space Division Multiple Access(SDMA) schemes. With SDMA, several mobiles share simultaneously the same frequency channel by creating virtual channels in the spatial domain. Another important argument in favor of using beamforming in cellular radio is that beamforming yields flexible signal processingschemesthat properlyhandlemultipatheffects whicharetypicalin radio communications. Multipath is the term given when the same signal arrives at the destination through different paths. This may arise when signals bounce off obstacles in their path of propagation. At the receiver, these arrivals are correlated. Their recombination causes severe signal distortions and fading. In limiting cases, the power of the received signal can become so small that the reliability of the data communications link is completely lost. In this section we design a multichannel beamformer to combat multipath effects. The receiver uses a base station antenna array which handles several radio links operating simultaneously at the same carrier frequency, while preserving the reliability of the communications. The approach relies on statistical signal processingmethods, yielding a solution that operates in a blind mode with respect to the parameters that specify the propagation channel. This means that, except for a few quantities related to system specifications, e.g., link budget and array geometry, the receiver that we describe here does not assume any prior knowledge about the locations of the sources and of the structures of the ray arrivals, including directions of arrival and correlations. The simulation results show the excellent performance of this multichannel beamformer in SDMA schemes. The chapter is organized as follows. In Section 69.2 we introduce the beamforming problem (see also [20]), and classical beamformers such as the delay-and-sum beamformer, the minimum output noise power beamformer, and the minimum variance beamformer. We show that these beamformers presentseveredrawbacks when operating inmultipathenvironments. Section69.3 presentsasolution to the beamforming problem for the case of correlated arrivals. This solution is based on a minimum mean square error (MMSE) approach. We compare the performance of this beamformer with the performance of the beamformers introduced in Section 69.2. We emphasize, in particular, the case of multipath propagation. In this section, we also discuss issues regarding the implementation of the minimum mean square error beamformer. In Section 69.4, we describe a method to implement the minimum mean square error beamformer in the context of a digital mobile communications system. The method operates in a blind mode and strongly exploits the structure of the received multipath data. Since the propagation channel parameters, e.g., angles of arrivals of the multiple paths, are not known, we estimate them with a maximum likelihood approach supported on a finite mixture distribution model of the array data. We maximize the likelihood function with an iterative scheme. We describe an efficient procedure to initialize the iterative algorithm. In general, this procedure convergesrapidly to the global maximum of the likelihoodfunction. Section 69.5 presents simulation results obtained with data synthesized by a simple mobile communications simulator. These results confirm the excellent performance of the MMSE beamformer described in the paper. 69.2 Beamforming Beamforming is an array processing technique for estimating a desired signal waveform impinging on an array of sensors from a given direction. This technique applies to both narrowband and wideband signals. Here, we will consider only the narrowband case. Let s(t) be the complex envelope of the source radiated signature. Under the farfield assumption the signal at the receiving array is a planar wavefront, see Fig. 69.1. In this case, and according to the c  1999 by CRC Press LLC FIGURE 69.1: Source/receiver geometry. model derived in [20], the complex envelope of the signal received at each sensor of a uniform and linear array of N omnidirectional sensors is s n (t) = s(t − τ n )e −jω 0 τ n , (69.1) where ω 0 is the carrier frequency and τ n is the intersensor propagation delay. Let d, c, and θ 0 be, respectively, the distance between sensors, the propagation velocity, and the direction of arrival (DOA). The intersensor delays are then τ n = (n − 1)d c sin θ 0 ,n= 1, 2, .,N . (69.2) Because of the narrowband assumption, we can make the simplification s(t − τ n )  s(t) in Eq. (69.1). This means that, for the values of τ n of interest, the source complex envelope s(t) is slowly varying when compared with the carrier e jω 0 t . We model each array sensor by a quadrature receiver, its output being given by z n (t) = s n (t) + n n (t), (69.3) where s n (t) is the complex envelope of the signal component and n n (t) is a complex additive dis- turbance, such as sensor noise, ambient noise, or another signal interfering with the desired one. Collecting in a vector z(t) all the responses of the N sensors of the array to a narrowband source coming at the array from the DOA= θ 0 ,wegettheN-dimensional complex vector z(t) = a(θ 0 )s(t) + n(t ) . (69.4) The vector a(θ 0 ) is referred to as the steering vector for the DOA θ 0 . The elements of the steering vector a(θ 0 ) are given by a n (θ 0 ) = e −jω 0 τ n ,n = 1, 2, .,N.The noise vectorn(t ) isan N-dimensional complexvector collecting the N sensor noises n n (t). In general, it includes components correlated with the desired signal as in multipath propagation environments. With multipath, several replicas of the same signal, each one propagatingalong a different path, arrive at the array with distinct DOAs. In beamforming, the goal is to estimate the source signal s(t) given a(θ 0 ). The narrowband beamformer is illustrated in Fig. 69.2. The output of the beamformer is y(t) = w H z(t) , (69.5) where w =[w 1 ,w 2 ,w 3 , .,w N ] T is a vector of complex weights. We use the notation {·} T to denote vector and matrix transposition, and {·} H for transposition followed by complex conjugation. The beamformer is completely specified by the vector of weights w. c  1999 by CRC Press LLC FIGURE 69.2: Narrowband beamformer. In the absence of the noise term n(t ) in (69.4), it is readily seen that choosing w = (1/N )a(θ 0 ), the beamformer output is y(t) = s(t). This corresponds to the simplest implementation of the narrowband beamformer, known as the delay–and–sum (DS) beamformer: it combines coherently the signal replicas received at each sensor after compensating for their corresponding relative delays. The interpretation of the DS beamformer operation is rather intuitive. However, we may ask ourselves the following question: is DS the best we can do to estimate the desired signal when the disturbance n(t ) is present? To answer the question satisfactorily, we begin by noting that, in the presence of noise, the output of the DS beamformer is y(t) = s(t) + (1/N)a H (θ 0 )n(t) . The influence of the error term on the estimate y(t) of s(t) depends basically on the structure of n(t). The optimal design of beamformers depends now on the choice of an adequate optimization criterion that takes into account the disturbance vector, with the goal of improving in some sense the quality of the desired estimate. In the sequel, we will consider several cases of practical interest. 69.2.1 Minimum Output Noise Power Beamforming (MNP) To reduce the effect of the error term at the beamformer output, we formulate the beamforming problem as follows: find the weight vector w such that the noise output power E    w H n(t))   2  is minimized subject to the constraint w H a(θ 0 ) = 1 , where E{·} denotes the statistical average. The cost function is E     w H n(t))    2  = w H R n w (69.6) with R n the covariancematrix of the disturbance vector n(t ), i.e., R n = E{n(t)n H (t)}. The constraint guarantees that the signal along the look direction θ 0 is not distorted. The solution to this constrained optimization problem is obtained by Lagrange multipliers tech- niques. It is given by w = (a H (θ 0 )R −1 n a(θ 0 )) −1 R −1 n a(θ 0 ). (69.7) The vector w in Eq. (69.7) is the gain of the MNP beamformer [20]. When the source signal is uncorrelated with the disturbance, E{s(t)n H (t)}=0 , c  1999 by CRC Press LLC it can be shown that the weight vector (69.7) of the MNP beamformer takes the form w = (a H (θ 0 )R −1 a(θ 0 )) −1 R −1 a(θ 0 ), (69.8) where R = E  z(t)z H (t)  (69.9) is the covariance matrix of the array data vector z.Thevectorw in Eq. (69.8) is the gain of the minimum variance (MV) beamformer [20]. The MV beamformer minimizes the total output power E     w H z(t))    2  = w H Rw subject to w H a H (θ 0 ) = 1. The MV beamformer presents an important advantage over the MNP beamformer. While to implement the MNP beamformer we need to know the covariance matrix R n of the disturbance vector n, in general, to implement the MV beamformer it is sufficient to estimate the array covariance matrix R using the available data z. We discuss how to estimate R.LetT time samples (snapshots) of the array response vector z(t) be available. An estimate of R is the data sample covariance matrix R s : R s = 1 T T  t=1 z(t)z H (t) . (69.10) Under technical conditions that we will not discuss here, the sample covariance matrix, R s , converges (in the appropriate sense) to the array covariance matrix R when T approaches infinity. This means that, for a large enough number T of snapshots, we can replace R in Eq. (69.8)byR s , without a significant performance degradation. We provide an alternative interpretation to the MNP beamformer. Using Eq. (69.7)inEq.(69.5), and taking into account Eq. (69.4), we see that the output of the MNP beamformer has a signal component s(t) and an error term w H n(t) with average power P o = w H E  n(t)n H (t)  w =  a H (θ 0 )R −1 n a(θ 0 )  −1 . (69.11) Since the power of the signal is preserved and the MNP beamformer minimizes the power of the noise at its output, the MNP beamformer maximizes the output signal-to-noise ratio (SNR). FIGURE 69.3: (a) Single source in white noise; (b) uncorrelated interference; (c) correlated interfer- ence. We will not discuss in detail the behavior of the MNP and MV beamformers. The reader is referred to the work in [2]. We list some of the properties of the MNP and MV beamformers in two scenarios of practical interest. c  1999 by CRC Press LLC Case 1: Single Source in White Noise Here (see Fig. 69.3(a)), we assume that the noise n(t ) is sensor noise. We model it as n(t ) = u(t ) where the components of u(t ) are jointly independent and identically distributed samples of zero mean white noise sequences with variance σ 2 , i.e., R n = R u = σ 2 I , where I is the identity matrix. The sensor noise models the thermal noise generated at each receiver and is assumed independent of (thus, uncorrelated with) the source signal. If S is the power of the desired signal, the SNR at each sensor is SNR i = S/σ 2 . Also, from Eq. (69.7), we conclude that when the additive noise is white, as in this case, the MNP beamformer reduces to the DS beamformer. Moreover, computingthe powerattheoutputof the beamformer with Eq.(69.11)forthisparticular situation yields P o = σ 2 /N. This means that, at the output of the beamformer, the signal-to-noise ratioisSNR o = N SNR i . We conclude, for the case of a single source in white noise, that the DS beamformer is optimum in the sense of maximizing the output signal-to-noise ratio SNR o . Further, SNR o increases linearly with the number N of array sensors. Case 2: Directional Interferences and White Noise Now, we assume that the disturbance n(t ) is the superposition of possibly several directional interferences and white noise. Without loss of generality, we consider the case of a single interferer: n(t) = a(θ i )i(t) + u(t ) , (69.12) where i(t) is the signal radiated by the interferer, θ i is the DOA of the interference signal, and u(t ) is the white noise vector. In general, we assume that u(t ) is uncorrelated with i(t). Case 2.1: Uncorrelated Arrivals This is the case where the desired signal and the interference are generated by distinct sources, see Fig. 69.3(b). It is clear that under this assumption, s(t) and n(t) are uncorrelated. As we emphasized before, this is the situation where the MNP beamformer (69.7) is equivalent to the MV beamformer (69.8). The covariance of the noise n(t ) is now R n = a(θ i )S i a H (θ i ) + σ 2 I , where S i is the average power of the interference i(t). At the DOA=θ i , the beamformer has an amplitude response |w H a(θ i )|= |β| 1 + (1 −|β| 2 )INR , (69.13) where INR = S i /(σ 2 /N) is the interference-to-noise ratio (INR), and β = (1/N)a H (θ i )a(θ 0 ) measures the spatial coherence between the desired source and the interference. Well Separated Arrivals When the signal and interference are well separated, their spatial coherence is small, i.e., |β|1. In Eq. (69.13), the denominator is approximately given by 1 + INR. The net effect is that the beamformer output along the interference direction decreases when INR increases. In other words, the MNP and the MV beamformers direct a beam with gain 1 towards the DOA of the desired signal c  1999 by CRC Press LLC and null the interference. The interference canceling property is reflected on the average power of the beamformer output error which is evaluated to P o = σ 2 N 1 1 −|β 2 | INR 1+INR . (69.14) For large INR and well-separated DOAs, P o  (σ 2 /N). This means that the interference contributes little to the estimation error at the output of the beamformer. Close Arrivals When the source and the interferer are spatially close, their spatial coherence is large, |β|1, and the output at the interference DOA is  1. This means that the MNP and MV beamformers no longer have the ability to discriminate the two sources. We conclude from the simple analysis of these two cases that the DOA discrimination capability is strongly related to the spatial resolution of the array geometry through the parameter β. In practice, to improve upon the resolution of a linear and uniform array, we increase, when feasible, the number N of sensors. This results in narrower beamwidths which can resolve closer arrivals. Case 2.2: Correlated Arrivals This is the case where the interference i(t) is correlated with the desired signal s(t), i.e., E{s(t)i ∗ (t)}=ρ =|ρ|e jφ ρ = 0 . We denote complex conjugate by (·) ∗ . With reference to Fig. 69.3(c), we discuss a simple example where the interference results from a secondary path generated by a reflector (multipath propagation) i(t) = γs(t). (69.15) The complex parameter, γ , accounts for the relative attenuation and delay of the reflected path. The correlation factor, ρ, between i(t) and s(t) is, in this case, given by ρ = γ/|γ | . The desired signal and the disturbance vector n(t) are now correlated and the MV beamformer is no longer equivalent to the MNP beamformer. Recall that the MV beamformer attempts to minimize the total output power under the constraint of a unitary gain at the DOA of the desired source. As the array output vector has a correlated signal component at a different DOA, to minimize the output powermaycause the desired signal itself to be strongly attenuated. This is the signal cancellation effect, typical of MV beamforming when operating in multipath environments like the one just considered. On the contrary, the behavior of the MNP beamformer is independent of the correlation degree between the desired signal and the disturbance: the MNP beamformer filters out correlated arrivals just as if they were uncorrelated interferences. To implement the MNP beamformer, besides the DOA of the desired signal, we also need to know the covariance matrix R n of the disturbance vector. In general, this covariance is not known a priori. It has to be estimated using the available data, and this can be a rather complicated task, not discussed here. In this section, we discussed the MNP solution to the beamforming problem. The MNP beam- former is optimum in the sense of maximizing the output SNR. When the noise is white, the DS beamformer is recovered as the optimum solution for the single source case. It points a beam towards the source DOA and reduces the sensor noise power by a factor of N, see Fig. 69.4(a). We also saw c  1999 by CRC Press LLC that, in the more general situation where the disturbance vector includes directional interferences, the MNP beamformer acts like an interference canceler: it points a beam towards the DOA of the desired signal, while nulling the remaining arrivals regardless of their correlation degrees, see Figs. 69.4(b) and (c). FIGURE 69.4: Artistic representation of alternative solutions to the beamforming problem. We comment on the adequacy of the MNP solution to the correlated arrivals scenario. By treating the correlated arrivals as an interference, the MNP beamformer neglects the information about the desired signal that may be provided by the reflectedpath. It is clear that anysolution that can combine coherently the information contents of all the correlated arrivals will be more effective in recovering the desired signal from the background noise. This type of solution should behave as a combiner of the outputs of different beams steered towards the DOAs of the correlated replicas of the desired signal, see Fig. 69.4(d). In the following section, we will see how this solution can be designed using a different optimization criterion when solving the beamforming problem. 69.3 MMSE Beamformer: Correlated Arrivals In this section, we study a different beamforming technique that, as we will see, is specially suited to multipath propagation environments. The approach is similar to that used in the previous section. ThebeamformerisstillgivenbyEq.(69.5), but we choose a different optimization criterion to solve the beamforming problem. We formulate the problem in the following way: find the weight vector w such that the output error power E    w H z(t) − s(t)   2  is minimized , i.e.,wewanttofindtheweightvectorw thatminimizesthemeansquareerrorbetweenthebeamformer output y(t) = w H z(t) and the desired signal s(t). The general solution, which we call the Wiener solution, is [2] w = R −1 r z s , (69.16) where r z s = E{z(t )s ∗ (t)} is the correlation between the array vector response z(t) and the desired signal s(t). To understand the behavior of the MMSE beamformer (69.16), we address the same alternative configurations considered in Case 2 of the previous section. Directional Interference and White Noise In this scenario, the disturbance n(t) is like in Eq. (69.12), so the received array signal is z(t) = a(θ 0 )s(t) + a(θ i )i(t) + u(t ) , c  1999 by CRC Press LLC where, as before, i(t) and u(t ) are the interference and the array sensor noise vector, respectively. Uncorrelated Arrivals The signal s(t)and the interference i(t)are uncorrelated. The correlation between the array vector and the desired signal is r z s = a(θ 0 )S , where S is the average power of s(t). The MMSE beamformer weight vector in Eq. (69.16) takes the particular form w = SR −1 a(θ 0 ). (69.17) Comparing (69.17) with (69.8), we conclude that in the present situation, except for a scale factor, the MMSE and the MNP (or MV) beamformers are equivalent. Thus, the MMSE beamformer cancels uncorrelated interferences and directs a beam towards the DOA of the desired signal. However, contrary to what happens with the MNP (or MV) beamformer, the gain of the MMSE beamformer at the look direction is not unity. On the other hand, the MMSE beamformer provides a stronger noise rejection and a smaller output error power than the MNP beamformer. In fact, it can be shown [2] that P o (MMSE) ≤ SNR o 1 + SNR o P o (MNP), (69.18) where SNR o = S/(σ 2 /N), and P o (MNP) is the average power of the MNP beamformer output error givenby(69.14). Equation (69.18) is particularly significant in low SNR environments. Correlated Arrivals We take again the interference to be like in Eq. (69.15), i.e., i(t) = γ s(t). It leads to the correlation r z s = (a(θ 0 ) + γ a(θ i ))S and to the weight vector w = SR −1 a(θ 0 ) + SγR −1 a(θ i ). (69.19) It is clear from Eq. (69.19) that the MMSE beamformer directs distinct beams towards the DOAs θ 0 and θ i of the correlated arrivals in order to combine coherently their respective outputs. If R has contributions of other sources uncorrelated with s(t), then these will be filtered out by both beams. This simple example shows how the MMSE beamformer uses the correlated arrivals to improve the output error power. To have an idea of how the behavior of the MMSE beamformer compares with that of the MNP beamformer, we represent in Fig. 69.5 the output error power as a function of the INR and the SNR, for the case of spatially close arrivals. We see from Fig. 69.5 that the MMSE beamformer outperforms the MNP beamformer in all scenarios considered. This is particularly apparent in the limiting case of coherent arrivals, when the correlation between the signal and the interference is such that |ρ|=1. The upper and lower bounds of the output error power shown in the figure are determined by −1 ≤ cos(φ ρ ) ≤ 1. To implement the MMSE beamformer in a multipath propagation environment, we need to know the correlation vector r z s or, alternatively, the DOAs of all correlated replicas of the desired signal as well as their relative attenuations and propagation delays. In practice, r z s can only be estimated from the data if we have available a reference signal whose correlation with the array vector is similar to that of the desired signal. This is the basic idea underlying adaptive implementations (least mean squares approach); see, e.g., [8, 9]. In these implementations, a time sequence, which is known bythe receiver, is transmitted by the source and used to adapt the beamforming weights until some error threshold is achieved. In many applications, a reference signal is not available or, as in communications, the c  1999 by CRC Press LLC [...]... matrix of steering vectors A defined in Eq (69. 25) is a rank 4 (N × 4) matrix We now motivate how to estimate Define the matrix † Q = A† (R − σ 2 I)AH c 1999 by CRC Press LLC (69. 30) From Eq (69. 21), we get R − σ 2 I = HHH Substituting this equation in (69. 30), we get † Q = A† HHH AH But by (69. 24) Q = A† A Clearly, if A = A, then H † (69. 31) H Q= AH AH (69. 32) Apparently, to estimate , we should... structure of H defined in Section 69. 4.1 For the simple example that we have been considering, we write (69. 24) H = A(θ ) , where A(θ) = [a(θd1 ), a(θr1 ), a(θd2 ), a(θr2 )] (69. 25) is the (N × 4) matrix of the steering vectors associated with each incoming path, θ being the vector of the unknown DOAs The remaining unknown parameters of our model are collected in   γd1 0  γ 0   (69. 26) =  r1  0 γd2 ... eigenvectors associated to these eigenvalues From Eq (69. 26) defining , we see that has itself orthogonal columns In general, the columns of have different norms Using the eigenvalue decomposition in Eq (69. 33), we conclude that, except for a phase difference, the columns of V 1/2 equal the columns of Thus, we can conclude that Q1/2 = = V 1/2 (69. 34) Diag(ej φm ) (69. 35) where Diag(·) is a diagonal matrix and... equality in Eq (69. 35) is true only when A = A We now consider how to resolve this phase uncertainty We show that this uncertainty can be resolved up to a sign difference This means that, at least theoretically, we can achieve an estimate whose columns are colinear with those of Using Eq (69. 34) and Eq (69. 28), it can be shown after some algebra that † f = Q1/2 A† xi = Diag(e−j φm )si (69. 36) The elements... In summary, the algorithm to estimate is as follows: • use Rs to compute Q as defined by Eq (69. 30); • compute the singular value decomposition of Q and then form the matrix Q1/2 as given by Eq (69. 34); • compute f as given by Eq (69. 36) and use its elements to form the diagonal matrix F; • compute defined in Eq (69. 37) Using the results of Steps (1) and (2), we compute the estimate used to initialize... In this case, where S = I, this estimate is R = HHH + σ 2 I (69. 21) This estimate is better than the sample covariance matrix computed by Eq (69. 10) because it avoids eventual mismatches between the estimate of H and its actual value embedded in Rs In the following section, we will describe a method that provides accurate estimates of H 69. 4.2 Maximum Likelihood Estimation of H Suppose that T is the... = Hs(t) + u(t) , (69. 20) where u(t) represents the complex sensor noise vector The noise u(t) will be assumed to be zero mean Gaussian with known covariance matrix σ 2 I The goal is to estimate the signal vector s(t) using a MMSE beamformer which, as we saw in Section 69. 3, is the appropriate approach for multipath propagation environments However, to implement the MMSE beamformer (69. 16), we need to... in Fig 69. 8 the amplitude response of the MMSE and the MNP beamformers as a function of the angle of arrival (beampatterns) The MMSE beampatterns were computed using the weights obtained from the synthesized data, while in the case of the MNP we used the ideal weights We see that the behavior of both beamformers confirm what was predicted by the simple analysis carried out in Sections 69. 2 and 69. 3 The... considered c 1999 by CRC Press LLC FIGURE 69. 8: MMSE and MNP beampatterns: (a), (c)-channel M1 ; (b), (d)-channel M2 In this section, we showed the adequacy of the multichannel MMSE beamformer for combating multipath propagation effects as occurring in mobile communications systems The simulations illustrated the efficiency of the implementation proposed in Section 69. 4 69. 6 Conclusions In this article, we... factorization of the form (69. 32) This is not so easy because, in general, a unique factorization of a Hermitian matrix like Q does not exist Nevertheless, Q plays a key role in the procedure for estimating This is essentially based on the singular value decomposition of Q, as we see now Notice that Q can be computed from the data using the data sample covariance matrix Rs given by Eq (69. 10) instead of the . Mellon University 69. 1 Introduction 69. 2 Beamforming Minimum Output Noise Power Beamforming (MNP) 69. 3 MMSE Beamformer: Correlated Arrivals 69. 4 MMSE Beamformer. beamformer weight vector in Eq. (69. 16) takes the particular form w = SR −1 a(θ 0 ). (69. 17) Comparing (69. 17) with (69. 8), we conclude that in the present