Optimal Beamforming 1 Introduction

1 Optimal Beamforming1 IntroductionIn the previous section we looked at howfixedbeamforming yields significant gains in communi-cation system performance. In that case the Beamforming wasfixed in the sense that the weightsthat multiplied the signals at each element were fixed (they did not depend on the received data).We now allow these weights to change oradapt, depending on the received datato achieve a certaingoal. In particular, we will try to adapt these weights tosuppress interference. The interferencearises due to the fact that the antenna might be Signal w0*w1*wN-1*Figure 1: A general Beamforming Array Weights and the Weighted ResponseFigure 1 illustrates the receive Beamforming concept. The signal from each element (xn) is multi-plied with aweightw n, where the superscript represents the complex conjugate.

2 Note that theconjugate of the weight multiplies the signal, not the weight itself. The weighted signals are addedtogether to form the output signal. The output signalris therefore given byr=N 1Xn=0w nxn,=wHx,(1)wherewrepresents the lengthNvector of weights,xrepresents the lengthNvector of receivedsignals and the superscriptHrepresents the Hermitian of a vector (the conjugate transpose), ,wH= w 0, w 1, .. w N 2, w N 1 = wT .1 Using these weights, we can define aweighted response. If the received data is from a singlesignal arriving from angle , the received signal isx=s( ). The output signal, therefore is,r( ) =wHs( ).(2)Plotting this functionr( ) versus results in the weighted response or theweighted array, ofNelements, receives message signals fromM+ 1 users. In addition, the signal ateach element is corrupted by thermal noise, modelled as additive white Gaussian noise (AWGN).

3 The received signals are multiplied by theconjugatesof the weights and then summed. The systemset up is illustrated in Fig. 1. The weights are allowed to change, depending on the received data,to achieve a purpose. If the signals at theNelements are written as a length-Nvectorxand theweights as a length-Nvectorw, the output signalyis given byy=N 1Xn=0w nxn=wHx.(3)The signal received is a sum over the signals from multiple users,one of which we will designatethe desired signal. The received data is a sum of signal, interference and h0+n,(4)n=MXm=1 mhm+ noise.(5)The goal of Beamforming or interference cancellation is to isolate the signal of the desired user,contained in the term , from the interference and noise. The vectorshmare thespatial signaturesof themthuser. Note that, unlike in direction of arrival estimation,we arenotmaking anyassumptions as to the structure of this spatial signature.

4 In perfect line-of-sight conditions, thisvector is the steering vector defined earlier. However, in more realistic setting, this vector is a singlerealization of a random fading process. The model above is valid because are assuming the fadingis slow (fading is constant over the symbol period or severalsymbol periods) and flat (the channelimpulse response is a -function).3 Optimal BeamformingWe begin by developing the theory for Optimal Beamforming . We will investigate three techniques,each for a different definition of optimality. Define the interference and data covariance matricesasRn= E nnH and R = E xxH , Minimum Mean Squared ErrorThe minimum mean squared error (MMSE) algorithm minimizes the errorwith respect to a referencesignald(t). In this model, the desired user is assumed to transmit thisreference signal, , = d(t), where is the signal amplitude andd(t) is known to the receiving base station.

5 The outputy(t) is required to track this reference signal. The MMSE finds the weightswthat minimize theaverage power in the error signal, the difference between thereference signal and the output signalobtained using Eqn. (3)wMMSE= arg minwEn|e(t)|2o,(6)whereEn|e(t)|2o= En wHx(t) d(t) 2o,= E wHxxHw wHxd xHwd+dd ,=wHRw wHrxd rHxdw+dd ,(7)whererxd= E{xd }.(8)To find the minimum of this functional, we take its derivativewith respect towH(we have seenbefore that we can treatwandwHas independent variables). En|e(t)|2o wH=Rw rxd= 0, wMMSE=R 1rxd.(9)This solution is also commonly known as theWiener emphasize that the MMSE technique minimizes the error with respect to a reference technique, therefore, does not require knowledge of the spatial signatureh0, but does requireknowledge of the transmitted signal. This is an example of atraining based scheme: the referencesignal acts to train the beamformer Minimum Output EnergyThe minimum output energy (MOE) beamformer defines a different optimality criterion: we min-imize the total output energy whilesimultaneouslykeeping the gain of the array on the desiredsignal fixed.

6 Because the gain on the signal is fixed, any reduction in the output energy is obtained3by suppressing interference. Mathematically, this can be written aswMOE= arg minwE |y|2 ,wHh0=c, arg minwEn wHx 2o,wHh0=c.(10)This final minimization can be solved using the method of Lagrange multipliers, finding minw[L(w; )],whereL(w; ) = En wHx 2o+ wHh0 c ,= E wHxxHw + wHh0 c ,=wHRw+ wHh0 c ,(11) L wH=Rw+ h0 wMOE= R 1h0(12)Using the constraint on the weight vector, the Lagrange parameter can be easily obtained bysolving the gain constraint, setting the final weights to bewMOE=cR 1h0hH0R 1h0.(13)Setting the arbitrary constantc= 1 gives us the minimum variance distortionless response (MVDR),so called because the output signalyhas minimum variance (energy) and the desired signal is notdistorted (the gain on the signal direction is unity).

7 Note the MOE technique does not require a reference signal. This is an example of ablindscheme. The scheme, on the other hand, does require knowledge of the spatial signatureh0. Thetechnique trains on this Maximum Output Signal to Interference Plus Noise Ratio -Max SINRG iven a weight vectorw, the output signaly=wHx= wHh0+wHn, wherencontains bothinterference and noise terms. Therefore, the output signalto interference plus noise ratio (SINR)is given bySINR = En| |2o wHh0 2En|wHn|2o=A2 wHh0 2En|wHn|2o,(14)4whereA2= En| |2ois the average signal power. Another (reasonable) optimality criterion is tomaximize this output SINR with respect to the output SINR= arg maxw{SINR}.(15)We begin by recognizing that multiplying the weights by a constant does not change the outputSINR. Therefore, since the spatial signatureh0is fixed, we can choose a set of weights such thatwHh0=c.

8 The maximization of SINR is then equivalent to minimizing the interference power, SINR= minwEn wHn 2o=wHRnw,wHh0=c.(16)We again have a case of applying Lagrange multipliers, as in Section , withRreplaced ,w=cR 1nh0hH0R 1nh0.(17) Equivalence of the Optimal WeightsWe have derived three different weight vectors using three different optimality criteria. Is there a best amongst these three? No! We now show that, in theory, all three schemes are equivalent. Webegin by demonstrating the equivalence of the MOE and Max SINR criteria. Using the definitionsof the correlation matricesRandRn,R=Rn+A2h0hH0(18)Using the Matrix Inversion Lemma1R 1= Rn+A2h0hH0 1, R 1=R 1n R 1nh0hH0R 1nhH0R 1nh0+A 2,(19) R 1h0=R 1nh0 R 1nh0hH0R 1nh0hH0R 1nh0+A 2,=R 1nh0 hH0R 1nh0 R 1nh0hH0R 1nh0+A 2,= A 2hH0R 1nh0+A 2 R 1nh0,=cR 1nh0,(20)1[A+BCD] 1=A 1 A 1B DA 1B+C 1 1DA , the adaptive weights obtained using the MOE and Max SINR criteria are proportional to eachother.

9 Since multiplicative constants in the adaptive weights do not matter, these two techniquesare therefore show that the MMSE weights (Wiener filter) and the MOE weights are equivalent, we startwith the definition ofrxd,rxd= E{xd (t)},= E{[ h0+n]d (t)}.(21)Note that the term = d(t), where is some amplitude term andd(t) is the reference ,rxd= |d|2h0+ E{n}d (t)= |d|2h0 h0 wMMSE wMOE(22) , the MMSE weights and the MOE weights are also equivalent. Therefore, theoretically, all threeapproaches yield the same weights starting from different criteria for optimality. We will see thatthese criteria are very differentin Suppression of InterferenceWe can use the matrix inversion lemma to determine the response of the Optimal beamformer to aparticular interference source. Since all three beamformers are equivalent, we can choose any one ofthe three.

10 We choose the MOE. Within a constant, the Optimal weights are given byw=R asQthe interference correlation matrix without the particular interference source (assumedto have amplitude i, with E | i|2 =Ai, and spatial signaturehi. In this case,R=Q+ E | |2 hihHi=Q+AihihHi R 1=Q 1 A2iQ 1hihHiQ 11 +A2ihHiQ 1hi6 The gain of the beamformer (with the interference) on the interfering source is given the definition of the weights,wHhi=hH0R 1hi=h0Q 1hi A2i hH0Q 1hi hHiQ 1hi 1 +A2ihHiQ 1hi=hH0Q 1hi1 +A2ihHiQ 1hi(23)The numerator in the final equation is the response of the Optimal beamformer on the interferenceif the interferer were not present. The denominator represents the amount by which this responseis reduced due to the presence of the interferer. Note that the amount by which the gain is reducedis dependent on the power of the interference (corresponding toA2i).)