Random Beamforming in Correlated MISO Channels for Multiuser Systems Andreas Senst, Peter Schulz-Rittich, Ulrich Krause, Gerd Ascheid, and Heinrich Meyr Institute for Integrated Signal Processing Systems (ISS) RWH Aachen University Germany Email: senst@iss.rwth-aachen.de Abstract We examine the technique of random beamforming to exploit multiuser diversity in the downlink of a wireless cellular communication system with an antenna array at the basestation. In random beamforming systems, the scalar signal is multiplied by a weight vector and the resulting signal vector is transmitted by the antenna array. By varying the weight vector in time, one can increase the dynamic of the effective channel resulting in faster fading and a larger variance of the effective channel. his can be exploited by a scheduler at the basestation which selects users for transmission that momentarily have a good channel. Our work focusses on the generation of the weight vectors for a uniform linear array in the basestation. We especially consider correlation between the antenna elements, but assume that the correlation matrices for the different users are not known to the basestation. hree random beamforming approaches are compared with the simplest possible case of using equal weights, where the weight vector is a constant which serves only to normalize the radiated power. We derive the mean of the received power for the different beamforming techniques and confirm our results by Monte Carlo simulations, where we evaluate the mean power of the actually scheduled user, if a proportional fair scheduling algorithm is used. One important result is the good performance of the rotating beam approach even in a not fully correlated environment. I. INRODUCION In a recent publication, P. Viswanath et al. showed that the average throughput in a cellular mobile communication system can be considerably increased if multiuser diversity is exploited []. his refers to the fact that there are usually several mobile terminals in a cell waiting for data transmitted over the downlink from a basestation (BS) or central access point. If the mobile terminals estimate their instantaneous channel quality (Signal-to-Noise Ratio, SNR) and feed it back to the BS, a scheduler in the BS can use this information to schedule a user that momentarily has an above-average channel quality and can thereby increase the system throughput. In order to enable all terminals to get their fair share of the channel, it must be ensured that all terminals have a good channel every once in a while. his is either naturally the case, if the channel dynamic is large enough, i.e. the channel state varies fast enough, or a sufficient dynamic must be induced artificially by multiplying the signal with a time varying weight vector. For this principle, Viswanath et al. have coined the term Opportunistic Beamforming. It is important to note that this concept does not assume knowledge about the instantaneous channel vectors to the different users in order to calculate the weight vector. Instead, the weights are chosen at random. In [], the authors examine both the case of uncorrelated and fully correlated antennas at the basestation and one antenna at each mobile terminal. We also consider several antennas at the basestation and terminals with one antenna each, but we assume a fixed array geometry (ULA) and can thus examine a more realistic correlation between the antenna elements for the case that there are many scatterers around the terminals but none close to the basestation, as is the case for a basestation located above rooftop level []. he basestation knows neither the instantaneous channel vectors to the different users nor the covariance matrices of these channel vectors. What we assume to know in the basestation is the structure of the covariance matrices due to the special array geometry (ULA). he outline of the paper is as follows. In section II, the signal and channel models are introduced. he random beamforming schemes are presented and analyzed in section III. Section IV explains the simulated scenarios, the results of which are presented in section V, followed by a conclusion in section VI. II. SIGNAL AND SPAIAL CHANNEL MODEL One base station equipped with three antenna arrays consisting of antennas each serves a number of mobile terminals distributed in a cell. We assume that the cell is divided into three sectors, each sector being served by one array []. In each time slot, the unit labelled antenna weight generator generates a ( ) weight vector w. he data sequence a(k) is multiplied by w = [w,w,,w ] H, where w designates the complex conjugate and w H the transposed complex conjugate of w. he resulting vector a(k)w is transmitted. We set the variance of the transmitted symbols a(k) to be one. Let h = [h,h,,h ] be the channel between the antenna array and one terminal, i.e. h i is the channel between the terminal and antenna element i of the BS. he received signal r(k) can then be written as r(k) = h w a(k)+...+h w a(k)+n(k) = w H ha(k)+n(k)
where n(k) is white Gaussian noise with variance σ. It can be seen that after the introduction of the weight vector we can describe a single link in the system as a SISO system with an effective channel c := w H h. We consider a uniform linear array (ULA) as the antenna geometry at the BS and model the channel fading by assuming a ring of uniformly spaced scatterers around the mobile terminal. his model is called Lee s model []. It is valid for a macrocell scenario, where the base station is mounted at roof-top level and the mobile terminal is surrounded by scatterers, e.g. in an urban environment. hese scatterers cause the angular spread γ max, defined here as the ratio of the radius of the ring of scatterers around the terminal and the distance between the BS and the terminal: γ max = r scatterers d BS M () he correlation between the antenna elements in a scenario as the one described above has been derived in []. With increasing number of scatterers and no line-of-sight components present, it approaches ρ(j i θ) := E{h i h j θ} = J (π(j i) yγ max cos(φ))e jπ(j i) y sin(φ) () where J is the Bessel function of the first kind of order zero, and θ = { y,φ,γ max } is the set of parameters describing the scenario ( y is the antenna spacing in wavelengths and φ the angle of arrival, defined as the angle between the terminal and the broadside direction, which is orthogonal to the antenna array). hus, the covariance matrix of the ULA is a ( x ) hermitian oeplitz matrix with elements R i,j = E{h i h j θ} = ρ(j i θ) (3) III. EXAMINED BEAMFORMING SCHEMES A. he simplest possibility to generate the weight vector w is to only normalize the transmitted power. his is achieved by his results in w = [ ] () E{ c } = R i,j (5) which is in a scenario with uncorrelated antennas (R I) and i,j E{ c } = + ( i) J (πi yγ max cos(φ)) cos(πi y sin(φ)) (6) if a correlation according to (3) is assumed. Note that the second term in (6) can be both positive and negative, resulting in an increased or decreased channel SNR. his effect can also be seen in Fig. and Fig.. In both figures, the average received signal power is plotted over the angle of arrival E[ c φ].5 3.5 3.5.5.5 = 5, γ max =.3, y =.5 Eigen BF Phased Array Uncorrelated RV 6 6 Angle of Arrival φ [ o ] Fig.. Mean received signal power over angle of arrival. In case of the phased array and eigenbeamforming, the beam points towards the user. E[ c φ].5.5.5 = 5, γ max =.3, y =.5 6 6 Angle of Arrival φ [ o ] (A) Uncorrelated RV (A) Phased Array (A) Phased Array (S) Eigen BF (S) Fig.. Mean received signal power over angle of arrival for random beamforming averaged over time, A: analytically computed, S: simulated (direction of the users) in the broadside sector ( φ 6 ). he received power increases for users close to the broadside direction ( ) and decreases near the borders of the sector. B. Phased Array When a ULA is employed, a beam can be formed into a direction Φ, being the angle relative to broadside, by using the weight vector which leads to w = [,e jπ y sin(φ),...,e jπ( ) y sin(φ) ] (7) E{ c Φ} = + ( i)j (πi yγ max cos(φ)) in the direction Φ the beam points to. his is also shown in Fig.. Note that in this figure φ = Φ, i.e. this plot shows the average SNR of a user in direction φ if the array actually forms a beam into this direction. We see that a user near the border of the sector profits more from this technique than one in the center. his results from the fact that the correlation between the antenna elements increases with increasing angle of arrival. (In the endfire direction, the antennas would be fully correlated.) In our system, however, we do not want to beamform to a specific user (in fact, we do not even know where the users
are in the sector). herefore, we use a rotating beam, i.e. we have a Φ which is uniformly distributed between +6 and 6. his is actually the beamforming technique proposed in [] for the case of fully correlated antennas in Rayleigh fading. If we average over time (i.e. over the time varying Φ) and set u = sin φ sin Φ, we obtain (cf. [3]) E{ c } = + ( i)j (πi yγ max cos(φ)) sin(φ)+ 3 3 cos(πi yu) sin(φ) 3 π du (8) (sin(φ) u) which is shown in Fig.. Compared to using equal weights, the mean received signal power in the center of the sector is reduced in favor of users near the border of the cell. Compared to Fig., we note that the increase in mean SNR near the borders of the sector is reduced. In fact, the mean SNR after averaging is almost equally distributed over angle of arrival. his is due to the fact that a user in the center of the cell is always more or less in the focus of the beam (the difference between the angle of his position and the direction of the beam is always less than 6 ) and therefore he always profits a little bit. A user at the border of the sector on the other hand is not focussed so often (in 5 % of the time, the difference between the angle of his position and the direction of the beam is more than 6 ), but when the rotating beam comes close to his direction, he profits all the more. C. Eigenbeamforming In their paper, Hochwald et al. [6] show that the average data rate in a multiuser system with a scheduler that transmits always to the user having the momentarily best channel is a function increasing monotonously with both the mean and the variance of the mutual information of the links to the single users. It can be shown that both the expectation and the variance of the mutual information increase with increasing w H Rw/σ. herefore, we expect the average data rate to increase with w H Rw/σ and propose to maximize this quantity. Note that this corresponds basically to eigenbeamforming [5], which maximizes the average SNR at the receiver, if R is known. he solution to this problem is to use an eigenvector corresponding to the largest eigenvalue λ max of R as weight vector w. his leads to E{ c θ} = λ max at the mobile the beam is formed to. Note that θ denotes again the set of parameters describing the scenario, including angle of arrival and angular spread of that user. his is also plotted in Fig., where we have assumed an angular spread γ max of.3 for all hey show this for a system in which the pdf of the mutual information of the links to the single users can be approximated by identical normal distributions. he eigenbeamforming approach only attempts to maximize the expectation of the mutual information, we want to maximize a function of both the expectation and the variance. As both quantities increase with w H Rw/σ, it suffices to maximize the expectation in order to also maximize the variance and thus the average data rate. directions of arrival 3. We see that if we consider not only the angle of arrival φ as in the last subsection, but also the angular spread γ max, the mean SNR increases slightly, especially in the broadside direction, where the correlation between antenna elements is lowest. As in the case of the phased array, in the system under consideration we cannot (and do not want to) eigenbeamform to a specific user because we cannot expect to know the covariance matrices of the users. hus, we must use a technique similar to a rotating beam. herefore, we assume a certain distribution of the covariance matrices in the cell, e.g. by assuming that all users are distributed uniformly in a disk around the BS and that all circles of scatterers around the mobiles have identical radii. We can then determine (numerically) the distribution of optimal eigenvectors. By means of optimum quantization we select N of them (N being the number of quantization steps) and store them in a lookup table. he beamforming algorithm randomly picks one of the vectors stored in this table and uses it as weight vector. his results in a rotating eigenbeam. he numerically calculated mean SNR is plotted in Fig.. We now see that, compared with the rotating beam, the increase in mean SNR that we observed for the users in the broadside direction almost vanishes. his is due to the averaging over the complete sector of. herefore we expect that both beamforming schemes perform approximately equally well in a complete system incorporating a scheduler. he similar performance of eigenbeamforming and phased array beamforming in correlated scenarios can also be explained by the following. Note that the phase of ρ in () depends linearly on the argument: arg(ρ) = (j i) α with α = π y sin(φ). herefore the eigenvectors w of R can be written as w = [w...w ] = [ w, w e jα... w e j( )α] i.e. their phase depends on α. his shows that the eigenbeamforming approach is similar to the phased array technique in so far, as the phase of the optimum weight vector is completely determined by the direction we want to beamform to. In contrast to the phased array, however, the eigenbeamforming approach varies the distribution of the transmitted power among the antennas depending on the angle of arrival and angular spread. We can thus implement the rotating eigenbeam approach by randomly selecting an angle Φ uniformly distributed between +6 and 6 and then reading the absolute values of the elements of one weight vector corresponding to that angle from a lookup table. D. Uncorrelated Gaussian Weights he last random beamforming scheme that we examine here is the use of uncorrelated Gaussian weights, normalized to yield a transmitted power of one. his is the scheme proposed in [] for the case of slow channel fading. his results in E{ c Φ} = E{ c } = (9) 3 his value has been selected arbitrarily from the range of γ in the system simulation in section IV, cf. able I.
ABLE I MACROCELL MODEL PARAMEERS FOR SCENARIO Path loss exponent Radius of ring of scatterers r scatterers [m] Minimum distance to BS d min [m] Maximum distance to BS d max [m] 3 Average mean SNR [db] Minimum mean SNR L(d max) [db] -8.33 Maximum mean SNR L(d min ) [db] +. Minimum angular spread Maximum angular spread ( ) rscatterers d max ( ) rscatterers d min IV. EXAMINED SCENARIOS.3 In order to assess the performance of the beamforming techniques analyzed in the last section in a complete system, we use Monte Carlo simulations. We assume a block fading model. his means, that for each user, a channel vector h is drawn once per slot according to this user s covariance matrix R. he channel quality in terms of SNR is measured by the mobile terminals and is fed back to the scheduler in the base station. he estimation of the channel quality is beyond the scope of this paper and is treated in detail in [7]. Finally, the scheduler determines which user will receive the next data packet. he mean scheduled SNR determines the actual throughput of the system and is used to compare the different approaches. We examine two different scenarios. In the first, the users are distributed uniformly on a circle segment centered around the broadside direction of the antenna array at the BS. hus, only the angle between the BS and the Ms varies, the path loss and the angular spread are assumed to be equal. In the second, more realistic scenario, the users are distributed uniformly in a disk segment centered around the broadside direction of the antenna array at the BS. he model parameters are given in able I. In the second scenario, the rotating eigenbeam technique picks one eigenvector stored in a lookup table with 6 entries in each time slot. hey correspond to 6 discrete directions (Φ) and different angular spreads per direction. In the first scenario, we have stored eigenvectors corresponding to 6 discrete angles of arrival, as all users have the same angular spread. he rotating beam approach forms a beam into one of 6 discrete directions in each time slot in both scenarios. We assess the performance of the random beamforming schemes by using two different scheduling algorithms, Maximum Carrier-to-Interference-Ratio (Max C/I) Scheduling and Proportional Fair Scheduling (PFS) as proposed in []. Max C/I Scheduling maximizes the overall throughput in the cell by always scheduling the user whose instantaneous channel is best. hus, users close to the BS are preferred to users at the edge of the cell. PFS, on the other hand, introduces the concept of fairness by scheduling not the user who has the absolutely best he results were obtained using CoCentric System Studio by Synopsys.. Mean of Scheduled SNR 8 6 Eigen BF Phased Array 3 5 Number of Antennas Fig. 3. Mean scheduled SNR for different numbers of antennas and and users, bars for users are left and in front of the bars for users channel, but the one who has the relatively best channel. his is accomplished by normalizing the instantaneous channel quality by a measure of the channel quality in the past. hus, the PFS scheduler selects the user that has the largest metric m u (k) := c u(k) u (k) () where the index u {,...,U} indicates the different users and u (k) is the average SNR of user u calculated using an exponentially weighted moving average filter. his value is updated in every time slot k according to u (k+) = ) ( tc u (k) + t c c u (k) ) ( tc u (k) if u scheduled, otherwise. () where t c is the parameter of the PFS algorithm that determines the forgetting rate of the moving average filter. he larger t c, the longer is the history of the channel quality that the instantaneous channel quality is compared to. V. RESULS A. Scenario : Fixed distance between BS and terminals Some simulation results for the first scenario, where the users are located in a broadside sector with fixed distance to the BS and uniformly distributed angle of arrival are shown in Fig. 3. Here, we assume a fixed angular spread of.3 and E{ h i } =. We compare the achieved SNR gain for equal weights, rotating beam, and rotating eigenbeamforming for and users in the sector, when we use the Max C/I scheduler. his particular scheduler was chosen to investigate the maximum SNR gain achievable without trading SNR gain against fairness. We see that all three schemes yield very similar results, especially that the simple equal weights approach reaches similar SNR values and thus throughput as the more complex techniques. A closer inspection shows that this is achieved by scheduling mainly users in the broadside direction (φ = ), and almost neglecting users at the border of the sector. Due to the flattening of the array characteristics (as seen in Fig. ), the three beamforming schemes allow a more balanced channel allocation between users, which means that they increase
Mean of Scheduled SNR Proportional Fair, t c =5 =5, y =.5 d = 3 m, r Scatter = m 8 RR Coherent BF RR 7 Uncorr. Gauss. Weights Phased Array, 6 States Eigen BF, 6 States 6 5 3 3 Number of Users in Cell Fig.. Beamforming Schemes in Macrocell-Scenario fairness compared with using equal weights at virtually no mean SNR loss. he mean scheduled SNR increases with the number of users (e.g. by a factor of more than three for only users), which shows that we can indeed exploit multiuser diversity even given only a limited number of users. In addition, it increases with a larger number of antennas and a smaller element spacing. his last effect is not astonishing, as it is well known that the random beamforming schemes actually gain with increasing correlation between antenna elements (cf. []). B. Scenario : erminals uniformly distributed on a disk In real world environments, all users in a cell experience different path losses, angular spreads and have different angles of arrival. his is modelled in the second simulation scenario. We saw in the last subsection that the equal weights technique can achieve a performance comparable to the beamforming schemes by scheduling only users that are in the broadside direction of the array. In order to avoid this effect and to introduce fairness, we now replace the Max C/I scheduler by the PFS scheduler. he simulation results can be seen in Fig.. As expected, the equal weights technique is now outperformed by all other beamforming schemes. he results for uncorrelated Gaussian weights, the approach originally proposed in [] for slow fading channels, are a little better. However, we see clearly that we can increase the performance by taking correlation between antenna elements into account and that this performance gain increases with the number of users in the cell. he slashed line shows the mean SNR if the scheduler picked one user after the other (round robin) and a beamformer which knows the covariance matrices formed a beam towards this user. he most important result is that the two examined approaches that explicitly use knowledge about the structure of the correlation matrices between antenna elements hardly differ at all in terms of scheduled SNR. his can be explained by noting that the eigenbeamforming approach and the rotating beam technique perform almost equally well if the angular spread is small and thus the correlation between the antennas is large (cf. Fig. ). In our scenario, the angular spread reaches values of up to., but as the users are distributed uniformly in the cell, the number of users with a small angular spread is larger. his effect can be further amplified by using smaller antenna separations. In addition, averaging over the different directions that we beamform to reduces the differences further, as we see in Fig.. his shows that even in not fully correlated scenarios it is sensible to use the simpler phased array approach instead of the more complex eigenbeamforming technique. VI. CONCLUSION In order to increase the spectral efficiency of cellular wireless systems, all available forms of diversity should be exploited. One novel concept to exploit multiuser diversity is random beamforming as proposed in [], where the signal is multiplied by a time varying weight vector to increase the natural dynamic of the channel and then transmitted by an antenna array. In this paper, we compared different approaches to generate this weight vector in a scenario where the antenna elements are correlated, but where the correlation matrices of the different users are unknown to the BS. At first, we examined analytically how the different beamforming schemes affect the mean and variance of the effective SNR. We then showed by means of system simulations of different scenarios that these theoretical results are reflected in the scheduled SNR in a multiuser system. Especially interesting is the result that even in a not fully correlated scenario, the rotating beam technique achieves virtually the same mean SNR and thus throughput as the more complex rotating eigenbeamforming. Further investigations which examine the dynamic aspects of the different beamforming schemes, such as the effect of the rate of weight vector change on coherent channel estimation, were recently published [8]. REFERENCES [] P. Viswanath, D. N. C. se, and R. Laroia, Opportunistic Beamforming using Dumb Antennas, IEEE rans. Inform. heory, vol. 8, pp. 77-9, June. []. Fulghum and K. Molnar, he Jakes fading model incorporating angular spread for a disk of scatterers, Proc. 8th IEEE Vehicular echnology Conf., pp. 89-93, May 998. [3] U. Krause, Design of Opportunistic Beamforming Algorithms for Highrate Multiuser Systems, Diploma thesis, Institute for Integrated Signal Processing Systems, Aachen University, 3. [] J. C. Liberti, Jr. and. S. Rappaport, Smart Antennas for Wireless Communications, IS-95 and hird Generation CDMA Applications, Upper Saddle River: Prentice Hall, 999. [5] C. Brunner, W. Utschick, and J. A. Nossek, Exploiting the short-term and long-term channel properties in space and time: Eigenbeamforming concepts for the BS in WCDMA, European ransactions on elecommunication, vol., no. 5, pp. 365-378, Sept. - Oct.. [6] B. M. Hochwald,. L. Marzetta, and V. arokh, Multi-Antenna Channel-Hardening and its Implications for Rate Feedback and Scheduling, submitted to IEEE rans. Info. heory,. Download available at http://mars.bell-labs.com/cm/ms/what/ mars/papers/ratefeedback. [7] P. Schulz-Rittich, A. Senst,. Bilke and H. Meyr, he Effect of Imperfect SNR Knowledge on Multiantenna Multiuser Systems with Channel Aware Scheduling, Proc. IEEE GLOBECOM 3, pp. 53-57, Dec. 3. [8] P. Schulz-Rittich, A. Senst, U. Krause and H. Meyr, Increasing System hroughput by ime-varying Beamforming in Multiuser Systems with Slowly Varying Fading Channels, Proc. 58th IEEE Vehicular echnology Conf., Oct. 3.