Impact of Antenna Geometry on Adaptive Switching in MIMO Channels

Impact of Antenna Geometry on Adaptive Switching in MIMO Channels Ramya Bhagavatula, Antonio Forenza, Robert W. Heath Jr. he University of exas at Austin University Station, C0803, Austin, exas, 787-040 {bhagavat, rheath}@ece.utexas.edu Rearden LLC 355 Bryant Street, Suite 0, San Francisco, CA 9407 antonio@rearden.com Abstract Adaptive switching between multiple-input multiple-output MIMO transmission schemes like diversity and spatial multiplexing yields significantly higher capacity gains over fixed transmission schemes. Previous work has shown the impact of channel correlation and SNR on the switching between different schemes like beamforming and spatial multiplexing. his paper explores the influence of antenna geometry on the switching points between these transmission schemes. Closed-form expressions for the switching points for a two element uniform linear array ULA as a function of antenna geometry and channel statistics are derived. he impact of the channel parameters and antenna geometry on the switching points is also shown by simulations. he paper is concluded with a discussion on the impact of varying antenna array configurations on the switching points between beamforming and spatial multiplexing. I. INRODUCION Multiple-input multiple-output MIMO systems can offer significant capacity gains over single-input single-output SISO systems through the use of multiple transmit and multiple receive antennas. he additional spatial dimension offered by MIMO technology can be exploited by adaptively switching between techniques like diversity or spatial multiplexing to obtain higher link robustness or capacity gains [], [], [3], [4]. he adaptive switching can occur using either instantaneous channel state information [] or spatial correlation [3]. As the spatial correlation of a channel plays a dominant role in determining the optimal transmit strategy and varies more slowly than the instantaneous channel, adaptation based on the spatial correlation matrix provides a balance between performance and reduced feedback requirements. Previous work on correlation-based switching considered adaptive switching between diversity modes like statistical beamforming or space-time block coding, hybrid multiplexing double space time transmit diversity, and spatial multiplexing [3]. Switching points that indicate when to switch between these three modes of operation were derived using capacity as a switching criterion in [3]. hese switching points are a function of the transmit and receive correlations in the channel, which in turn are a function of the channel and antenna parameters [5], [6]. Antenna geometry in particular has a significant impact on the achievable capacity gains [6]. Unfortunately, previous work did not address the impact of antenna geometry on adaptive switching between transmission schemes. In this paper, we explore the impact of antenna geometry and channel parameters on the switching points between statistical beamforming BF and spatial multiplexing SM. For analytical tractability, we focus on the uniform linear array ULA with two active antenna elements, at the transmitter side. he antenna geometry is varied by varying the inter-element spacing between the antennas in the ULA. his is equivalent to selecting a subset of two transmit antennas. We derive closed form expressions for the switching points of the ULA between BF and SM, using bounds on the ergodic capacity. his gives us an insight into how the antenna geometries might be reconfigured adaptively in response to the channel parameters to improve performance. In particular, BF benefits from using antennas that are close together as the higher correlation between the antenna elements allows the channel to have a stronger single spatial direction. SM works better with antennas that are further apart. his causes the multiple streams to have lower correlation between them, which in turn is beneficial for SM. Using numerical simulations we study the switching between BF and SM for other array configurations, like the uniform circular array UCA and star configurations, to show the impact of antenna configurations on adaptive switching between BF and SM. his motivates the development of reconfigurable MIMO antenna arrays. II. CHANNEL MODEL Consider a wireless link with M transmit antennas and M R receive antennas. he channel impulse response for the open loop MIMO system is represented by an M R M matrix, H. he channel is also assumed to be frequency flat. he received signal, y is given by Es y = Hx + n M where y C MR, E s is the average transmit power per symbol, x C M is the input signal vector, subject to the power constraint E x = M, n is

Fig.. Array geometries considered: a ULA with an interelement spacing of λ, and b ULA with an inter-element spacing of 5λ. he active elements are shown in black, and the inactive element is in grey. the zero-mean additive white Gaussian noise vector with covariance matrix given by Enn = N o I MR, where N o is the noise power. γ o = E s /N o represents the signal-tonoise ratio SNR. he statistical correlation between the entries of the channel matrix, H can be expressed as R tot = EvecH vech. Assuming a Kronecker model for the correlation [5], R tot can be written as R tot = R R R 3 where R and R R stand for the transmit and receive covariance matrices respectively and stands for the kronecker product. o simplify our analysis, we consider a single-cluster channel with single-sided correlation and set R R = I MR. hus, we are essentially studying the impact of using different array configurations at the transmitter side. III. ANALYSIS For our analysis, we use the ULA configuration consisting of two antenna elements. Fig. shows the three-element configuration that uses two active antenna elements at a time for a given configuration. he two array geometries - ULA and ULA have inter-element spacings of λ and 5λ, respectively as shown in Fig.. We analyze the performance of each of these arrays in terms of capacity by considering two MIMO transmit strategies - beamforming BF and spatial multiplexing SM. In [3], an adaptive algorithm to switch between BF and SM in spatially correlated MIMO channels was proposed based on channel correlation and SNR. In this paper, we demonstrate that the SNR switching thresholds between BF and SM are a function of array geometries i.e the inter-element spacing and the channel parameters. For small values of the angle-of-departure AoD and angular spread, the correlation coefficients of a ULA with a Laplacian distributed power azimuth spectrum, are approximated as [7] e kodm lsinφc r l,m 4 + σ φ [k odm lcosφ c ] where k o = π/λ is the wavenumber λ is the wavelength at which the antenna operates, σ φ is the standard E stands for expectation,. stands for transposition and conjugation, I R stands for a R R identity matrix and. is the transpose. deviation of the power azimuth spectrum and φ c is the mean AoD of the single cluster and d is the interelement spacing of the antenna elements in the ULA. he eigenvalues for a ULA [8] for low values of angular spread and around the broadside direction of AoD φ c = 0 degrees can be derived from 4 as λ ULA, = ± 5 + σ φ k odcosφ c where λ ULA, are the two eigenvalues of the transmit correlation matrix. It is to be noted that as we are considering a single-sided correlated channel model, we let λ ULA R, =, where λ ULA R, are the eigenvalues of the receive correlation matrix. A. Capacity of the MIMO System Using Beamforming he closed-form expression of the capacity of a MIMO system using BF in [3] is not valid for a single-sided correlated channel. he general capacity of a MIMO channel that employs beamforming using a maximum ratio combining receiver is given in [3] as C BF γ o = E [log + γ ] oλ tmax η 6 where where λ tmax is the maximum eigenvalue of the transmit covariance matrix, R. η = M R i= ε i, where ε i s are i.i.d. exponentially-distributed random variables. For a MIMO system, η = ε + ε. he p.d.f. of η is given as fη = 4 ηe η/, for η > 0. 7 he capacity of the MIMO system using beamforming can then be calculated as C BF γ o = log + γ oλ tmax η fηdη.8 0 his can in turn be expressed in closed-form as 4Λ o + [ Λ o ] e Λo 0, Λ o Γ C BF Λ o = 9 4Λ o ln where Λ o = γ o λ tmax / and Γ.,. is the incomplete Gamma function. For a two-element ULA, as per 5, for small values of AoD and angular spread, λ tmax can be written as λ ULA = +. 0 + σ φ k odcosφ c BF needs a single strong spatial direction, which in turn implies a large maximum eigenvalue at the transmit side. From 0, it is evident that as the inter-element spacing, increases. his in turn increases the value of Λ o, which will increase capacity. herefore, it can be said that for a constant SNR, BF will perform best for smaller values of d. Intuitively, this can be explained by recognizing that the correlation between antenna elements is maximum for smaller d, leading to better performance using BF. d reduces, the value of λ ULA

B. Capacity of the MIMO System Using Spatial Multiplexing he capacity of a MIMO channel employing spatial multiplexing and using a zero-forcing ZF receiver, is given by the following closed form expression [3] C SM γ o = exp k= R kk R γ o ln Γ 0, Rkk R γ o where R kk corresponds to R with the k th row and k th column removed. In the case of the two-element ULA, R and R =. can be obtained from [8] and [7] as R = R Using R ULA = λ ULA λ ULA, we obtain R ULA = + σ φ k odcosφ c. For a MIMO system, using ULA at the transmit side, can be rewritten as CSM ULA γ o = ln exp R ULA Γ 0, γ o R ULA γ o 3 where R ULA is given by. It is to be noted here that and hence, 3 are approximations that are valid only for small values of AoD and angular spread. From, it is seen that as d increases at constant SNR, the value of R ULA increases, thereby increasing the value of the capacity of the MIMO system. his can also be explained by considering that larger interelement spacing will cause the multiple data streams from the different antenna elements to have lower correlation among them, which is desired for SM. C. Switching Points between BF and SM From the previous sections, we saw that a ULA with closely spaced antenna elements will produce larger capacity than a ULA with widely spaced antenna elements using BF. We also saw that the converse is true for SM. Hence, we consider in our analysis, switching between the ULA -BF combination to ULA -SM combination as a means to obtain maximum capacity gains. he theoretical crossing points between BF and SM are derived in [3] for both the transmission schemes using the same antenna geometry. We extend this analysis to obtain the switching point for the case when antenna geometry is varied along with the transmission scheme, i.e. from ULA -BF to ULA -SM. We define Mode to be the ULA -BF combination and Mode to be the ULA -SM combination. Let R,ULA and R,ULA be the transmit covariance matrices of the two configurations shown in Fig.. he switching point for a MIMO system that switches Fig.. Variation of the switching point with respect to the AoD and angular spread for an inter-element spacing at ULA and ULA of 5λ. between the transmission schemes with the same antenna geometry is given in [3] as γ CP = 4 4 R λ tmin where λ tmin is the minimum eigenvalue for the transmit covariance matrix, R. he switching point for a MIMO system that switches from Mode ULA - BF to Mode ULA -SM is γ CP = 4 λ ULA R,ULA R,ULA. 5 It is to be noted that these switching points have been derived from the upper limit of the capacities of BF and SM, as given in [3]. IV. RESULS In this section, we show the impact of channel parameters and antenna geometry on the switching points between Mode and Mode. We finally show the impact of varying antenna configurations on the switching points, by means of simulations. A. Impact of Channel Parameters on Switching Points In this section, we show that the influence of channel parameters AoD and the angular spread on the switching points between BF and SM. 5 shows the impact of σ φ and φ c on the eigenvalues of the transmit correlation matrix of the ULA and hence on the switching point, as shown in 5. In this section, we explore this effect by using the ULA configuration with an inter-element spacing of d = 5λ. he impact of AoD and angular spread on the switching point is shown in Fig.. he plot shows the variation

of the switching point only for AoD and angular spread up to 30 degrees, because 4 is valid only for small values of AoD and angular spread. It is seen from Fig. that the switching point decreases with increasing angular spread. As the angular spread increases for a given AoD, in 5,which determines the SM-ZF performance, gets larger. Hence, from 4, the crossing point reduces with an increase in angular spread. It is also seen from Fig. that for a constant angular spread, the crossing point increases slightly with an increase in AoD. his can be explained from 5 by noting that as the value of λ ULA t min and in turn R,ULA increases. his causes the crossing point to reduce, as per 4. AoD increases for constant angular spread, λ ULA t min B. Impact of Antenna Geometry on Switching Points he impact of the antenna geometry employed at the transmit side is evident from 5. his has been illustrated in this section. We considered the two ULA configurations shown in Fig.. For the purpose of analysis we consider a single cluster channel with AoD = 0 degrees and angular spread = 5 degrees. We compute the switching points between BF and SM for the two configurations in Fig., and then compare the results with those obtained by means of the closed-form expressions derived. Mode performs best in the case of low SNR, which is expected, as Mode corresponds to beamforming and closely spaced antennas are preferred for beamforming. Mode, which is a widely spaced antenna array configuration performs best at the higher SNRs. Hence, the best performance in terms of capacity is obtained by switching between Mode to Mode. Fig. 3 shows the crossing point between the two modes. It is to be noted here that the upper bounds of the capacities for Modes and have been plotted here to show that the crossing point obtained here is 8.5 db, which is in close correspondence with that obtained by means of the closed- form expression in 5, which is 8.8 db. We also show the impact of the spacing between the antenna elements in Fig. 4. It is seen from here that as the distance between the antenna elements increases, the switching point reduces. his is because with all the other parameters constant, the as d increases, the value of R,ULA reduces. his causes the crossing point to reduce, as shown in 5. V. IMPAC OF ANENNA CONFIGURAION ON SWICHING POINS We show by simulation results, the impact of different antenna configurations on the switching points between BF and SM. For this purpose, we consider three commonly used array configurations - ULA, UCA and star configurations. o illustrate the concept of reconfigurable MIMO antenna arrays, we consider an antenna configuration with nine antenna elements shown in Fig. 5. At any time, only four of the antenna elements are active shown in black. Fig. 3. Capacities of the two transmission modes: Mode - ULA and BF, and Mode - ULA and SM, as a function of the SNR. he crossing point between the two modes is also shown. Fig. 4. Variation of the switching point with respect to the interelement spacing in wavelengths at ULA and ULA for AoD = 0 degrees and angular spread = 5 degrees. he diameter of the circular access point is taken as 3λ for the purpose of analysis. Using the array response vectors described in [6], the capacity of each of the three antenna array configurations was obtained for the two transmission schemes - beamforming and spatial multiplexing. A single-sided correlated clustered channel model was considered here with the AoD = 30 degrees, and an angular spread = 00 degrees. he performance of beamforming and spatial multiplexing for each of the three different antenna types is reported in Fig. 6. It is seen that for BF, the ULA performs the best among all the three configurations. his is because the

inter-element spacing for the ULA is the small enough that no grating lobes are present. he inter-element spacing in other configurations is comparatively larger leading to the presence of grating lobes that degrade performance. For the case of SM, the scenario is reversed. his is again explained by means of the inter-element spacing, which is least for the ULA. his causes the performance of the ULA to be the worst of all the three configurations. he other two, having relatively larger inter-element spacing, have better performance, as seen in Fig. 6. It is also seen that the different array configurations give a significant performance difference for SM, as against BF, where the performance does not vary too much. It is seen from [3] that for BF, capacity is a function of just the maximum eigenvalue of the transmit covariance matrix. For SM with the ZF receiver, capacity is a function of all the eigenvalues of the transmit covariance matrix. his makes SM with the ZF receiver more sensitive to changes in the array configurations as compared to BF. Finally, Fig. 6 reveals that the SNR switching thresholds are a function of the array configurations. In the case of a single MIMO antenna array system, the switching threshold is only a function of the channel parameters. In reconfigurable antenna array systems, capacity can be maximized by switching between different antenna arrays for different schemes based on the channel parameters. For the current channel model, an ideal adaptive algorithm for reconfigurable MIMO arrays would select BF with ULA for SNRs lower than 7dB and SM with UCA configuration for SNRs greater than 7dB. VI. CONCLUSIONS In this contribution, we showed the impact of the antenna geometry and channel parameters on the switching points between BF and SM. Considering a MIMO system and a single-sided correlated clustered channel model that uses ULA configuration at the transmit side, we derived closed form expressions for the switching points between the transmission schemes as a function of both the channel parameters and array geometries. We also showed that antenna array configurations also impact the switching points obtained between BF and SM. Future work involves Fig. 5. Array configurations considered: a ULA, b UCA, and c Star configurations. he reference system for the angle-ofdeparture AoD has been shown. he active elements for each configuration are shown in black, and the inactive elements, in grey. Fig. 6. Comparison of the performance of beamforming and spatial multiplexing for three different antenna array configurations. analyzing the impact of varying antenna configurations on the switching points between BF and SM. Future work in this area is aimed towards developing an adaptive algorithm that will give the best performance with respect to capacity. VII. ACKNOWLEDGEMENS his material is based upon work supported by the National Science Foundation under Grant No. CCF-5494, the Office of Naval Research under grant number N0004-05--069. REFERENCES [] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, From heory to Practice: An Overview of MIMO Spaceime Coded Wireless Systems, IEEE Jrnl. on Select. Areas of Comm., vol., no. 3, pp. 8-30, Apr. 003. [] R. W. Heath, Jr. and A. J. Paulraj, Switching Between Diversity and Multiplexing in MIMO Systems, IEEE rans. on Comm., vol. 53, no. 6, pp. 96-968, June 005. [3] A. Forenza, M. R. McKay, A. Pandharipande, R. W. Heath Jr., and I. B. Collings, Adaptive MIMO ransmission for Exploiting the Capacity of Spatially Correlated Channels, accepted for publication in IEEE rans. on Veh. ech., Apr. 005. [4] S. Catreux, V. Erceg, D. Gesbert and R. W. Heath Jr., Adaptive Modulation and MIMO Coding for Broadband Wireless Data Networks, IEEE Comm. Mag., vol. 40, no. 6, pp. 08-5, June 00. [5] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading Correlation and its Effect on the Capacity of Multielement Antenna Systems, IEEE rans. on Comm., vol. 48, no. 3, pp. 50-53, Mar. 000. [6] A. Forenza and R. W. Heath Jr., Impact of Antenna Geometry on MIMO Communication in Indoor Clustered Channels, Proc. of AP-S Intern. Symp., vol., pp. 700-703, June 004. [7] A. Forenza, D. J. Love, and R. W. Heath Jr., A Low Complexity Algorithm to Simulate the Spatial Covariance Matrix for Clustered MIMO Channels, Proc. of the IEEE Veh. ech. Conf., vol., pp. 889-893, May 004. [8] A. Forenza and R. W. Heath, Jr., Benefit of Pattern Diversity Via -element Array of Circular Patch Antennas in Indoor Clustered MIMO Channels, IEEE rans. on Comm., vol. 54, no. 5, pp. 943-954, May 006.