Effect of Signal and Noise Mutual Coupling on MIMO Channel Capacity

Wireless Personal Communications (2007) 40: 317 328 DOI 10.1007/s11277-006-9193-8 c Springer 2006 Effect of Signal and Noise Mutual Coupling on MIMO Channel Capacity SNEZANA M. KRUSEVAC 1,RODNEYA.KENNEDY 1 and PREDRAG B. RAPAJIC 2 1 Research School of Information Science and Engineering, The Australian National University and National ICT Australia, ACT 0200, Australia E-mails: snezana.krusevac@rsise.anu.edu.au, rodney.kennedy@anu.edu.au 2 University of Greenwich, Chatham Maritime, United Kingdom E-mail: P.Rapajic@greenwich.ac.uk Abstract. In this paper we analyze the impact of mutual coupling on MIMO channel capacity, considering its effect on both signal and thermal noise. We calculate noise correlation matrix in the multi-antenna system with closely spaced antenna by applying Nyquist s thermal noise theorem. Then, we employ the noise correlation matrix in the channel capacity formula, which enables the identification of thermal noise correlation contribution on the MIMO channel capacity. In addition, we examine the variations in the mean branch signal-to-noise ratios (SNR) due to the noise correlation. Our simulation results corroborate the theoretical analysis that mean and outage MIMO channel capacity is underestimated if noise correlation due to mutual coupling effect is not accounted for. Keywords: mutual coupling, thermal noise, MIMO channel capacity 1. Introduction MULTIPLE-INPUT multiple output (MIMO) wireless systems, characterized by multiple antennas at the transmitter and receiver, have demonstrated the potential for increased capacity by exploiting the spatial properties of the multi-path channel [1]. If the channel matrix coefficients are independent identically distributed (i.i.d.) complex Gaussian variables, a linear increase in capacity with the number of antenna is possible. The mutual independence of channel coefficients is generally achieved by wide inter-element spacings in multi-element antenna systems. However, it is not practically achievable due to limitations in the physical size of subscriber units. Close antenna spacing in multi-antenna systems results in mutual coupling which affects the capacity performance of the MIMO system [2]. Principally, the investigation of mutual coupling is through the channel matrix coefficients and their cross-correlation [3 5]. Furthermore, in order to corroborate the theoretical analysis, experimental results have been presented [6]. Adiscussionaboutthemutualcouplingimpactonthesignal-to-noiseratio(SNR),radiation pattern, antenna gain, spatial correlation, and their mutual influence on the channel capacity has been presented in [7]. Finally, the RF end-to-end network analysis has been presented in two papers [8, 9]. In order to provide an extensive analysis of mutual coupling on MIMO channel capacity, authors investigate a great variety of parameters that are affected by mutual coupling, but one important parameter is still missing the frontend noise. Though, in [9, 10], the receive

318 S.M. Krusevac et al. amplifier noise is included into their analysis, antenna noise is taken to be white and gaussian. In summary, antenna thermal noise has been neglected from the mutual coupling effect investigation. Although, the radiation characteristic of thermal noise was discussed in [11] and the mutual coupling effect in two closely spaced antenna was presented in [12]. To our knowledge, the modeling of spatial noise correlation due to mutual coupling and its effect on MIMO channel capacity has not been addressed. In this paper, we extend the analysis of the effect of mutual coupling on MIMO system performance, considering its effect on both signal and noise and showing its combined impact on channel capacity of MIMO wireless systems. 1. We derive the noise correlation matrix, which enables the investigation of the effect of noise mutual coupling on the MIMO channel capacity. We apply Nyquist s thermal noise theorem [14] to calculate the noise correlation matrix. 2. We develop a MIMO channel capacity formula considering the mutual coupling effect on both signal and noise. We apply the eigenvalue decomposition to get better insight into the capacity performance of MIMO system with coupled antennas. The advantage of the eigenvalue decomposition is that it directly provides information about the MIMO channel capacity co-factors: the intensity of each MIMO sub-channel (evaluated by SNR level) and the effective active number of MIMO sub-channels. 3. Our simulation results show that ergodic and outage channel capacity is underestimated for antenna spacing up to 0.35λ and slightly overestimated for antenna spacing range [0.35λ, 0.5λ], ifmutualcouplingeffectonthermalnoiseisnotaccountedfor.additionally, we show that the mutual coupling affects thermal noise in such a way that it acts as an increasing factor to the mean branch signal-to-noise ratio of MIMO systems in the antenna spacing range [0, 0.35λ]. The rest of this paper is organized as follows. In Section 2, the MIMO system model is given. The signal correlation due to the mutual coupling and its impact on the MIMO channel capacity is presented in Section 3. Then, the noise correlation matrix is derived in Section 4. The theoretical analysis of the noise mutual coupling effect on the MIMO channel capacity is presented in Section 5. Simulation results are presented in Section 6. Concluding remarks are given in Section 7. 2. MIMO System Model For MIMO system with n T transmit and n R receive antennas, transmitted and received signals are related by y = Hx + n where x and y are (n T 1) transmitted and (n R 1) received signal vectors, respectively. Noise is represented by a (n R 1) noise vector n and H is a (n R n T ) channel matrix. (1) 3. Mutual Coupling Effect The principal feature of an antenna is to convert an electromagnetic field into an induced voltage or current. However, for closely spaced antenna elements, the total (measured)

Impact of Mutual Coupling on MIMO 319 voltage on each antenna element is a function not only of the excited field but also of the voltages on the other elements. The phenomenon is known as mutual coupling and can be included in the received voltage model, by inserting coupling matrix [3] y = MHx + n Thus, the new channel matrix that considers the antenna mutual coupling effect in the multiantenna system is H mc = MH [3]. The coupling matrix for the multi-antenna systems, based on the electromagnetic and circuit theory for the multi-antenna system, can be calculated as [5] M = (Z A + Z L )(Z + Z L I) 1 (3) where Z A is the isolated antenna impedance and Z the mutual impedance matrix. While, Z L the impedance at the receiver loads of each element and it is chosen as the complex conjugate to Z A. The impedance mismatch due to the mutual coupling effect [9] is not considered in this paper. In addition, as the receiver antenna elements are closely spaced, the fading correlation [13] has to be included into the channel matrix. Thus, the channel matrix that considers both spatial correlations, fading, and electromagnetic can be written H c = MH f. (2) 4. Noise Correlation In this Section, we calculate the noise correlation matrix N c,whichexistsduetothemutual coupling. We apply the generalized Nyquist s thermal noise theorem [14] to derive an expression for this matrix. 4.1. Generalized Nyquist s Thermal Noise Theorem The generalized Nyquist s thermal noise theorem [14] allows us to determine thermal noise power of coupled antenna in a multi-antenna system. The magnitude of correlation between two closely spaced antenna with isolated receivers is given in [12]. The theorem states that for apassivenetworkinthermalequilibriumitispossibletorepresentthecompletethermal-noise behavior by applying Nyquist s theorem independently to each element of the network. In the case of the multi-antenna system these elements are self-impedances and mutual-impedances. General, even nonreciprocal, networks with a system of internal thermal generators all at absolute temperature T is equivalent to the source-free network together with a system of noise current generators I r and I s with infinite internal impedance [14]. Noise currents are correlated and their cross-correlation is given by I s I r = 2kT ( Y sr + Y rs ) where Y rs and Y sr are the mutual admittances and k is Boltzmann s constant and T is absolute temperature. Correlation is zero when the mutual coupling is purely reactive. 4.2. Noise Correlation Matrix In order to derive the noise correlation matrix, we use a nodal network for the multi-antenna system with n R antenna elements, as shown in Figure 1. (4)

320 S.M. Krusevac et al. Figure 1. Nodal network representation for multi-antenna system. The spectral density of total thermal noise current for the multi-antenna system (Figure 1) is given by J = j + j L = (Y + Y L ) V (5) where J is the column vector of spectral density of total noise current and V is the column vector of spectral density of noise voltage. Additionally, Y is the mutualadmittancematrix of the multi-antenna system; Y L is the diagonal admittance matrix of the receiver loads. Then, the spectral density of thermal noise voltage of the multi-antenna system is V = (Y + Y L ) 1 J = (Y + Y L ) 1 ( j + j L ) The power spectral density matrix of thermal noise N c ( f ) is (6) N c ( f ) = 1 2 (Y + Y L) 1 VV H (7) where (.) H denotes the Hermitian transpose.

Impact of Mutual Coupling on MIMO 321 In order to calculate (7), we derive the spectral squared noise voltage matrix VV H,inthe following form VV H = (Y + Y L ) 1 JJ H ( (Y + Y L ) 1) H Based on (4) the following relations are valid: ) 1. j j jk (y = 2kT jk + ykj ; 2. j Lj j k = 0 ; 3. j Lj j Lk = 0 ; j = k where (.)* denotes complex conjugate. Now, the spectral squared current matrix of thermal noise becomes JJ H = ( ) ( ) H j + j L j + j L = 2kT [(Y + Y H) ( )] + Y L + Y H L Substituting (10) in (8) the spectral noise voltage matrix of thermal noise can be expressed as VV H = 2kT (Y + Y L ) 1 ((Y + Y L ) + (Y + Y L ) ) ((Y + Y L ) 1) H Combining (11) and (7), the power spectral density matrix of thermal noise for the multiantenna system with closely spaced antenna can be expressed as N c ( f ) = kt ( Y L + Y ) 1 L [(Y + Y L ) 1 ((Y + Y L ) + (Y + Y L ) ) ((Y + Y L ) 1) H ] 4.2.1. Two-antenna case In order to get better insight into the noise correlation due to mutual coupling, we present atwo-antennaexample:oneelementofthepowerspectraldensitymatrixofthermalnoise N c ( f ) for the two-antenna system is given by ( YL1 + YL1) N c ( f ) [1, 1] = 2kT 2 D D (y 22 +Y L2 ) ( y22 +Y L2)(( YL1 +Y ( L1) + y11 +y )) ( 11 y 21 y 22 + YL2)( y12 +y12) y 12 (y 22 +Y L2 ) ( y 12 +y12 ) +y 21 y21 (( YL2 +YL2 ) ( + y22 +y22)) (13) where y 11 and y 22 are the self-admittance of first and second antenna, respectively and y 12 and y 21 are the mutual admittances. Here, the mutual admittance y 12 quantifies the contribution from the second antenna to total noise power of the first antenna. One can understand that the first term in (13) is thermal noise that originates in its own resistive element and the last three terms represent the noise induced from the adjacent antenna element. Now, based on (12), the noise power matrix of the multi-antenna system with closely spaced antenna is given by N c = N c ( f ) (14) B where B is the frequency band of receiver band-pass filter. (8) (9) (10) (11) (12)

322 S.M. Krusevac et al. For narrow-band system, the noise power matrix (14) becomes N c ( f ) = ktb ( Y L + Y ) 1 L [(Y + Y L ) 1 ((Y + Y L ) + (Y + Y L ) ) ((Y + Y L ) 1) H ] In (15), we assume that the mutual admittance matrix Y and the load admittance matrix Y L are invariant within the narrow frequency band B. Additionally, for the purpose of the channel capacity analysis, we define the normalized noise correlation matrix N nc = N C N 1.Here,N is the uncorrelated noise matrix and its elements represent noise powers of isolated, widely spaced, antenna elements. (15) 5. Channel Capacity Analysis We analyze the channel capacity of MIMO system with closely spaced antenna elements when the transmitter does not know the channel characteristics. Transmitted power is equally distributed across all n T transmit antenna elements. Additionally, we assume that transmitted signals are independent and that their covariance matrix is Q = P TX n T I nt where P TX is total emitted power and I nt is identity of order n T. Defining N R as a correlation matrix of the noise samples at the receiver and H c as the channel matrix that includes spatial (fading [13] and electromagnetic [7]) correlation, the ergodic channel capacity for (n T n R ) MIMO system based on [1] and [15] can be expressed as [16] { ( )]} C = E H log 2 [det I nr + N 1 R H cq H H c (16) where E H is expectation over all channel realizations. For MIMO system with closely spaced antenna elements, the correlation matrix of the noise samples at the receiver N R is equal to the noise correlation matrix N C. Thus, the channel capacity becomes C = E H { log 2 [det ( I nr + N 1 c H c Q H H c )] } By substituting the noise correlation matrix N C = N nc N,thechannelcapacitybecomes { ( )]} C = E H log 2 [det I nr + (N nc N) 1 H c Q H H c (18) (17) Furthermore, { ( C = E H log 2 [det I nr + ρ )] } Nnc 1 n H c Q H H c T (19) where ρ/n T is the mean signal-to-noise per one receiving antenna and ρ is defined by ρ = P TX /N.

Applying an eigenvalue decomposition to H c H H c becomes { n C = E H log 2 i=1 Impact of Mutual Coupling on MIMO 323 and N 1 nc,theergodicchannelcapacity ( 1 + ρ ) } { n ( λ i = E H log n T ν 2 1 + ρ ) } λ i (20) i n i=1 T ν i where n = rank ( N 1 nc H C H H C) min (nt, n R ),λ i is eigenvalue of matrix H c H H c and ν i is eigenvalue of normalized noise correlation matrix N nc.relation(20)istrueastheeigenvalues of N nc matrix compose a diagonal matrix, and the inverse of diagonal ( matrix is) diagonal matrix of its inverse elements A = diag(a 1,...,a n ) A 1 = diag a1 1,...,a 1 n The eigenvalues ν i of the noise correlation matrix N nc are given by ν i = N c (i, i) N In [17], it is shown that coupled thermal noise power N C (i) of ith antenna element is lower than thermal noise power of isolated antenna element N for the antenna spacing up 0.35λ. Following that, one can write ν i 1 1 ν i 1 (21) where inequality holds true for antenna spacing lower then 0.35λ [17]. Now based on (20) and (21), one can conclude that as the eigenvalues of channel matrix represent virtual channel gains, the eigenvalues of the noise correlation matrix appear as factors which increase the channel gains. In that way, they increase the MIMO channel capacity. 6. Simulation Experiments To corroborate the presented analytical analysis, we use simulation models consisting of uniform linear arrays (ULA) with two, three and four half-wave dipoles. The analysis of the multi-antenna system with two, three and four antenna elements enables significant conceptual insight to be gained into the multi-antenna system behavior with many antenna elements. Mutual impedance and admittance matrices are calculated by using SONNET software [18]. The channel capacity is estimated over the 10000 channel realizations. The spatial fading correlation is taken into account into the channel matrix calculation [19]. It is presupposed that the transmitted antennas are widely spaced, thus transmitted signals are not spatially correlated. First, we present the variations in the mean signal-to-noise ratio (SNR) per MIMO sub-channel (branch) due to the mutual coupling on thermal noise for antenna spacing up to 0.5λ (Figure 2). As SNR is one of two cofactors that determine the MIMO channel capacity, it is crucial to present its variations due to the mutual coupling on thermal noise. We compare SNR per receiving antenna for the case when the mutual coupling affects both signal and thermal noise with the case when the mutual coupling on thermal noise is neglected. In order to get more realistic insight into the SNR behavior for closely spaced antenna elements, we include the fading correlation as well as electromagnetic correlation in analysis [5]. Our simulation results show that the branch signal-to-noise ratio of MIMO systems is underestimated by about 6% at antenna spacing 0.1λ and about 3% at the antenna spacing

324 S.M. Krusevac et al. 0.2λ, ifthemutualcouplingonthermalnoiseisneglected.inaddition,forantennaspacing [0.35, 0.5λ] it is overestimated for less than 1%, if the mutual coupling on thermal noise is not accounted for. For antenna spacing wider then 0.5λ, themutualcouplingeffectisnegligible. The mean SNR variations are presented for the receive multi-antenna systems with two, three and four antennas. Figure 3 depicts the ergodic capacity of 2 2, 3 3and4 4 MIMO systems as a function of antenna element spacing. We analyze channel capacity for the case when the receiver antenna elements of MIMO system are closely spaced, presupposing wide inter-element spacing at the transmitter side. In order to explore the noise correlation effect due to the mutual coupling on channel capacity, the MIMO channel capacity is estimated under three assumptions: (1) signal and thermal noise are affected by mutual coupling (mcstn), (2) the mutual coupling effect is considered Variation of the Mean SNR per MIMO sub-channel [%] 7 6 5 4 3 2 1 0 two-antennae system three-antennae system four-antennae system -1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Antenna spacings (wavelengths) - d/λ Figure 2. The variations in the mean branch signal-to-noise ratio due to the thermal noise mutual coupling. 26 4 x 4 MIMO system 24 Mean (ergodic) Capacity [bps/hz] 22 20 18 16 14 12 3 x 3 MIMO system 2 x 2 MIMO system 10 8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Antenna spacings (wavelengths) - d/λ Figure 3. Mean (ergodic) MIMO channel capacity versus antenna element spacing. mcstn mcs nmc

Impact of Mutual Coupling on MIMO 325 only for signal (mcs) and (3) the mutual coupling effect is neglected either on signal or on noise (nmc). From Figure 3 we corroborate our theoretical analysis that the correlated noise intensifies eigenvalues of channel matrix, improving the strength of sub-channels. Therefore, it appears that the channel capacity calculated under the assumption that mutual coupling correlates both signal and noise (mcstn) is higher then (mcs) and (nmc). In fact, we provide more accurate estimation of channel capacity if we include all factors, fading and electromagnetic signal correlation and electromagnetic noise correlation. Additionally, we corroborate our theoretical analysis for the multi-antenna system with two, three and four antenna elements. In such a way, one can conclude that the MIMO channel capacity is underestimated for antenna spacing range [0, 0.35λ] for any number of antenna elements at the receiver side, if the mutual coupling on thermal noise is not taken into account. Figure 4 depicts the 1% outage capacity C 0.01. The outage capacity C 0.01 means that there is a probability of 0.01 that the capacity is less then what is displayed in the figure. We compute the outage capacity with no mutual coupling (nmc), mutual coupling only for signal (mcs) and mutual coupling on both signal and thermal noise (mcstn). From Figure 4 we show that outage capacity of MIMO system is underestimated when the thermal noise correlation (due to the mutual coupling effect) is not considered. We show that for over 99% of channel realizations, the MIMO channel capacity is underestimated if the mutual coupling on thermal noise is neglected at the receiver side. 7. Conclusions In this paper, we analyzed the mutual coupling effect on channel capacity of MIMO system, taking into account both signal and noise correlation. We showed that ergodic and outage channel capacity of MIMO systems is underestimated if the noise correlation is not accounted for. Based on the presented results, one can conclude that thermal noise mutual coupling effect is a significant issue in the analysis of the mutual coupling effects on MIMO channel capacity. 16 15 Outage capacity - C 0.01 14 13 12 11 10 mcstn mcs nmc 9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Antenna spacings (wavelengths) - d/λ Figure 4. 1% Outage channel capacity for 3 3MIMOsystemsversusantennaelementspacing.

326 S.M. Krusevac et al. Acknowledgment The authors acknowledge support by National ICT Australia which is funded through the Australian Government s Backing Australia s Ability initiative, in part through the Australian Research Council. References 1. G. Foschini, On Limits of Wireless Communication in Fading Environment when Using Multiple Antennas, Wireless Personal Commun.,Vol.6,pp.311 355,Mar.1998. 2. I.J. Gupta, A.K. Ksienski, Effect of the Mutual coupling on the Performance Of the Adaptive Arrays, IEEE Trans. Antenna Propagat., Vol. 31, No. 5, pp. 785 791, Sep. 1983. 3. T. Sventenson, A. Ranheim, Mutual coupling effects on the capacity of the multielement antenna system, in Proc. IEEE ICASSP 2001,SaltLakeCity,UT,2001,pp.2485 2488. 4. B. Clerckx, D. Vanhonacker-Janvier, C. Oestegs and L. Vanderdorpe, Mutual Coupling Effect on the Channel Capacity on the Space-time Processing of MIMO Communication Systems, in Proc. IEEE International Conference on Communications, (ICC 03),Anchorage,AK,May2003,pp.2638 2642. 5. R. Janaswamy, Effects of Element Mutual Coupling on the Capacity of Fixed Length Linear Arrays, IEEE Antennas Wirless Propagat. Lett.,Vol.1,pp.157 160,2002. 6. V. Jungnickel, V. Pohl and C. von Helmolt, Capacity of MIMO Systems with Closely Spaced Antennas, IEEE Commun. Lett., Vol. 7, No. 8, pp. 361 363, Aug. 2003. 7. M. Ozdemir, E. Arvas and H. Arslan, Dynamics of Spatial Correlation and Implications on MIMO systems, IEEE Commun. Mag., Vol. 42, No. 6, pp. S14 S19, June 2004. 8. C. Waldschmidt, S. Schulteis and W. Wiesbeck, Complete RF System Model for Analysis of Compact MIMO arrays, IEEE Trans. Veh. Technol., Vol.53,No.3,pp.579 586,May2004. 9. J.W. Wallace and M.A. Jensen, Mutual Coupling in MIMO Wireless Systems: A rigorous Network Theory Analysis, IEEE Trans. Wireless. Commun., Vol. 3, No. 4, pp. 1317 1325, July 2004. 10. M.L. Morris and M.A. Jensen, Network Model for MIMO Systems with Coupled Antennas and Noisy Amplifiers, IEEE Trans. Antennas Propagat., Vol. 53, issue 1, part 2, pp. 545 552, Jan. 2005. 11. S.M. Rytov, Yu.A. Krastov and V.I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields, Berlin, Heidelberg, NewYork: Springer, 1987. 12. R.Q. Twiss, Nyquist s and Thevenin s Generalized for Nonreciprocal Linear networks, J. Applied Phys., Vol. 26, pp. 559 602, May 1955. 13. D. Shiu, G. Foschini, M. Gans and J. Kahn, Fading correlation and its effect on the capacity of multi-element antenna system, IEEE Trans. Commun., Vol.48,pp.502 513,Mar.2000. 14. George E. Valley and Jr. Henry Wallman, Vacuum Tube Amplifier, Volume 18 of MIT Radiation Laboratory Series,McGraw-Hill,NewYork,1948. 15. P. Rapajic, Information Capacity of the Space Division Multiple Access Mobile Communication System, Wireless Personal Communication, Vol.50,pp.131 159,1999. 16. L. Schumacher, K.I. Pedersen and P.E. Mogensen, From Antenna Spacings to Theoretical Capacities - Guidelines for Simulating MIMO Systems, in Proc. IEEE Int. Symp. On Personal Indoor Mobile and Radio Commun.- PIMRC,Lisbon,Portugal,Vol.2,pp.578 502,Sep.2002. 17. S. Krusevac, P.B. Rapajic and R.A. Kennedy, Mutual Coupling Effect on Thermal Noise in Multi-element Antenna Systems, Progress In Electromagnetics Research (PIER), PIER 59, pp. 325 333, 2006. 18. SONNET, Full wave 3D electromagnetic simulator, www.sonnet.com 19. D. Gesbert, H. Bolcskei, D. Gore and A. Paulraj, Outdoor MIMO Wireless Channel: Model and Performance, IEEE Trans. Commun., Vol.50,No.12,pp.1926 1935,2002.

Impact of Mutual Coupling on MIMO 327 Snezana M. Krusevac received the B.S. degree in electrical engineering and telecommunications from University of Belgrade, Serbia in 1996. Since 2003, she has been working towards the Ph.D. degree in Research School of Information Sciences and Engineering at the Australian National University. Her current research interest includes spatial correlation and polarization diversity in the multi-antenna systems, channel modeling and MIMO channel capacity. Rodney A. Kennedy received his BE from the University of New South Wales, Australia, ME from the University of Newcastle, and Ph.D. from the Australian National University. He worked 3 years for CSIRO on the Australia Telescope Project. He is head of the Department of Information Engineering, Research School of Information Sciences and Engineering at the Australian National University. His research interests are in the fields of digital signal processing, spatial information systems, digital and wireless communications, and acoustical signal processing.

328 S.M. Krusevac et al. Predrag Rapajic received his Ph.D. from The University of Sydney, Australia in 1994. He held full time academic positions at The University of Sydney, The Australian National University and The University of New South Wales, Australia. Predrag is Professor of Communication Systems and Head of Research Program in Computing and Communications, The University of Greenwich, UK. His research interests include: adaptive multi-user detection, equalization, error control coding, mobile communication systems and multi-user information theory.