MULTIPLE -input multiple-output (MIMO) wireless

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1 Mutual Coupling in MIMO Wireless Systems: A Rigorous Network Theory Analysis Jon W. Wallace and Michael A. Jensen Abstract A new framework for the analysis of multipleinput multiple-output (MIMO) wireless systems is introduced to account for mutual coupling effects in the antenna arrays. The multi-port interactions at transmit and receive are characterized by representing the channel using a scattering parameter matrix. A new power constraint that limits the average radiated power is also introduced. The capacity of the MIMO system with mutual coupling is defined as the maximum mutual information of the transmit and receive vectors over all possible transmit signaling and receive loading. Full-wave electromagnetic antenna simulations combined with a simple path-based channel model are used to demonstrate the utility of the method. I. Introduction MULTIPLE -input multiple-output (MIMO) wireless systems have demonstrated the potential for increased capacity in rich multipath environments -3. For example, when the channel matrix coefficients are wellrepresented as independent identically distributed (i.i.d.) complex Gaussian random variables, linear increase in capacity with the number of antennas is possible. This independence of the channel coefficients is generally achieved by constructing antenna arrays with wide inter-element spacings (several wavelengths). For many mobile subscriber units, such separations are often unrealistic. Close antenna element spacing inevitably leads to mutual coupling 4, 5. Generally speaking, this coupling means that current induced on one antenna either due to a driving generator or a received electromagnetic wave produces a voltage at the terminals of nearby elements. Intuition suggests that high mutual coupling leads to a higher correlation in channel fading coefficients. Surprisingly, however, studies have demonstrated that two closely-spaced coupled dipoles exhibit a lower correlation coefficient than identically spaced uncoupled dipoles 6, 7. This correlation decrease stems from the modification of the antenna radiation and reception patterns as the antennas are brought into close proximity, providing a degree of pattern diversity. Other studies have been performed to examine the effect of mutual coupling on pattern characteristics for a variety of communications applications 8-. While these prior studies have presented important findings concerning the effect of array mutual coupling on MIMO system performance, they have neglected two key concepts. First, coupling at the transmitter produces coherent field interactions that render existing transmitted The authors are with the Department of Electrical and Computer Engineering, 459 Clyde Building, Brigham Young University, Provo, UT This work was supported by the National Science Foundation under Wireless Initiative Grant CCR and Information Technology Research Grant CCR power constraints inappropriate. Second, the power collection capability and pattern characteristics of coupled antennas depend on impedance matching of the array to the communication electronics. Since real systems do not simply sample the electromagnetic field but rather collect power into a load, this implies that the effective channel transfer matrix depends on loading. Therefore a true definition of capacity requires the selection of optimal loading at the receiver. This work overcomes these difficulties by applying an exact network theory framework to automatically account for mutual coupling in MIMO system analysis. This framework includes a new power constraint that limits the actual radiated power when mutual coupling is present. New expressions for capacity are derived that maximize the mutual information of transmit and receive signals over all possible loading networks, providing a true upper bound on system performance. The network theory framework also allows more flexibility in specifying the dominant source of noise in the system. Consequently, two realistic noise models are presented, and conclusions concerning their impact on capacity are provided. This new framework offers the appropriate tools for definitively answering questions about the impact of mutual coupling on MIMO capacity. The organization of the paper is as follows. Section II reviews MIMO capacity analysis and presents a modified capacity solution appropriate for the new radiated power constraint. Section III presents the network models for transmit and receive systems and formulates the radiated power constraint for coupled transmit antennas. Section IV explains the network model required to define capacity in systems with mutual coupling and presents two realistic noise models with accompanying closed-form capacity expressions. In order to demonstrate the application of the new capacity expressions, Section V describes fullwave Finite-Difference Time-Domain (FDTD) simulations of coupled -element arrays and uses the results to compute capacity as a function of element spacing for a simple stochastic channel model. The results are also compared to existing methods for computing capacity in MIMO systems with mutual coupling. II. Simple Narrowband MIMO Capacity Assessing the impact of mutual coupling on MIMO system capacity requires careful definition and formulation of the capacity bounds. To facilitate this discussion, consider the narrowband MIMO wireless system depicted in Figure consisting of N T transmit and N R receive antennas. The vector of complex baseband transmit signals X and

2 H ij CHANNEL σ /S ii. For the average power constraint P T = Tr(K X) P max, the classical water-filling solution maximizes (4) by making K X diagonal. The power P i to each orthogonal channel is determined by finding ν that satisfies P i = K X,ii = (ν N i ) +, (5) Fig.. N T N R Simple representation of the MIMO wireless channel. receive signals Y are related by Y = H X + N, () where N is a noise vector, usually assumed to consist of i.i.d. complex Gaussian elements with variance σ. For the optimal case of Gaussian transmit signaling, the mutual information of the vectors Y and X is expressed as H K X H H I(Y ; X) = log + I, () where K X = E { X X H} and { } H is the Hermitian or conjugate transpose operator. The Shannon capacity is then defined as the maximum of () over all possible K X subject to a transmit power constraint 3. The following subsections formulate the capacity for three relevant constraints. A. Uninformed Transmit Capacity We first assume that the transmitter has no knowledge of the channel matrix H and is constrained by P T = Tr(K X ) P max, where P T is total average transmit power and P max represents an imposed upper bound. The optimal signaling scheme divides power equally among the N T transmit antennas in uncorrelated streams such that K X = I/N T. The capacity computed from () becomes σ C = log H H H N T σ B. Standard Water-Filling + I. (3) With perfect knowledge of the channel matrix, the transmitter can form orthogonal communication channels and achieve a higher mutual information than the uninformed transmitter 3. Substituting the singular value decomposition (SVD) H = U S V H into () results in I(Y ; X) = log S K XS σ + I, (4) where K X = V H K X V. Since S is diagonal, the channel now consists of N T independent Gaussian channels, with the effective noise variance on the ith channel of N i = where (z) + = resulting in a capacity of C = i C. Modified Water-Filling { z, z 0, 0, otherwise, (6) ( log + P ) i. (7) N i As shown in Section III-B, a constraint of the form P T = Tr(K X A) P max is appropriate when the transmit array exhibits mutual coupling. Using the eigenvalue decomposition (EVD) A = ξ A Λ A ξ H A, we can rewrite the power constraint as P T = Tr(K X) with K X = Λ/ A ξh A K X ξ A Λ/ A. (8) Solving for K X and substituting into () leads to an expression that can be maximized using the standard waterfilling approach. Unfortunately, this computation may be numerically sensitive when A has poor conditioning. To avoid these numerical instabilities, we instead first perform the SVD of the channel matrix to obtain (4) and express the resulting power constraint as Tr(K XA ) = P T, where K X = V H K X V, and A = V H A V. If S = diag(s), and the singular values are ordered from largest to smallest, we now consider only the N S largest singular values that satisfy S /S i < ρ, where ρ is some maximum allowed conditioning number. The matrices K X, S, and A are then truncated to contain only the leading N S rows and columns. In any model where A and H are consistent, this truncation ensures that A has suitable conditioning, since signaling vectors yielding low radiated power at the transmit will also yield low gain through the channel. Next, we apply the substitution K X = S ξ Z Λ / Z KZ X Λ/ Z ξh Z S, (9) where we have performed the EVD of S A S = ξ Z Λ Z ξ H Z. The mutual information expression in (4) becomes Λ / Z I(Y ; X) = log KZ X Λ/ Z σ + I (0) with the power constraint Tr(K Z X) = P T. This expression may now be maximized using the water-filling solution approach.

3 b 0 S RR a R b R b a a b S M b L = 0 a L Z 0 Z 0 the network is lossless, S H M S M = I. If it is reciprocal, (S M = S t M, where { } t is the transpose operator) we also have S M S H M = I. It can be shown that the singular values of each S ij matrix lie on the range 0,. Also, if S is set to be any matrix with singular values on the range 0,, one can use the matrix properties for lossless, reciprocal networks to specify the necessary form of S, S, and S as shown in Appendix A. Insertion of a lossless matching network between the source and the loads can increase the power collection if S RR 0. In this case, the forward wave into the loads is Fig.. Network model for the receive subsystem along with incident and reflected wave definitions. III. Network Analysis Array mutual coupling studies typically use impedance matrices to represent the antenna network. However, we have found the use of scattering parameter (S-parameter) matrices to be a more natural representation for the capacity analysis formulated here. This S-parameter description can be generally expressed as b = S a, () where the vectors a and b denote the complex envelopes of the inward-propagating and outward-propagating waves, respectively, and S is the S-parameter matrix. We will adopt the standard convention that the total voltage and current on the nth port are given as v n = Z / 0 (a n + b n ) and i n = Z / 0 (a n b n ), where Z 0 is a chosen reference impedance used for computing the S-parameters. This normalization leads to the expression that the total power flowing into the nth port is simply a n b n. A. Receive Subsystem: Matching Networks We first consider the network model depicted in Figure for the receive subsystem. This model treats the antenna as a source with N R ports that creates the source wave vector b 0 due to the received electromagnetic wave. This implies that if a load of characteristic impedance Z 0 is placed on each source port, the total power collected in the loads is equal to b 0. The source is further characterized by a (full) S-parameter matrix S RR such that b R = b 0 +S RR a R. A matching network with S-parameter matrix S M is used to maximize the power transfer from the source to the N R loads of resistance Z 0. We partition this matrix as S S M = S, () S S where the subscripts and refer to input and output ports, respectively. Ideally, the matching network is formed with passive, reactive elements so that it is lossless and reciprocal. If b = S (I S RR S ) b 0, (3) and the total power collected is proportional to P (S) = b (4) = b H 0 (I S RR S ) ( )H S H S (I S RR S ) b 0. For a lossless network, we have the condition that S H S + S H S = I, and the expression becomes where P (S) = b H 0 W (S )b 0, (5) W (S ) = (I S RR S ) ( )H (I S H S )(I S RR S ). (6) Appendix A proves that for a fixed (but arbitrary) b 0, this expression is maximized when S = S H RR. In this case the expression reduces to W (S ) = (I S RR S H RR) ( )H. (7) Setting the input S-parameters of the matching network equal to the conjugate transpose of the source S-parameters is the multivariate extension of conjugate matching. If S is a passive (non-amplifying) network, the singular values of S will all lie on the range 0,. In this case, one can show that a reciprocal, lossless network exists such that S = S H RR is satisfied and the collected power is maximized. B. Transmit Subsystem: Constrained Radiated Power Traditional analyses of MIMO wireless systems have generally ignored the effect of mutual coupling on radiated power. Consider a transmit antenna array with N T elements and network S-parameters S T T. The net power flowing into the network is a T b T which, for lossless antennas, equals the instantaneous radiated transmit power PT inst. Since b T = S T T a T, we have P inst T = a T S T T a T = a H T a T (S T T a T ) H S T T a T = a H T (I S H T T S T T ) a T, (8) X H A X 3

4 b T a R b T a R b R a T b R a T S H b R S M N T N R N R + Z 0 v R Z 0 S H Fig. 4. Network model for the entire MIMO communication system. N T N R Fig. 3. Network model for the channel between the transmit and receive antenna ports. where A is defined as the coherence matrix and X denotes a transmit signal vector. For zero mean signals, the average radiated power is given by P T = E { PT inst } = Tr(KX A), (9) which corresponds to the constraint considered in Section II-C. It is noteworthy that while the water-filling solution must be modified to incorporate this power constraint, the uninformed transmit solution remains unchanged for uncorrelated transmit streams. IV. Network Channel Models Capacity analysis of MIMO systems requires that the communication channel be formulated within the network description adopted in this work. The following subsections describe the channel modeling framework and present two basic noise models for analysis. A. Channel Representation Figure 3 depicts transmit and receive arrays consisting of N T and N R antenna elements, respectively, embedded in a linear scattering medium. The inward-traveling and outward-traveling waves at the transmitter are defined as a T and b T, respectively, while those at the receiver are defined as a R and b R. The S-parameter channel matrix may be partitioned into the signal representation bt S = T T S T R at. (0) b R S RT S RR a R S H For this analysis, we assume that S T R = 0, which means that power reflected from the receive antennas does not couple significantly back into the transmit antennas. B. Communication System Model Figure 4 depicts a realistic communication system incorporating elements discussed thus far. The N T transmit antennas (the N T input ports to S H ) are excited by generators with arbitrary phases and magnitudes. A unit gain element that is matched to the reference impedance Z 0 is included to allow the addition of noise in the receiver. Each port in the chain is then terminated by a matched load Z 0, and the voltage across this load is sampled to obtain v R. Because the output ports of the matching network (S M ) are terminated in Z 0, only the outward-traveling wave b R will exist at this point. In the noiseless case, the sampled voltages are related to the transmit signal according to / v R = Z0 S (I S RR S ) S RT Y R H(S M ) a T, () X R where the underbraces indicate the relationship to the simple MIMO model in Section II. This relationship indicates that the effective channel is a function of the matching network employed at the receiver. Thus, a true definition of capacity will in general require a maximization of the mutual information of X and Y not only over all possible transmit excitations, but also over all allowed matching networks. This maximization is dependent on the type of noise model assumed. Therefore, we consider two realistic noise models for existing microwave systems. C. Channel Noise Model If the dominant source of noise in the system is from the channel (co-channel interference, channel instability, cosmic radiation, etc.), we may neglect noise additions in the receiver. When no signal is present and the receive antenna ports are terminated in Z 0, we define the resulting forward traveling noise wave on the ith receive port as b RN,i = Z / 0 N i, where N i is an effective noise voltage. With the matching network inserted, the forward traveling wave becomes b RN = Z / 0 (I S RR S ) N. () Superimposing the signal and noise vectors yields the result b R = (I S RR S ) (S RT a T + Z / 0 N), (3) 4

5 leading to the channel equation v R Y = S (I S RR S ) 0 S RT P (Z / H a T X +N). (4) Assuming complex Gaussian signaling at the transmitter and Gaussian channel noise, the mutual information of X and Y is expressed as 3 I(Y ; X) = h(y ) h(y X) = h(y ) h(p N) (5) P S RT K X S H RT P H + P K N P H = log, P K N P H where h( ) represents differential entropy and K N = E{N N H }. If P is non-singular, this expression can be simplified to the form S RT K X S H RT + K N I(Y ; X) = log. (6) K N This analysis indicates that, since the signal and noise undergo the same transformation in the matching network, matching does not change the mutual information. The only exception to this observation occurs when P is singular, which implies loss of information in the network. For simplicity, the physical matching network can be removed (S = 0 and S = I) and (6) can be used to compute the mutual information. This expression is equivalent to () with the channel matrix expressed as S RT, and therefore capacity may be computed using the methods in Section II. D. Receiver Noise Model In single-user point-to-point transmission systems, the receiver front end is often the major source of noise. In this case, the amplifiers in Figure 4 contribute the noise vector N at the output, leading to the relation v R Y = Z / 0 S (I S RR S ) S RT H(S M ) a T X +N. (7) In this case, the mutual information expression is H(S M )K X H(S M ) H I(Y ; X) = log σ + I W (S )M = log σ + I, (8) where W (S ) is given in (6), the noise vector is i.i.d. complex Gaussian with single element variance σ, and M = S RT K X S H RT, (9) In general, M is a Hermitian positive semi-definite matrix, so that we can use the EVD of M to write M = ξ M Λ M ξ H M = M / M (/)H (30) with M / = ξ M Λ / M. (3) Thus, maximization of the mutual-information for a fixed (but arbitrary) K X requires maximization of M (/)H W (S )M / I(Y ; X) = log + I (3) over all possible values of S and K X. This multi-variate maximization is simplified by the fact that a simple conjugate match will always maximize (3) for fixed but arbitrary K X. To show this, we use the result from Section III-A that σ x H W (S H RR)x x H W (S )x (33) for all possible values of S and x. Letting x = M / y and W (S) = M (/)H W (S)M /, we obtain y H W (S H RR)y y H W (S )y and therefore y H W (S H RR) σ + I y y H W (S ) σ + I y. (34) For two positive definite matrices A and B, x H A x x H B x, x, (35) if and only if the singular values of B A are all less than or equal to 4. Hence, relation (35) implies that A B, (36) leading to the conclusion (compare to (34)) that W (S H RR) σ + I W (S ) σ + I. (37) Therefore, for arbitrary K X, S = S H RR will maximize (3) and the mutual-information expression reduces to Z 0 I(Y ; X) = log σ (I S RRS H RR) S RT K X S H RT + I. (38) Finding the value of K X that maximizes this equation will therefore lead to an expression for channel capacity. Since (I S RR S H RR) is Hermitian and positive definite, (38) can be manipulated into the form Q K X Q H I(Y ; X) = log + I, (39) where σ Z 0 (I S RR S H RR) = ξ D Λ D ξ H D (40) is an EVD and Q = Λ / D ξh DS RT. (4) The mutual information expression (39) is identical to () with the channel matrix replaced by the effective channel Q. Capacity may therefore be computed using the methods in Section II. 5

6 λ 4 λ λ 50 λ z Position λ 4 d PML ŷ ẑ ˆx x Position e-05 Fig. 6. Near fields of two dipoles spaced at d = 0.5λ computed using FDTD. The arrows indicate electric field direction while the shading indicates the field strength. Fig. 5. FDTD simulation volume showing the geometry of the coupled dipole antennas Re(Z ) V. Capacity Simulations To demonstrate application of the analysis framework developed in this paper and to illustrate the impact of mutual coupling on the capacity of MIMO systems, we will explore transmit and receive arrays consisting of two coupled dipoles. Specifically, we focus on the receiver noise capacity expression from Section IV-D. Antenna network S-parameter descriptions and radiation patterns obtained from full-wave FDTD simulations are combined with a simple path-based channel model to construct the effective channel matrix. A. FDTD Antenna Simulations Figure 5 depicts the FDTD geometry used for the coupled antenna simulations. Half-wave (total-length) dipoles with wire radius 0.0λ and separated by a distance d are located at the center of the domain. Because we are considering narrowband systems, single-frequency antenna excitation is used. The FDTD grid uses 80 cells per wavelength in the ẑ direction and 00 cells per wavelength in the ˆx and ŷ directions. This finer resolution is required to adequately model the current variations on the finite-radius wire for close antenna spacings. A quarter-wavelength buffer region is placed between the antennas and the terminating 8-cell perfectly matched layer (PML) absorbing boundary condition. Figure 6 shows the near field patterns for two dipoles spaced at d = 0.5λ, with the arrows indicating electric field direction and the background brightness indicating field intensity. Here, the left dipole is excited with a constant voltage across its gap, and the right dipole is left open (no excitation). These results reveal a high degree of interelement coupling. This coupling is confirmed in Figure 7 which plots the self and mutual impedance as a function of the element spacing. It is particularly noteworthy that the mutual impedance remains significant even for antenna spacings greater than λ. Finally, Figure 8 compares the Impedance (Ω) Re(Z ) Im(Z ) Im(Z ) Spacing (wavelengths) Fig. 7. Self and mutual dipole antenna impedances as a function of dipole separation d. radiated far-field patterns for four different dipole spacings. For this computation, the left antenna is driven by a source having an internal impedance of Z while the right antenna is terminated in Z 0. Each pattern is normalized to have unit mean. These results indicate that the coupling noticeably impacts the patterns, particularly near a spacing of d = 0.5λ. B. Path-based Channel Model When the transmit and receive arrays and scattering objects are all in the far-field of one another, a simple path-based model can be used to approximate the channel. Here we derive a simple two-dimensional single-polarization path-based model for use in the subsequent simulations. Assuming a transmitter with N T antennas, we can write the total radiated far-field in the azimuthal plane as N T E T (θ T ) = Ej T (θ T )Ij T, (4) j= where θ T is the transmit azimuthal angle, Ej T (θt ) is the transmit radiation pattern of the jth antenna, and Ij T is a transmit antenna current. We also assume a channel with N A paths for propagation from transmit to receive, with the nth path characterized by departure and arrival angles 6

7 λ 0.5λ 0.5λ.0λ N T Z T β n N A Z R N R Fig. 9. Network channel model explicitly showing the far-field ports used to represent the path-based propagation model Fig. 8. Far-field coupled dipole patterns in the azimuthal plane for four different antenna spacings. The second antenna is terminated in Z 0. θn T and θn R, respectively, and a complex channel gain β n. When an N R element array is placed in the incident receive field, the voltage on the ith receive element may be written as N T N A Vi R = Ei R (θn R )β n Ej T (θn T ) Ij T (43) j= n= } {{ } Z RT,ij where Ei R(θR ) is the reception pattern of the ith receive antenna. The expression in (43) indicates that Z RT is a transimpedance matrix relating receive antenna voltages to transmit antenna currents. This impedance may be expressed as Z RT = Z RF β Z F T, (44) where β is the diagonal matrix of multipath gains, and Z RF and Z F T represent reception and transmission far-field patterns, respectively. By reciprocity, we may write Z F R = Z t RF and define each pattern matrix as the impedance matrix relating the complex amplitude of the far-fields radiated in the nth azimuthal direction to the currents forced on the antenna elements, or Z F Q,ni = EQ (θ Q n ) I Q i (45) I Q k =0, k i, where Q {T, R}. Relation (44) may now be interpreted in a network theory context according to Figure 9. The transmit and receive arrays are split into separate networks that are connected by N A far-field propagation paths represented by the interior ports. In the transmit block, excitation of the input ports leads to an azimuthal field radiation pattern E T (θ T ). The nth far-field output port samples the complex field envelope in the θ T n direction to produce an output voltage. Thus the Z-parameter transmit block has the form Z T = Z T T Z F T Z T F Z F F, (46) where Z T T is the transmit Z-parameter matrix for the isolated transmit array and the subscript F denotes far-field ports. For physical considerations, no interference of the far-field ports is allowed (Z F F = 0). Similarly, the receive block samples the current at the nth far-field port and generates an incident plane wave with the same complex envelope propagating in the θ R n direction. The Z-parameter block has the form Z R = Z RR Z F R Z RF Z F F, (47) where Z RR is the isolated receive array Z-parameter matrix and Z F F = 0. The far-field ports of transmit and receive are connected by inserting an ideal unilateral gain element between the nth transmit and receive ports such that I out,n = β n V in,n. (48) This assignment ensures that the cascade of the network blocks represents the path-based channel model that generated relation (44). While this formulation makes use of Z-parameters, the capacity derivations in Section IV assume S-parameters, necessitating a conversion between the two 5. Because we are neglecting feedback through the channel (S T R = 0), we can individually convert Z T T and Z RR into S T T and S RR. The block S RT is found by exciting an inwardtraveling wave at the transmitter a T and measuring the outward-traveling wave at the receiver b R. To ensure a R = 0, the receive ports are terminated with loads Z 0. This procedure yields the relation ( ) ZRT b R = I + Z RR (I S T T ) a T. (49) Z 0 Z 0 S RT 7

8 Correlation Coefficient Jakes Model Open Circuit Self Match MP Conj. Match Antenna Spacing Relative Collected Power Self Impedance Match Multiport Conj. Match Antenna Spacing Fig. 0. Correlation coefficient versus dipole spacing for different antenna terminations compared to Jakes model (neglecting mutual coupling). Fig.. Power collection (normalized to the power collected by a single dipole) versus antenna spacing for two dipoles assuming two different antenna terminations. C. Correlation and Power Collection The correlation coefficient of the complex baseband signals on two antennas is often used as a metric for assessing diversity performance. Correlation has also been used to draw conclusions about capacity, since a channel with transfer matrix coefficients that are complex Gaussian with low correlation exhibits high capacity. Here, we model the -element array with matching networks creating open-circuit, self-impedance matched load (Z), and optimal multi-port conjugate match terminations. The voltage correlation coefficient is computed as E{v v}, where v i, i {, } is the voltage at the ith output port of the antenna matching network. The expectation is taken over plane wave azimuthal arrival angles which are uniformly distributed on 0, π. The resulting magnitudes of the correlation coefficients are plotted versus antenna spacing in Figure 0. Surprisingly, the matching network offering a conjugate match always has an output correlation of zero for nonzero spacing. For any two-element array with omnidirectional illumination and a multi-port conjugate match, this result may be confirmed analytically. Physically, this occurs because the conjugate match modifies the radiation patterns for the individual elements such that they are orthogonal over 0, π in azimuth 6. The other curves show that the correlation behavior is load dependent, with a selfimpedance match exhibiting more decorrelation than the open circuit. The correlation curve assuming Jakes propagation scenario (mutual coupling is neglected) is also shown for comparison 7. Because capacity depends upon received signal power, the power collection capability of compact arrays is also of interest. Physical arguments suggest that as two antennas approach each other, the total collected power approaches that for a single antenna. To assess this, we examine the power collected by the coupled dipoles for plane wave illumination. Figure plots this collected power averaged over all azimuthal arrival angles for optimal conjugate and sub-optimal self-impedance match. Interestingly, only for very small spacings does the power collection drop signif- icantly. The results also show that two closely-spaced antennas can actually collect more power than two widely separated (non-interfering) ones. Physically, this can be understood by recognizing that each receive antenna will re-radiate a portion of the incident energy due to the excited antenna currents. Some of this scattered energy can be recaptured by an adjacent antenna, particularly when the matching network is appropriately implemented. D. Channel Capacity To demonstrate application of the analysis framework developed in this work, we examine a system employing two transmit and two receive dipoles ( ) and a channel consisting of N A = 4 paths. The simple path-based model was combined with the full-wave FDTD simulations in order to compute the channel matrix S H. The ray arrival and departure angles were independent and uniformly distributed on 0, π. The path gains β n were Rayleigh i.i.d. in amplitude and uniformly distributed in phase. For each channel realization, the noise power was fixed by first computing the average signal strength that would be received if one transmit and one receive antenna were present, with the averaging performed over random placement of the single transmit and receive antennas. The noise power was then computed to achieve a signal-to-noise ratio (SNR) of 0 db relative to the average signal power. Mean capacity was computed over 7000 realizations for each antenna spacing. To illustrate the effect of different transmit power constraints, simulations were run with ideal (no mutual coupling) receive antennas separated by λ. The transmit array was simulated with (mc) and without (nmc) mutual coupling, and the capacity was computed using standard (wf) and modified water-filling (mwf) solutions. The results of this study are shown in Figure. For large spacings, the mutual coupling is low leading to identical capacity results. However, for very close spacings, the capacity computing using the modified water-filling approach is actually higher than that obtained using the standard waterfilling solution. Furthermore, the modified water-filling solution properly accounts for the power loss observed as the 8

9 Capacity (bits/use) nmc (wf) mc (wf) mc (mwf) Capacity (bits/use) nmc (mwf) mc (mwf) mcsi (mwf) Transmit Antenna Spacing (wavelengths) Antenna Spacing (wavelengths) Fig.. Mean capacity versus transmit dipole antenna spacing for a MIMO system with different assumptions on mutual coupling and transmit power constraints. The receive antennas are ideal and spaced at d = λ. Capacity (bits/use) nmc (norm) mc (norm) nmc (mwf) mc (mwf) mcsi (mwf) Receive Antenna Spacing (wavelengths) Fig. 3. Mean capacity versus receive antenna spacing for two different capacity computations and different antenna loads. The transmit antennas are ideal and spaced at d = λ. antennas collapse to a single element, leading to the reduced capacity at d = 0. Because the traditional power constraint does not incorporate the coherent interactions, it fails to properly predict this behavior. We consider next the effect of mutual coupling at the receiver. In this case the receive antenna spacing was varied while the (ideal) transmit antennas were fixed at a separation of λ. A normalized channel analysis (norm) was performed by terminating the antennas with a selfimpedance match and normalizing the channel transfer matrix to obtain exactly 0 db average single-input singleoutput (SISO) SNR 8 for each channel realization. This normalized channel analysis is similar to that considered in previous work 7. Capacity was also computed with the new mutual coupling analysis with a constant noise giving an average of 0 db SNR for the single antenna case with random placement as discussed at the beginning of this section. Figure 3 plots the average capacity versus spacing assuming mutual coupling with an optimal match (mc), no mutual coupling with an optimal match (nmc), and mutual coupling with a sub-optimal self-impedance match (mcsi). A small offset between the normalized and full analyses ex- Fig. 4. Mean capacity versus transmit and receive antenna spacing for different coupling assumptions. ists for large spacings since the SNR constraints are slightly different. In the normalized analysis, mutual coupling always provides a capacity benefit. In the full analysis, mutual coupling appears to provide nearly the same benefit as suggested by the normalized analysis. However, for very close spacings (< 0.λ), the capacity curve for the full analysis rolls off sooner than the normalized analysis. Finally, we note that although the simple self-impedance match performs well for large spacings, capacity degradation is apparent for spacings smaller than 0.5λ. Figure 4 demonstrates the combined effect of mutual coupling at transmit and receive. Here, the transmit and receive antenna spacings were equal and capacity was computed for ideal antennas (nmc), mutual coupling at transmit and receive with an optimal match (mc), and mutual coupling with a sub-optimal self-impedance match (mcsi). For spacings between 0.λ and 0.3λ, mutual coupling provides an obvious capacity benefit. For spacings below 0.λ, mutual coupling can actually degrade capacity. Finally, the sub-optimal matching network yields a modest capacity degradation. VI. Conclusion This paper has presented a rigorous network-theory framework for the analysis of mutual coupling in MIMO wireless communications. A detailed network model was used to develop a new mutual information expression and radiated power constraint accounting for this antenna coupling. Closed form derivation of the system capacity was made possible by relating the mutual information maximization problem to the multi-port conjugate matching solution. Unlike previous analyses, this new method includes the effect of mutual coupling, and the resulting capacity expression provides a true upper bound on system performance. The framework was used to analyze the impact of mutual coupling in a simple yet realistic MIMO system by combining full-wave FDTD simulations with a path-based channel model. This simple example demonstrated the usefulness of the technique and provided some insight into the impact of coupling on MIMO performance. However, be- 9

10 Z 0 b S a R a 0 a S b R S b a S In order to solve this problem, we require expressions that relate the sub-blocks of S for a lossless reciprocal network. One may ensure that S is lossless (S H S = I) with the relations S RR = U RR Λ / RR V H RR S SR = U SS Θ(I Λ RR ) / V H RR Z 0 Fig. 5. Network model for the equivalent receive impedance matching problem. fore more general conclusions can be drawn concerning the effect of mutual coupling, more extensive simulations using increased array sizes and various array configurations must be performed. Specifically, it is anticipated that mutual coupling will more significantly impact the performance of larger planar arrays. Fortunately, the tools developed here provide a comprehensive framework for systematically conducting these important studies. Appendix I. Multi-Port Conjugate Impedance Matching Consider the problem depicted in Figure. For any given b 0, the power available from the source block S RR is fixed. We wish to choose the lossless matching block S M such that the power delivered to the load is maximized, regardless of the choice of b 0. We can cast this problem into the equivalent problem depicted in Figure 5. Everything to the right of reference plane has been replaced with the block S (since b L = 0 in Figure ). The source block is now represented with as many input ports as output ports, and the complete S-parameter matrix is given by S = S SS S RS S SR S RR. (50) To make this problem equivalent to the initial one, we choose S SS, S SR, and S RS to make S a lossless reciprocal network and set a 0 = a S = S RS b 0. The relation for waves just to the left of plane is b R = S RR a R + S RS a S = S RR a R + b 0, (5) which is precisely the same relation we had for the initial problem. Since S is lossless, the power available to the load is a 0 = S RS b 0. The load will collect all of this available power if we can choose S such that the reflection b S = 0. Relating the various inward- and outward-traveling waves, we have b S = S SS S RS + S SR(I S S RR ) S b 0. (5) S RS = U RR Θ H (I Λ RR ) / V H SS S SS = U SS Λ / RR V H SS, (53) where the first equation is the SVD of S RR, U SS and V SS are arbitrary unitary matrices, and Θ is a complex diagonal matrix whose elements have unit magnitude with arbitrary phase. If we desire S to be reciprocal (S = S t ), we have the additional requirement that U RR = U RR, U SS = V SS, and Θ = ji. Using (53) and letting S = S H RR = V RR Λ / RR U H RR in (5) results in b S = U SS M U H RR b 0 with M = Λ / RR V H SSV SS Θ(I Λ RR ) / (54) + Θ(I Λ RR ) / V H RRV RR (I Λ RR ) V H RRV RR Λ / RR. Upon canceling the unitary matrices, we find that M is indeed the zero matrix, which ensures that b S is the zero vector for any choice of b 0. Therefore, the assignment S = S H RR ensures that all available power is dissipated in the load, thus maximizing the collected receive power. References J. H. Winters, On the capacity of radio communication systems with diversity in a Rayleigh fading environment, IEEE Journal on Selected Areas in Communications, vol. SAC-5, pp , June 987. G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol. 6, pp , Mar G. G. Raleigh and J. M. Cioffi, Spatio-temporal coding for wireless communication, IEEE Transactions on Communications, vol. 46, pp , Mar J. L. Allen and B. L. Diamond, Mutual coupling in array antennas, Tech. Rep. 44 (ESD-TR ), Lincoln Laboratory, M.I.T., C. A. Balanis, Antenna Theory: Analysis and Design, Wiley, J. Luo, J. R. Zeidler, and S. McLaughlin, Performance analysis of compact antenna arrays with MRC in correlated Nakagami fading channels, IEEE Transactions on Vehicular Technology, vol. 50, pp , Jan T. Svantesson and A. Ranheim, Mutual coupling effects on the capacity of multielement antenna systems, in IEEE ICASSP 00, Salt Lake City, UT, May 7-00, vol. 4, pp W. C. Y. Lee, Effect of mutual coupling on a mobile-radio maximum ratio diversity combiner with a large number of branches, IEEE Trans. Communications, vol. COM-0, pp , Dec R. R. Ramirez and F. De Flaviis, Mutual coupling study of linear polarized microstrip antennas for use in BLAST wireless communications architecture, in IEEE AP-S International Symposium Digest, Salt Lake City, UT, July 6-000, vol., pp

11 0 G. V. Tsoulos, Experimental and theoretical capacity analysis of space-division multiple access (SDMA) with adaptive antennas, IEE Proceedings: Communications, vol. 46, no. 5, pp , 999. K. R. Dandekar, H. Ling, and G. Xu, Effect of mutual coupling on direction finding in smart antenna applications, Electronics Letters, vol. 36, pp , Oct A. M. Wyglinski and S. D. Blostein, Mutual coupling and scattering effects on cellular CDMA systems using smart antennas, in IEEE Vehicular Technology Conference (IEEE Fall VTC000), Boston, MA, Sep , vol. 4, pp T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, R. Horn and C. Johnson, Matrix Analysis, chapter 7, Cambridge Univ. Press, D. M. Pozar, Microwave Engineering, chapter 4, John Wiley & Sons, R. G. Vaughan and J. B. Andersen, Antenna diversity in mobile communications, IEEE Trans. Vehicular Tech., vol. VT-36, pp. 49 7, Nov W. C. Jakes, Microwave Mobile Communications, IEEE Press, J. W. Wallace and M. A. Jensen, Characteristics of measured 4x4 and 0x0 MIMO wireless channel data at.4-ghz, in IEEE AP-S 00, Boston, MA, July , vol. 3, pp