Fundamental Capacity Limits on Compact. MIMO-OFDM Systems. Pawandeep S. Taluja, Brian L. Hughes. Department of Electrical and Computer Engineering.
IEEE ICC 2012 - Communications Theory
Fundamental Capacity Limits on Compact MIMO-OFDM Systems Pawandeep S. Taluja, Brian L. Hughes Department of Electrical and Computer Engineering North Carolina State University Raleigh, NC 27695-7914 {pstaluja, blhughes}@ncsu.edu Abstract—We undertake an information-theoretic approach to characterize the optimal design of a broadband multi-antenna system in the presence of mutual coupling. It was shown recently that mutual coupling effectively decomposes otherwise spectrallyidentical spatial modes of an antenna array into spectrally non-identical eigen-modes. We shall use Shannon’s information theory and Fano’s broadband matching theory to develop optimal transceiver designs for a compact broadband MIMO system. It will be shown that in the presence of channel state information, optimal transmit power allocation and matching characteristic follow a mutual space-frequency water-pouring solution.
I. I NTRODUCTION Multiple-input-multiple-output (MIMO) systems have the capability to achieve significantly high date rates [1] and can alleviate the problem of signal fading arising from multipath propagation in wireless channels [2]. However, current portable wireless devices require an antenna spacing on the order of a fraction of wavelength in order to make MIMO communications possible. Several current and upcoming wireless standards like LTE and 802.11n, increasingly require the support of broadband systems with multiple antennas in a compact form-factor, and understanding the fundamental limitations of such physically constrained devices is key to the evolution of wireless systems. At such small spacings, the voltages induced across the antenna terminals by the incident electro-magnetic (EM) signal become spatially correlated. Besides, the currents flowing in one antenna induces voltage across the other, commonly referred to as mutual coupling. Several studies have shown that performance degradation resulting from these impairments can be undone by use of transceiver architectures that employ optimal multiport matching networks [3]-[6]. However, these studies assume a narrowband system model, but for broadband systems, these matching networks designed at the center frequency suffer from severe RF bandwidth limitations [7], [8]. Whereas, a physically realizable broadband matching network built from lumped passive elements for such a coupled antenna array must obey the gain-bandwidth trade-offs regulated by Fano’s broadband matching theory [10].
In this paper, we seek to develop a unified informationtheoretic framework to support the design of broadband multiantenna wireless transceivers. The main idea is that Fano’s broadband matching theory provides a characterization of physically-realizable matching networks, while Shannon’s information theory provides a way to evaluate how each network could be used to communicate in the best possible way. By combining these theories, we hope to understand how antennas, matching networks, amplifiers and communications algorithms interact to determine overall system performance, and how best to jointly optimize these components. A study [11] has analyzed bounds on the spectral efficiency of an arbitrary antenna array inserted in a sphere, under Rayleigh fading conditions. It however, does not explicitly model the impact of coupling on the system bandwidth. It was recently shown [8] that mutual coupling in broadband multi-antenna systems, transforms the spectrally-identical spatial modes of an antenna array into spectrally non-identical eigen-modes. We aim to address information-theoretic limits on coupled multi-antenna systems by combining matching limitations from broadband matching theory. II. MIMO-OFDM S YSTEM M ODEL We begin by presenting the system model of a traditional MIMO-OFDM system with N transmit and N receive antennas [9]. It is assumed that the separation between the antennas is such that the coupling between them is negligible. In later sections, we shall build upon this basic model to incorporate the effects of coupling. It is well known that the use of orthogonal sub-carriers in OFDM with a cyclic prefix converts a frequency-selective MIMO channel into a set of parallel frequency-flat MIMO channels. A MIMO-OFDM system with K sub-carriers modulated by N × 1 vector symbols sk where k (k = 1, . . . , K) represents the k-th sub-carrier, is well modeled by rk = Hk sk + nk
where, rk is the N × 1 received vector symbol on the k-th sub-carrier and nk is the N × 1 additive white Gaussian noise (AWGN) vector at the receiver with zero mean and covariance1
This material is based upon work supported by the National Science Foundation under grant CCF-1018382.
978-1-4577-2053-6/12/$31.00 ©2012 IEEE
(1)
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1 E[·]
represents the expectation operator.
Rnk = E[nk nH k ], denoted by nk ∼ CN (0, Rnk ). The MIMOOFDM channel matrix for the k-th sub-carrier is given by ⎤ ⎡ h11 [k] . . . h1N [k] ⎥ ⎢ .. .. Hk = ⎣ ⎦ , k = 0, . . . , K − 1 . . . hN 1 [k]
...
hN N [k]
The transmit and receive spatial fading-correlation is modeled using the Kronecker model [13] such that the l-th tap timedomain MIMO channel (obtained via inverse-Fourier transform) can be expressed as wl R1/2 l = R1/2 H H T R wl represents the N × N white channel matrix where H having i.i.d. complex Gaussian entries with zero mean and wl ∼ CN (0, I). unit variance denoted by H Owing to the orthogonal decomposition of the frequency selective channel, the cumulative MIMO-OFDM system can be represented in matrix notation by r = Hs + n
(2)
where, H is a KN × KN block diagonal matrix consisting of Hk , k = 1, . . . , K, and the corresponding KN × 1 vectors are given by ⎡ ⎤ ⎤ ⎤ ⎡ ⎡ r1 s1 n1 ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ r = ⎣ ... ⎦ , s = ⎣ ... ⎦ , n = ⎣ ... ⎦ . rK
sK
nK
The Shannon capacity (in nats/s/Hz) is given by [14]
1 1 H
C= HR H max log
I + R−1 s n
K Rs N
(3)
where2 I is an N K × N K identity matrix, and Rs and Rn are the transmit-signal and noise covariance, respectively Rs = E[ssH ] , Rn = E[nnH ] .
(4)
There have been numerous extensions to the above model in order to account for some of the channel non-idealities, such as fading correlation at the transmitter and/or the receiver, either due to smaller antenna separation or non-richness of the multipath fading environment. But the broadband antenna behavior has mostly been assumed ideal. In the next section, we introduce a channel model that incorporates the impact of antenna coupling on the signal-model outlined in (1). III. C OUPLING IN M ULTI -A NTENNA T RANSCEIVERS Mutual coupling in a compact multi-antenna transceiver is often modeled by representing the antenna array via its impedance matrix over the bandwidth of interest. The electromagnetic (EM) field incident on the antenna array induces voltages and currents across the terminals of its elements. For a receiver with N antennas, the antenna array is essentially an N port network. This voltage-current relationship can conveniently be expressed using a Thevenin equivalent N ×N 2 |A|
represents the determinant of matrix A.
antenna impedance matrix ZA and open-circuit voltage vo . For illustrative purposes, we restrict ourselves to an N = 2 antenna array with identical elements placed a distance d normalized to the center-wavelength λc (wavelength corresponding to the center-frequency fc ) apart such that
z11 (jω) z12 (jω) ZA (jω) = z21 (jω) z22 (jω) where, ω = 2πf is frequency in radians-Hz, and the diagonal and off-diagonal entries represent self- and mutual-impedances of the array, respectively. However, when dealing with broadband systems it is rather convenient to work with scattering-parameter representation of the array, or the S-matrix: S(jω) = (ZA (jω) + I)−1 (ZA (jω) − I) , computed with 1 Ω terminations on its ports. The S-matrix essentially relates the inward and outward traveling wave vectors, a and b, respectively by
a1 b1 . (5) , a= b = Sa , b = a2 b2 The wave-vectors are normalized signals, a function of the voltage and current flowing through a given input or output port. The receive side of a multi-antenna transceiver can be represented by an S-matrix network model as shown in Fig. 1. It shows a cascade of two 2N -port networks - Na , representing a coupled lossless and reciprocal antenna array3 , and Nm representing the lossless and reciprocal matching network terminated into a bank of uncoupled load impedances zL . The load here is indicative of low noise amplifiers and other downstream components of an RF chain, primarily, demodulators and A/D converters. The 2N ×2N S-matrices for the antenna array and matching network (in N × N block-matrix format) normalized with respect to 1 Ω reference impedances are given by
S11m S12m S11a S12a , SA = , SM = S21m S22m S21a S22a where we have omitted the frequency-dependence for aesthetic reasons. It shall henceforth be assumed implied, unless stated otherwise. The overall 2N -port cascaded network can be represented using S-matrix as
S11c S12c SC = SA SM = S21c S22c where, represents the cascading operation and S11c S12c
= S11a + S12a (I − S11m S22a )−1 S11m S21a = S12a (I − S11m S22a )−1 S12m
S21c S22c
= S21m (I − S22a S11m )−1 S21a (6c) −1 = S22m + S21m (I − S22a S11m ) S22a S12m . (6d)
(6a) (6b)
3 It is important to point out that the antenna array is essentially an N port network appropriately extended to a 2N -port network for mathematical convenience [4].
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aT
a m1 = b R + S22 ab m1
bR
Transmit Antennas
Receive
Physical MIMO
S R = S 22 a
H
Load
Network
Channel
ST
Matching
v1
Antennas
a m 2 = b L = S La L
S L = ρ LI
SM
vN
S 22c bT = ST aT Fig. 1.
Equivalent MIMO channel
Here, S21c represents the transmission matrix of the cascaded network comprising the antenna array and matching network and forms an integral part of the effective MIMO channel matrix in the presence of mutual coupling. IV. MIMO-OFDM WITH C OUPLING In this work, we consider an N × N MIMO system with coupling only at the receiver. We skip the tedious network analysis and present the new MIMO channel matrix at the kth sub-carrier directly [15].4 The MIMO-OFDM signal model in the presence of mutual coupling for the k-th sub-carrier can be expressed as rk = Sk Hk sk + nk ,
(7)
where, Sk S21c,k = S21c (fk ) and effective MIMO channel matrix is given by Hk
= S21m,k (I − S22a,k S11m,k )−1 S21a,k Hk = S21c,k Hk .
(8)
The additive noise nk is usually modeled as a combination of noise from various sources in the RF chain [5]. We categorize the broadband noise into three types: (a) sky noise or antenna noise, consisting of thermal radiation, cosmic background, and interference from other devices, (b) amplifier noise, and (c) downstream noise, consisting of noise from the rest of the RF chain. However, based on our receiver model, we classify the noise as antenna noise, and load noise - a combination of amplifier and downstream noise. Furthermore, the load noise in general can be considered to be a combination of forward traveling noise nf , and reverse traveling noise nr [16, Chap. 1]. Thus, the total noise at k-th sub-carrier referenced to the load, is given by nk = S21c,k ns,k + nf,k + S22c,k nr,k . 4 This
b m2 = a L
b m1
(9)
model is based on the narrowband model presented in [16, Chap. 1].
The sky noise and load noise can be well modeled as statistically independent, ZMCSCG and spectrally white: ns ∼ CN (0, 4kTA BRA ) , nf ∼ CN (0, 4kTα BI) , nr ∼ CN (0, 4kTβ BI) , a reasonable assumption for bandwidths less than 10%. The cumulative MIMO-OFDM system in the presence of mutual coupling can be written in matrix format (similar to (2)), as r = H s + n ,
H = SH
(10)
where S is the KN × KN block-diagonal matrix consisting of Sk , k = 1, . . . , K. The Shannon capacity that incorporates the impact of mutual coupling can thus be represented by
1 1 H
C = H R H max log
I + R−1 s n
K N
1 1 H H
= SHR H S max log
I + R−1 s n
K S,Rs N where the optimization space now includes matching network design, besides the usual transmit signal covariance: Tr(Rs ) ≤ KP0 .
(11)
The broadband matching network SM can not be designed arbitrarily. Any physically realizable matching network must obey certain integral constraints imposed by the source and load impedance that it is trying to match. Fano [10] laid down the gain-bandwidth trade-offs for matching networks with resistive sources terminated into complex loads. Since then, numerous authors have extended the theory of broadband matching to include various kinds of source and load impedances and multiport networks. However, unlike our problem, these multiport networks are assumed uncoupled. Recently, broadband matching constraints in the presence of mutual coupling applicable to a special but practical class of antenna arrays - uniform circular arrays (UCA), have been
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1
proposed [8]. In the next section, we briefly summarize its main findings useful for the problem addressed here.
0.9 0.8 Antenna transfer function
V. B ROADBAND M ATCHING -N ETWORK C ONSTRAINTS To that end, consider a N = 2 element antenna array separated by a distance d. The antenna array impedance matrix ZA for such a symmetric system can be expressed in terms of its eigen-value decomposition (EVD) as ZA (jω) = QΛA (jω)QH ,
(12)
0.7 0.6 0.5 0.4 0.3 0.2
where the set of unitary eigen-vectors is given by
1 1 1 , Q = √ 2 1 −1
0.1 0 0.5
and the eigen-values (also referred to as eigen-impedances) by λ(jω) = {z11 (jω) + z12 (jω), z11 (jω) − z12 (jω)} .
Fig. 2.
1.5
Eigen-modes for a coupled 2 antenna system
(13)
The symmetry of the problem allows us to write block-wise EVD for the entries in SA , SM and SC in the fashion
H
0 Q Λ11a Λ12a Q 0 . SA = Λ21a Λ22a 0 Q 0 QH The frequency-independent unitary transformation Q here, is essentially an orthogonal beam-forming matrix implemented using RF networks. These beam-formers are capable of producing N spatially orthogonal beams, hence the operation represents a spatial discrete Fourier transform (DFT) which essentially transforms a coupled identical array to an uncoupled virtual array represented by the eigen-impedances. Having established the concept of virtual antennas, we can define broadband matching constraints for coupled arrays similar to Fano’s approach, which requires analytic impedance circuit models. For eigen-impedances of the form λ(jω) = R + jX(ω), where R and X represent the real and imaginary parts, respectively, we have Λ22a (jω) Λ21a (jω) Λ11a (jω)
1 Normalized frequency (f/B)
VI. I NFORMATION -T HEORETIC A PPROACH Traditionally, broadband matching networks have been designed using a frequency-flat (or box-car) characteristic for Γn (f ). Fano’s broadband matching theory proposes physically realizable networks using passive lumped elements given the matching constraints are met. In general, the analysis gets cumbersome as the number of elements used to model the source or load impedance grow. For the information-theoretic approach taken in this work, we rely on constraining the matching network design by (15a) alone, partly motivated by the observation that in the special case of box-car characteristics, it yields near-optimal results5 . Before evaluating the capacity limits of MIMO systems in the presence of mutual coupling, we address some simplifications. Observe that the EVD enables us to express the transmittance matrix Sk as Sk
−1
= (ΛA (jω) + I) (ΛA (jω) − I) = I − Λ22a (jω) = Λ22a (jω) , Λ12a (jω) = Λ21a (jω) . (14)
Using a series RLC model for an antenna (or equivalently expressing it through its quality factor Q and resonant frequency ω0 ), the set of broadband matching constraints on the reflection coefficient Γ can be derived as [11] 1 ω0n (a) log df = G0n , (15a) 2 |Γn (f )| Qn 1 4π 2 df = G1n . (15b) (b) f −2 log |Γn (f )|2 ω0n Qn The integral bound is inversely proportional to the antenna Q signifying that an antenna with a broader frequency response (lower Q) offers better gain-bandwidth trade-offs. Fig. 2 shows the virtual-antenna responses obtained using NEC [17] for a 2 antenna system consisting of dipole antennas with length 0.475 λc and radius 10−3 λc , spaced d = λc /4 apart. The antenna eigen-impedance resonance parameters (Q, f0 ) are computed using NEC (3.75, 1.0425fc ) and (16, 0.9675fc ) [8].
= QTk QH .
(16)
Since the noise samples among the orthogonal subcarriers are assumed to be spatially-correlated but spectrallyuncorrelated, we have (a)
E[ni nH j ] = 0 , i, j = 1, . . . , K , i = j
(b)
Rnk = E[nk nH k ] , k = 1, . . . , K
such that the noise covariance Rn for (10) has a blockdiagonal structure. The noise covariance admits a similar EVD Rnk
= QΣnk QH .
(17)
Also, the block diagonal matrices S and Rn can be expressed via block-wise EVD as S = QT QH , Rn = QΣn QH ,
(18)
where, T and Σn are block-diagonal matrices consisting of Tk and Σnk , respectively where k = 1, . . . , K, and Q is a block diagonal matrix of Q’s. 5 For a box-car Γ(f ) over bandwidth B, the effective constraints yield |Γ|2 = exp(−B(1 + B 2 /4)Q/2π), instead of |Γ|2 = exp(−BQ/2π).
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1
10
We now formulate the problem of information theoretic optimal broadband matching for coupled MIMO systems. With perfect channel state information (CSI) at the transmitter, our aim is to find the information theoretic limits of this system. From (3) and (10), the spectral efficiency of the system can be written as
1 1 −1 H H H
C = max log I + Σn T T Q HRs H Q
K N
1/w
1
1/w2 0
Eigen−mode gains (norm)
10
where the optimization is subject to the constraints on S and Tr(Rs ) ≤ KP0 . It can be easily shown that the optimal signal covariance Rs is such that H
−1
10
−2
10
−3
10
H
Q HRs H Q = Ψ ,
10
(19)
represents a diagonal matrix. This leads to an uncoupled system with capacity
1 1 H
C = ΨT T max log
I + Σ−1 n
, K Ψ,T N
Fig. 3.
20
30 40 Sub−carrier index
50
60
Effective MIMO eigen-mode gains
Using (19) and (21), the total power constraint can be expressed as a linear constraint: K N 1 wn (fk )ψn (fk ) ≤ P0 , K n=1
where Ψ represents power allocation for the space-frequency eigen-modes of the system.
(23)
k=1
A. Optimization The spectral efficiency above can thus be written as the sum of the capacities of KN space-frequency modes:
K
1 1 H
log
I + Σ−1 Ψ T T C = max k k k
nk K Ψk ,Tk N k=1 K N 1 1 ψnk 2 = 1 − |Γnk | max log 1 + K ψnk ,Γnk N σnk k=1 n=1 K N 1 1 ψnk 1 − e−Gnk = max log 1 + K ψnk ,Γnk N σnk n=1
where the effective MIMO channel gains are expressed as the inverse of (24) wn (fk ) = Ω−1 , Ωk = QH Hk HH k Q . k nn With a change of variable Pnk = wnk ψnk , χnk = N wnk σnk , the capacity can be expressed as C
k=1
k=1
where, ψnk = [Ψk ]nn , σnk = [Σnk ]nn , Γk ΓH k = I − Tk TH k , Γnk = [Γk ]nn , and Gnk log
1 |Γnk |2
(20)
is usually referred to as the return loss of the overall network. To convey the key ideas of the paper, we have assumed a relatively simpler noise model (without loss of generality), such that the noise covariance is independent of matching: Σnk = 4kTα BI = N0 I . Here, B represents the system bandwidth. The problem thus reduces to finding the optimal ψnk , Γnk that maximizes the Shannon capacity subject to the following constraints from (11) and (15a), respectively: (i) (ii)
1 Tr(Rs ) ≤ P0 , K K 1 1 Gn (k) = G0n , n = 1, . . . , N , K B k=1
(21) (22)
=
K N 1 Pnk −Gnk . 1−e max log 1 + K Pnk ,Gnk χnk n=1
B. Solution The optimal solution to the above problem is a spacefrequency mutual-water-pouring characteristic [12]:
+ 1 χn (k) Pn (k) = (25) − ξ (1 − e−Gn (k) ) + χn (k) −1 log 1 + μn − log 1 + (26) Gn (k) = Pn (k) for k = 1, . . . , K and n = 1, . . . , N and the term mutual denotes the inter-dependence of Pn (k) and Gn (k). The waterlevels are a function of ξ and μn computed according to the power and matching constraints, respectively. Observe that because of decoupling at the receiver, the matching characteristic for each antenna has an independent water-level, unlike power allocation. The above mutual-water-pouring solution can be solved iteratively using the algorithm outlined in Appendix D.5 [15]. It is important to point out that solution for Gn (k) is an upper bound since it only incorporates (15a). However, it does help us gain qualitative insights into the frequencyselectivity of the optimal solution.
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30 Optimal G1(k) (norm)
Optimal P1(k)
0.1 0.08 0.06 0.04 0.02 0
10
20
30
40
50
20
10
0
60
0.08
2
0.06 0.04 0.02 0
20
10
20
30
40
50
60
30 40 Sub−carrier index (k)
50
60
8 Optimal G (k) (norm)
Optimal P2(k)
0.1
10
10
Fig. 4.
20
30 40 Sub−carrier index (k)
50
6 4 2 0
60
Optimal power allocation Pn (k)
Fig. 5.
VII. N UMERICAL R ESULTS We consider a 2 × 2 MIMO-OFDM system with K = 64 sub-carriers spread over B = 20 MHz, with a relative bandwidth B/fc of 6.67%. The channel is represented using 15 taps and exhibits an exponential power delay profile. The transmit antennas are assumed sufficiently separated such that RT = I, while the receiver correlation RR is calculated using NEC, where the incident electric field is modeled as a superposition of 32 vertically polarized plane waves. The antenna resonance parameters are specified in Sec. V. Results are presented for a sample channel realization at an SNR (P0 /N0 ) of 10 dB. Fig. 3 shows the effective MIMO eigen-mode channel-gains (cf. (24)). The optimal solution for power allocation and matching characteristics are shown in Fig. 4 and Fig. 5, respectively. The eigen-mode with higher G0n clearly offers better matching characteristics for a given bandwidth. However, its impact on the matching efficiency (1−|Γn (k)|2 ) translates to little capacity increase compared to a box-car matching with appropriate decoupling networks. Due to scarce literature, little is known about existing solutions for broadband coupled MIMO systems, hence a fair comparison with designs that do not incorporate Fano’s matching theory is difficult. VIII. C ONCLUSION In this paper, we presented a model for compact MIMOOFDM systems and investigated optimal end-to-end design in the presence of mutual coupling. Using an informationtheoretic approach, we formulated the problem of jointlyoptimal power allocation and matching network by incorporating Fano’s broadband matching theory. It was shown that matching and power allocation follow a space-frequency mutual-water-pouring solution. Although we used a 2-element array for illustrative purposes, the results presented here are equally applicable to larger uniform circular arrays [8]. The use of OFDM makes it easy to water-pour power across frequency, however frequency-selective matching would require some kind of variable impedance matching using adaptive elements. The framework presented here can be
Optimal return loss Gn (k)
used to evaluate diversity systems and frequency-flat matching profiles with the use of appropriate decoupling networks. R EFERENCES [1] 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. 311-335, Feb. 1998. [2] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Comm., 1998. [3] J. W. Wallace and M. A. Jensen, “Termination-Dependent Diversity Performance of Coupled Antennas: Network Theory Analysis,” IEEE Transactions on Antennas and Propagation, vol. 52, no. 1, Jan. 2004. [4] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1317-1325, Jul. 2004. [5] C. P. Domizioli et al., “Receive diversity revisited: correlation, coupling, and noise” in Proc. 2007 IEEE Globecom, Washington, D.C., pp. 36013606. [6] C. P. Domizioli et al., “Optimal front-end design for MIMO receivers,” in Proc. 2008 IEEE Global Commun. Conf., New Orleans, LA. [7] B. K. Lau et. al., “Impact of matching network on bandwidth of compact antenna arrays,” IEEE Trans. Antennas Propag., vol 54, no. 11, pp. 32253238, Nov. 2006. [8] P. S. Taluja and B. L. Hughes, “Bandwidth limitations and broadband matching for coupled multi-antenna systems,” in Proc. 2011 IEEE Global Commun. Conf., Houston, TX. [9] H. Bolcskei, D. Gesbert and A. Paulraj, “On the Capacity of OFDMBased Spatial Multiplexing Systems,” IEEE Transactions on Communications, vol. 50, no. 2, pp. 225-234, Feb. 2002. [10] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” Journal of the Franklin Institute, vol. 249, 1950. [11] M. Gustafsson and S. Nordebo, “On the spectral efficiency of a sphere,” PIER, vol. 67, pp. 275-296, 2007. [12] P. S. Taluja and B. L. Hughes, “Information theoretic optimal broadband matching for communication systems,” in Proc. 2010 IEEE Global Commun. Conf., Miami, FL. [13] J. P. Kermoal et al, “A stochastic MIMO radio channel model with experimental validation,” IEEE Journal on Selected Areas in Communications, vol. 20, pp. 1211-1226, August 2002. [14] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecomm., vol. 10, pp. 585-595, 1999. [15] P. S. Taluja, “Information-theoretic limits on broadband multi-antenna systems in the presence of mutual coupling,” Ph.D. dissertation, North Carolina State University, Raleigh, NC, USA, 2011. [16] M. A. Jensen and J. W. Wallace, Space-Time Processing for MIMO Communications: Chapter 1: MIMO Wireless Channel Modeling and Experimental Characterization, Wiley, 2005. [17] G. J. Burke and A. J. Poggio, “Numerical Electromagnetics Code (NEC) – Method of moments,” Tech. Doc. 11, Naval Ocean Systems Center, San Diego, CA, Jan. 1981.
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