- Research Article
- Open Access
Antenna Selection for MIMO Systems with Closely Spaced Antennas
© Yang Yang et al. 2009
- Received: 1 February 2009
- Accepted: 28 June 2009
- Published: 19 August 2009
Physical size limitations in user equipment may force multiple antennas to be spaced closely, and this generates a considerable amount of mutual coupling between antenna elements whose effect cannot be neglected. Thus, the design and deployment of antenna selection schemes appropriate for next generation wireless standards such as 3GPP long term evolution (LTE) and LTE advanced needs to take these practical implementation issues into account. In this paper, we consider multiple-input multipleoutput (MIMO) systems where antenna elements are placed side by side in a limited-size linear array, and we examine the performance of some typical antenna selection approaches in such systems and under various scenarios of antenna spacing and mutual coupling. These antenna selection schemes range from the conventional hard selection method where only part of the antennas are active, to some newly proposed methods where all the antennas are used, which are categorized as soft selection. For the cases we consider, our results indicate that, given the presence of mutual coupling, soft selection can always achieve superior performance as compared to hard selection, and the interelement spacing is closely related to the effectiveness of antenna selection. Our work further reveals that, when the effect of mutual coupling is concerned, it is still possible to achieve better spectral efficiency by placing a few more than necessary antenna elements in user equipment and applying an appropriate antenna selection approach than plainly implementing the conventional MIMO system without antenna selection.
- Long Term Evolution
- User Equipment
- Spectral Efficiency
- MIMO System
- Receiver Side
The multiple-input multiple-output (MIMO) architecture has been demonstrated to be an effective means to boost the capacity of wireless communication systems , and has evolved to become an inherent component of various wireless standards, including the next-generation cellular systems 3GPP long term evolution (LTE) and LTE advanced. For example, the use of a MIMO scheme was proposed in the LTE standard, with possibly up to four antennas at the mobile side, and four antennas at the cell site . In MIMO systems, antenna arrays can be exploited in two different ways, which are : diversity transmission and spatial multiplexing. However, in either case, one main problem involved in the implementation of MIMO systems is the increased complexity, and thus the cost. Even though the cost for additional antenna elements is minimal, the radio frequency (RF) elements required by each antenna, which perform the microwave/baseband frequency translation, analog-to-digital conversion, and so forth, are usually costly.
These complexity and cost concerns with MIMO have motivated the recent popularity of antenna selection (AS)—an attractive technique which can alleviate the hardware complexity, and at the same time capture most of the advantages of MIMO systems. In fact, for its low user equipment (UE) complexity, AS (transmit) is currently being considered as a baseline of the single-user transmit diversity techniques in the LTE uplink which is a MIMO single carrier frequency division multiple access (SC-FDMA) system . Further, when it comes to the RF processing manner, AS can be categorized into two groups: (1) hard selection, where only part of the antennas are active and the selection is implemented in the RF domain by means of a set of switchs (e.g., [5–7]); (2) soft selection, where all the antennas are active and a certain form of transformation is performed in the RF domain upon the received signals across all the antennas (e.g., [8–10]).
A considerable amount of research efforts have been dedicated to the investigation of AS, and have solidly demonstrated the theoretical benefits of AS (see  for a tutorial treatment). However, previous works largely ignore the hardware implementation issues related to AS. For instance, the physical size of UE such as mobile terminals and mobile personal assistants, are usually small and invariable, and the space allocated for an antenna array is limited. Such limitation makes the close spacing between antenna elements a necessity, inevitably leading to mutual coupling , and correlated signals. These issues have caught the interest of some researchers, and the capacity of conventional MIMO systems (without AS) under the described limitations and circumstances was investigated, among others, in [12–17]. To give an example, the study in  shows that as the number of receive antenna elements increases in a fixed-length array, the system capacity firstly increases to saturate shortly after the mutual coupling reaches a certain level of severeness; and drops after that.
Form factors of UE limit the performance promised by MIMO systems, and can further affect the proper functionality of AS schemes. These practical implementation issues merit our attention when designing and deploying AS schemes for the 3GPP LTE and LTE advanced technologies. There exist some interesting works, such as [18, 19] which consider AS in size sensitive wireless devices to improve the system performance. But in general, results, conclusions, and ideas on the critical implementation aspects of AS in MIMO systems still remain fragmented. In this paper, through electromagnetic modeling of the antenna array and theoretical analysis, we propose a comprehensive study of the performance of AS, to seek more effective implementation of AS in size sensitive UE employing MIMO where both mutual coupling and spatial correlation have a strong impact. In this process, besides the hybrid selection [5–7], a conventional yet popular hard AS approach, we are particularly interested in examining the performance potential of some typical soft AS schemes, including the FFT-based selection  and the phase-shift-based selection , that are very appealing but seem not to have attracted much attention so far. At the meantime, we also intend to identify the operational regimes of these representative AS schemes in the compact antenna array MIMO system. For the cases we consider, we find that in the presence of mutual coupling, soft AS can always achieve superior performance as compared to hard AS. Moreover, effectiveness of these AS schemes is closely related to the interelement spacing. For example, hard AS works well only when the interelement spacing is no less than a half wavelength.
Additionally, another goal of our study is to address a simple yet very practical question which deals with the cost-performance tradeoff in implementation: as far as mutual coupling is concerned, can we achieve better spectral efficiency by placing a few more than necessary antenna elements in size sensitive UE and applying a certain adequate AS approach than plainly implementing the conventional MIMO system without AS? Further, if the answer is yes, how would we decide the number of antenna elements for placement and the AS method for deployment? Our work will provide answers to the above questions, and it turns out the solution is closely related to identifying the saturation point of the spectral efficiency.
This paper is organized as follows. In Section 2, we introduce the network model for the compact MIMO system and characterize the input-output relationship by taking into account the influence of mutual coupling. In Section 3, we describe the hard and soft AS schemes that will be used in our study, and also estimate their computational complexity. In Section 4, we present the simulation results. We discuss our main findings in Section 5, and finally conclude this paper in Section 6.
Let us firstly consider the transmitter side, which can be regarded as a coupled port network with terminals. We define and as the vectors of terminal currents and voltages, respectively, and they are related through
where denotes the impedance matrix at the transmitter side. The -th entry of , when , denotes the mutual impedance between two antenna elements, and is given by [20, Chapter 21.2]:
In the above expression, denotes the characteristic impedance of the propagation medium, and can be calculated by , where and denote permittivity and permeability of the medium, respectively. Likewise, denotes the propagation wavenumber of an electromagnetic wave propagating in a dielectric conducting medium, and can be computed through , where is the angular frequency. Finally , and are defined as
When , is the self-impedance of a single antenna element, and can also be obtained from (2) by simply redefining , and as follows:
Thus, the self-impedance for an antenna element with and for example, is approximately
Further, let us consider an example that antenna elements of such type are equally spaced over a linear array of length . The impedance matrix is given by
For , the terminal voltage can be related to the source voltage via the source impedance by . Define , and . Then, from Figure 2, we can obtain the following results: and . Therefore, the relationship between terminal voltages and source voltages can be written in matrix form as Similar to , we choose , which roughly corresponds to a conjugate match in the presence of mild coupling. In the case of uncoupling in the transmitter side, is diagonal, and its diagonal elements are all the same. Consequently, is also diagonal, and its diagonal element can be denoted as . To accommodate the special case of zero mutual coupling where is equal to , in our model we modify the relationship between and into
Denote as the vector of open circuited voltages induced across the receiver side antenna array, and as the voltage vector across the output of the receive array. Since we assumed high-input impedance of these LNAs, a similar network analysis can be carried out at the receiver side and will yield
where . is the mutual impedance matrix at the receiver side, and is a diagonal matrix with its th entry given by , . is given by It is noted that the approximate conjugate match  is also assumed at the receiver side, so that the load impedance matrix is diagonal with its entry given by , for .
In frequency-selective fading channels, the effectiveness of AS is considerably reduced , which in turn makes it difficult to observe the effect of mutual coupling. Therefore, we focus our attention solely on flat fading MIMO channels. The radiated signal is related to the received signal through
where is a complex Gaussian matrix with correlated entries. To account for the spatial correlation effect and the Rayleigh fading, we adopt the Kronecker model [22, 23]. This model uses an assumption that the correlation matrix, obtained as with being the operator stacking the matrix into a vector columnwise, can be written as a Kronecker product, that is, , where and are respectively, the receive and transmit correlation matrices, and denotes the Kronecker product. This implies that the joint transmit and receive angle power spectrum can be written as a product of two independent angle power spectrum at the transmitter and receiver. Thus, the correlated channel matrix can be expressed as
where is a matrix whose entries are independent identically distributed (i.i.d) circular symmetric complex Gaussian random variables with zero mean and unit variance. The -th entry of or is given by , where is the zeroth order Bessel function of the first kind, and denotes the distance between the , -th antenna elements.
Therefore, based on (8)–(11), the output signal vector at the receiver can be expressed in terms of the input signal at the transmitter through
where can be regarded as a compound channel matrix which takes into account both the Rayleigh fading in wireless channels and the mutual coupling effect at both transmitter and receiver sides, and is the thermal noise. For simplicity, we assume uncorrelated noise at the receiving antenna element ports. For the case where correlated noise is considered, readers are referred to [16, 17].
We describe here some typical hard and soft AS schemes that we will investigate, assuming the compact antenna array MIMO system described in Section 2. For hard AS, we focus only on the hybrid selection method [5–7]. For soft AS, we study two typical schemes: the FFT-based selection  which embeds fast Fourier transform (FFT) operations in the RF chains, and the phase-shift-based selection  which uses variable phase shifters adapted to the channel coefficients in the RF chains. For simplicity, we only consider AS at the receiver side with antennas being chosen out of the available ones, and we focus on a spatial multiplexing transmission.
We assume that the propagation channel is flat fading and quasistatic, and is known at the receiver. We also assume that the power is uniformly allocated across all the transmit antennas, that is, . We denote the noise power as , and the nominal signal-to-noise ratio (SNR) as . Then assuming some codes that approach the Shannon limit quite closely are used, the spectral efficiency (in bits/s/Hz) of this full-complexity (FC) compact MIMO system without AS could be calculated through 
It is worth noting that the length limits of transmit and receive arrays, and , enter into the compound channel matrix in a very complicated way. It is thus difficult to find a close-form analytical relationship between and ( ). Consequently, using Monte Carlo simulations to evaluate the performance of spectral efficiency becomes a necessity.
To avoid detailed system configurations and to make the performance comparison as general and as consistent as possible, we only use the spectral efficiency as the performance of interest. Moreover, all these AS schemes we study here are merely to optimize the spectral efficiency, not other metrics. Since each channel realization renders a spectral efficiency value, the ergodic spectral efficiency and the cumulative distribution function (CDF) of the spectral efficiency will be both meaningful. We will then consider them as performance measures for our study.
3.1. Hybrid Selection
Optimal selection that leads to requires an exhaustive search over all subsets of , which is evident by (14). Note that
Then, the matrix multiplication in (14) has a complexity of . Calculating the matrix determinant in (14) requires a complexity of . Thus, we can conclude that optimal selection requires about complex additions/multiplications. This estimated complexity for optimal selection can be deemed as an upper bound of the complexity of any hybrid AS scheme, since there exist some suboptimal but reduced complexity algorithms, such as the incremental selection and the decremental selection algorithms in .
3.2. FFT-based Selection
As for this soft selection scheme (e.g., ), a -point FFT transformation (phase-shift only) is performed in the RF domain firstly, as shown in Figure 4(b), where information across all the receive antennas will be utilized. After that, a hybrid-selection-like scheme is applied to extract out of information streams. We denote as the unitary FFT matrix with its th entry given by:
Accordingly, this system delivers a spectral efficiency of
The only difference between (14) and (17) is the -point FFT transformation. Such FFT transformation requires a computational complexity of . If we assume , then the computational complexity of optimal selection that achieves can be estimated as , which is the worst-case complexity.
3.3. Phase-Shift Based Selection
This is another type of soft selection scheme (e.g., ) that we consider throughout this study. Its architecture is illustrated in Figure 4(c). Let us denote as one matrix whose elements are nonzero and restricted to be pure phase-shifters, that we will fully define in what follows. There exists some other work such as  that also considers the use of tunable phase shifters to increase the total capacity of MIMO systems. However, in Figure 4(c), the matrix that performs phase-shift implementation in the RF domain essentially serves as a -to- switch with output streams. Additionally, unlike the FFT matrix, might not be unitary, and hence the resulting noise can be colored. Finally, this system's spectral efficiency can be calculated by 
Let us define the singular value decomposition (SVD) of as , where and are , unitary matrices representing the left and right singular vector spaces of , respectively; is a nonnegative and diagonal matrix, consisting of all the singular values of . In particular, we denote as the th largest singular value of , and as the left singular vector of associated with . Thus one solution to the phase shift matrix can be expressed as [10, Theorem 2]:
where gives the phase angles, in radians, of a matrix with complex elements, denotes the element-by-element exponential of a matrix.
The overall cost for calculating the SVD of is around [26, Lecture 31]. Computing the matrix multiplication in (19) requires a complexity around the order of . The matrix determinant has an order of complexity of . Therefore, the phase-shift-based selection requires around complex additions/multiplications.
Our simulations focus on the case when AS is implemented only on the receiver side, but mutual coupling and spatial correlation are accounted for at both terminals. However, in order to examine the mutual coupling effect on AS at the receiving antenna array, we further assume equally-spaced antennas at the transmitter array, and the interelement spacing is fixed at . This large spacing is chosen to make the mutual coupling effect negligible at the transmitting terminal. For the receiver terminal, we fix the array length at . We choose and for all the dipole elements. Each component in the impedance matrices and is computed through (2) which analytically expresses the self and mutual impedance of dipole elements in a side-by-side configuration. Finally, we fix the nominal SNR at dB.
As algorithm efficiency is not a focus in this paper, for both hybrid and FFT-based selection methods, we use the exhaustive search approach to find the best antenna subset. For the phase-shift-based selection, we compute the phase shift matrix through (19) given each . For each scenario of interest, we generate random channel realizations, and study the performance in terms of the ergodic spectral efficiency and the CDF of the spectral efficiency.
4.1. Ergodic Spectral Efficiency of Compact MIMO
4.2. Ergodic Spectral Efficiency of Compact MIMO with AS
Soft AS always performs no worse than hard AS. The phase-shift-based selection performs strictly better than the FFT-based selection.
With the same number of RF chains, the system with AS performs strictly better than the RFC system.
Interestingly, these conclusions that hold for this compact antenna array case are also generally true for MIMO systems without considering the mutual coupling effect (e.g., ). But a cross-reference to the results in Figure 5 can help understand this phenomenon. In Figure 5, it is shown that when increases from to , the ergodic spectral efficiency of the compact MIMO system behaves nearly the same as that of MIMO systems without considering the mutual coupling effect. Therefore, it appears natural that when AS is applied to the compact MIMO system with , similar conclusions can be obtained. It is also interesting that the FFT-based selection performs almost exactly the same as the hybrid selection.
Soft AS always outperforms hard AS. The phase-shift-based selection delivers the best performance among all these three AS schemes.
The phase-shift-based selection performs better than the RFC system when . The FFT-based selection performs better than the RFC system when . The advantage of using the hybrid selection is very limited.
We further increase the number of antennas to . Now the mutual coupling effect becomes more severe, and different conclusions are demonstrated in Figure 8. It is observed that , indicating the following.
Soft AS performs strictly better than hard AS.
The phase-shift-based selection performs better than the FFT-based selection when . After that, there is not much performance difference between them.
Also, similar to what we have observed in Figures 6 and 7, in terms of the ergodic spectral efficiency, none of the systems with AS outperforms the FC system with receive antennas (and thus RF chains). However, as for the RFC system with only antennas (and thus RF chains), in Figure 8 we observe the following.
The RFC system always performs better than the hybrid selection. The hybrid selection seems futile in this case.
The phase-shift-based selection performs better than the RFC system when . The benefit of the FFT-based selection is very limited, and it seems not worth implementing.
This indicates that due to the strong impact of mutual coupling in this compact MIMO system, only the phase-shift-based selection is still effective, but only for a limited range of numbers of the available RF chains. More specifically, when it is best to use the phase-shift-based selection, otherwise the RFC system with antennas when . Further increase in the number of RF chains, however, will not lead to a corresponding increase in the spectral efficiency, as demonstrated in Figure 5.
4.3. Spectral Efficiency CDF of Compact MIMO with AS
In our study, we also test different scenarios by varying the length of linear array , for example, we choose , , and so forth. For brevity, we leave out these simulation results here, but summarize our main findings as follows.
Suppose the ergodic spectral efficiency of a compact antenna array MIMO system saturates at . Our simulation results (e.g., Figure 5) indicate that
When , both soft and hard selection methods are effective, but the selection gains vary with respect to . Particularly, the phase-shift-based selection delivers the best performance among these tested schemes. Performance of the FFT-based selection and the hybrid selection appears undistinguishable.
- (2)When , there exist two situations:
When , the selection gain of the phase-shift-based selection still appears pronounced, but tends to become smaller when approaches . The advantage of using the FFT-based selection is quite limited. The hybrid selection seems rather futile.
When , neither soft nor hard selection seems effective. This suggests that AS might be unnecessary. Instead, we can simply use a RFC system with RF chains by equally distributing the elements over the limited space.
It is noted that in all these cases we examine, soft AS always has a superior performance over hard selection. This is because soft selection tends to use all the information available, while hard selection loses some additional information by selecting only a subset of the antenna elements. Our simulation results also suggest that, if hard selection is to be used, it is necessary to maintain . Otherwise, the strong mutual coupling effect could render this approach useless. Further, if the best selection gain for system spectral efficiency is desired, one can place or so elements along the limited-length linear array, use or less RF chains, and apply the phase-shift-based selection method. Therefore, it becomes crucial to identify the saturation point . This in turn requires the electromagnetic modeling of the antenna array that can take into account the mutual coupling effect.
In this paper, we proposed a study of some typical hard and soft AS methods for MIMO systems with closely spaced antennas. We assumed antenna elements are placed linearly in a side-by-side fashion, and we examined the mutual coupling effect through electromagnetic modeling of the antenna array and theoretical analysis. Our results indicate that, when the interelement spacing is larger or equal to one half wavelength, selection gains of these tested soft and hard AS schemes will be very pronounced. However, when the number of antennas to be placed becomes larger and the interelement spacing becomes smaller than a half wavelength, only the phase-shift-based selection remains effective and this is only true for a limited number of available RF chains. The same conclusions however, are not observed for the case of hard selection. Thus it seems necessary to maintain the interelement spacing no less than one half wavelength when the hard selection method is desired. On the other hand, if the best selection gain for system spectral efficiency is desired, one can employ a certain number of elements for which the compact MIMO system attains its maximum ergodic spectral efficiency, use or less RF chains, and deploy the phase-shift-based selection method. This essentially indicates, if the cost-performance tradeoff in implementation is concerned, by placing a few more than necessary antenna elements so that the system spectral efficiency reaches saturation and deploying the phase-shift-based selection approach, we can achieve better performance in terms of system spectral efficiency than the conventional MIMO system without AS. Overall, our study provides novel insight into the deployment of AS in future generation wireless systems, including the 3GPP LTE and LTE advanced technologies.
This material is based on research supported by the Air Force Research Laboratory under agreement FA9550-09-1-0576, by the National Science Foundation under Grant CCF-0829958, and by the U.S. Army Research Office under Grant W911NF-08-1-0449. The authors would like to thank Dr. Dmitry Chizhik and Dr. Dragan Samardzija of Bell Laboratories, Alcatel-Lucent for the helpful discussions on the modeling and implementation issues.
- Foschini GJ, Gans MJ: On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications 1998, 6(3):311-335. 10.1023/A:1008889222784View ArticleGoogle Scholar
- Overview of 3GPP Release 8 V0.0.3 November 2008, http://www.3gpp.org/Release-8
- Molisch AF, Win MZ: MIMO systems with antenna selection. IEEE Microwave Magazine 2004, 5(1):46-56. 10.1109/MMW.2004.1284943View ArticleGoogle Scholar
- Kotecha J: LTE:MIMO Techniques in 3GPP-LTE. Freescale Semiconductor, June 2008Google Scholar
- Win MZ, Winters JH: Analysis of hybrid selection/maximal-ratio combining in Rayleigh fading. IEEE Transactions on Communications 1999, 47(12):1773-1776. 10.1109/26.809693View ArticleGoogle Scholar
- Molisch AF, Win MZ, Winter JH: Reduced-complexity transmit/receive-diversity systems. IEEE Transactions on Signal Processing 2003, 51(11):2729-2738. 10.1109/TSP.2003.818211MathSciNetView ArticleGoogle Scholar
- Gorokhov A, Gore DA, Paulraj AJ: Receive antenna selection for MIMO spatial multiplexing: theory and algorithms. IEEE Transactions on Signal Processing 2003, 51(11):2796-2807. 10.1109/TSP.2003.818204MathSciNetView ArticleGoogle Scholar
- Collin L, Berder O, Rostaing P, Burel G: Soft vs. hard antenna selection based on the minimum distance for MIMO systems. Proceedings of the 38th Asilomar Conference on Signals, Systems and Computers, November 2002, Grove, Calif, USA 2: 1369-1373.Google Scholar
- Molisch AF, Zhang X: FFT-based hybrid antenna selection schemes for spatially correlated MIMO channels. IEEE Communications Letters 2004, 8(1):36-38. 10.1109/LCOMM.2003.822512View ArticleGoogle Scholar
- Zhang X, Molisch AF, Kung S-Y: Variable-phase-shift-based RF-baseband codesign for MIMO antenna selection. IEEE Transactions on Signal Processing 2005, 53(11):4091-4103.MathSciNetView ArticleGoogle Scholar
- Balanis CA: Antenna Theory: Analysis and Design. 3rd edition. John Wiley & Sons, New York, NY, USA; 2005.Google Scholar
- Janaswamy R: Effect of element mutual coupling on the capacity of fixed length linear arrays. IEEE Antennas and Wireless Propagation Letters 2002, 1: 157-160.View ArticleGoogle Scholar
- Waldschmidt C, Schulteis S, Wiesbeck W: Complete RF system model for analysis of compact MIMO arrays. IEEE Transactions on Vehicular Technology 2004, 53(3):579-586. 10.1109/TVT.2004.825788View ArticleGoogle Scholar
- Wallace JW, Jensen MA: Mutual coupling in MIMO wireless systems: a rigorous network theory analysis. IEEE Transactions on Wireless Communications 2004, 3(4):1317-1325. 10.1109/TWC.2004.830854View ArticleGoogle Scholar
- Abouda AA, Häggman SG: Effect of mutual coupling on capacity of MIMO wireless channels in high SNR scenario. Progress in Electromagnetics Research 2006, 65: 27-40.View ArticleGoogle Scholar
- Gans MJ: Channel capacity between antenna arrays—part I: sky noise dominates. IEEE Transactions on Communications 2006, 54(9):1586-1592.MathSciNetView ArticleGoogle Scholar
- Gans MJ: Channel capacity between antenna arrays—part II: amplifier noise dominates. IEEE Transactions on Communications 2006, 54(11):1983-1992.View ArticleGoogle Scholar
- Xu Z, Sfar S, Blum RS: On the importance of modeling the mutual coupling for antenna selection for closely-spaced arrays. Proceedings of IEEE Conference on Information Sciences and Systems (CISS '06), March 2006, Princeton, NJ, USA 1351-1355.Google Scholar
- Lu D, So DKC, Brown AK: Receive antenna selection scheme for V-BLAST with mutual coupling in correlated channels. Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '08), September 2008, Cannes, France 1-5.Google Scholar
- Orfanidis SJ: Electromagnetic Waves and Antennas. Rutgers University, Piscataway, NJ, USA; 2004.Google Scholar
- Hewes J: Impedance and reactance. The Electronics Club, February 2008, http://www.kpsec.freeuk.com/imped.htm
- Kermoal JP, Schumacher L, Pedersen KI, Mogensen PE, Frederiksen F: A stochastic MIMO radio channel model with experimental validation. IEEE Journal on Selected Areas in Communications 2002, 20(6):1211-1226. 10.1109/JSAC.2002.801223View ArticleGoogle Scholar
- Oestges C, Clerckx B, Vanhoenacker-Janvier D, Paulraj AJ: Impact of fading correlations on MIMO communication systems in geometry-based statistical channel models. IEEE Transactions on Wireless Communications 2005, 4(3):1112-1120.View ArticleGoogle Scholar
- Shiu D-S, Foschini GJ, Gans MJ, Kahn JM: Fading correlation and its effect on the capacity of multielement antenna systems. IEEE Transactions on Communications 2000, 48(3):502-513. 10.1109/26.837052View ArticleGoogle Scholar
- Nakaya Y, Toda T, Hara S, Takada J-I, Oishi Y: Incorporation of RF-adaptive array antenna into MIMO receivers. Proceedings of IEEE Topical Conference on Wireless Communication Technology (WCT '03), October 2003, Honolulu, Hawaii, USA 297-298.Google Scholar
- Trefethen LN, Bau D III: Numerical Linear Algebra. SIAM, Philadelphia, Pa, USA; 1997.MATHView ArticleGoogle Scholar
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