Capacity Limits of Dense Palm-Sized MIMO Arrays David Gesbert, T. Ekman, N. Christophersen Department of Informatics, University of Oslo Gaustadalleen 23, P.O.Box 1080 Blindern, 0316 Oslo, Norway Email: gesbert,torbjoee,
[email protected]
Abstract – We address the capacity limits of sizeconstrained multi-input multi-output (MIMO) arrays. While most work on MIMO focuses on arrays with of a small number of sufficiently spaced, low correlated antenna elements, we look at the case where a fixed small space is being filled up with antenna elements. We obtain a dense MIMO system whose rate performance is analyzed. We establish capacity limits and an equivalence with the capacity of MIMO arrays with unlimited aperture.
1.
INTRODUCTION
MIMO interest was recently revealed by information theory results [1, 2] showing the large wireless capacity gains that can be extracted from such system. So far, to the exception of work like [3], much of the examples of application in the literature have been on MIMO systems exhibiting appropriately low fading correlation so the MIMO channel matrix has full rank. Physically, this assumes antenna elements are placed sufficiently spaced apart, both at the transmitter and at the receiver. For example λ/2 (half a wavelength) to λ spacing is seen to be necessary to achieve decorrelation between two elements [4]. Additionally, typical MIMO examples have a low number of antennas (in most cases 3 to 4) in an attempt to keep both the RF cost low and the array size reasonable. However, at say 2GHz, 3 to 4 λ-spaced elements in a linear antenna will lead to an array 22cm to 29cm wide, which is above what is desirable for small multimedia devices. This paper takes a theoretical look at this problem by considering the performance limits of dense MIMO arrays. Dense arrays are obtained by cramming a large number of RF elements within a fixed given (e.g. wavelength-like) total aperture. In practice, beyond a certain point, mutual coupling effects arise between neighboring antennas. However for our purpose coupling can be viewed as an effect only altering the antenna correlation model, as suggested in e.g. [5]. In addition we arWork supported in part from Telenor Research.
gue that limiting capacity behavior is well approximated with a reasonable number of antennas. We analyze the Shannon capacity, limiting ourselves to 1D arrays here although the generalization to 2D arrays is clearly of practical interest. We investigate the capacity limits when the number of antennas is large and we compare with an ideal system unconstrained in size. We show that, in a normalized capacity sense, both dense and unlimited arrays converge to finite non-zero limits that can be compared. Although the performance of a dense array levels off quickly with the number of antennas, a smallsize MIMO array can yield a capacity equivalent to that of an uncorrelated array of a much larger size. The performance limit and dependence on the continuous correlation model is described analytically. The effect of correlation on MIMO capacity due to limited scattering or angle spread has been modeled in a number of recent papers (e.g. [3, 6]). In [7, 8], the impact of correlation models is studied on asymptotic arrays with unrestricted lengths. To our knowledge, however, the performance of dense arrays has not been so far addressed, to the exception of one recent independent contribution [9]. 2.
SYSTEM AND SIGNAL MODELS
Consider a MIMO system with N transmit antennas and M receive antennas. Transmission takes place in a possibly correlated fading channel represented by the M × N channel matrix H. The effects of correlation on the channel can be captured independently at the transmitter and receiver via the following expression [10]: 1/2
H = R1/2 rx H0 Rtx 1/2 Rrx
(1)
1/2 Rtx )
(resp. is the hermitian square root where of the M × M receive (resp. N × N transmit) antenna correlation matrix. H0 is an M × N MIMO matrix such as the one used in [1, 2], whose elements are independent, unit-variance random complex coefficients. The receive correlation matrix writes simply: [Rrx ]k,l = r(|d(k) − d(l)|) for k, l = 1..M
(2)
where d(i) denotes the position of the i-th receive antenna, i = 1..M and r(x) is the normalized correlation between two antennas spaced by x meters. A similar expression can be obtained for the transmit correlation matrix. Clearly, in the presence of full correlation, the correlation matrix converges to an all-ones matrix. If on the other hand the antennas are maintained far apart from each other, the correlation matrix coincides with the identity. Correlation model- There exists a number of models to characterize the correlation between the antennas including [3, 11]. Here we are particularly interested in the impact of antenna density (rather than spacing of just two antennas) on the capacity. Therefore this paper does not attempt to evaluate the correlation models. Simple correlation models function of antenna spacing include exponential models and the so-called Jakes model r(x) = J0 (2πx/λ)
(3)
where J0 is the zero-order bessel function of the first kind and λ is the wavelength. This model is often used to describe cases with rich omnidirectional multipath around the antenna array. The absolute value of the correlation is illustrated in Fig.1. Note that mutual coupling effects have been shown to help reduce correlation of closely spaced antennas [5] and can be incorporated in the form of a more realistic correlation function. Bessel Correlation model 1
0.9
0.8
absolute value of correlation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8 1 1.2 spacing in wavelengths
1.4
1.6
1.8
2
Figure 1: Jakes model for antenna correlation. 2.1. Uniform compact array We are primarily interested in applications with compact subscriber units and consider a receiving array limited in total aperture to d0 . For the sake of clarity, the transmitter is supposed to be unconstrained in size so that the transmit antenna correlation is identity matrix IN . This system is called RX-compact, with channel: H = R1/2 rx H0
(4)
There are various and interesting ways of filling a RXcompact array of size d0 . An obvious way is to use uniformly distributed antennas: d(i) = (i − 1)
d0 with i = 1..M M −1
(5)
However other approaches are also possible, yielding possibly different performance trade-offs. It turns out, however, that uniformly spatially sampled arrays give useful mathematical properties that can be exploited in an analytical analysis here. In that case, the receive correlation matrix is symmetric Toeplitz and can be expressed as: [Rrx ]k,l = r(
|(k − l)d0 | ) for k, l = 1..M M −1
(6)
It is seen that as the number of antennas is increased, neighboring antennas become more closely spaced and of course more correlated. On the other hand, the correlation of antennas located at each side of the array remains constant, possibly low if e.g. d0 ≈ λ. 3.
CAPACITY ANALYSIS
We here look at the capacity performance of a RXcompact MIMO array in the large number of antenna case (large M, N ). Because a compact MIMO array is limited in total aperture, the number of significant eigenmodes will also be limited, thereby posing an absolute limit on the otherwise linear capacity growth with the number of antennas. When the eigenmode saturation occurs it is expected that array gain effects (in log2 (M )) will dominate, as more receive antennas just keep capturing more energy, at least theoretically. It is however the eigenmode transmission limit that we wish to investigate here, independently from array gain effects. To do so, we introduce below an “array-gain normalized” capacity quantity. We assume that the receiver has full channel knowledge, unlike the transmitter. We first examine the case of a spatially unconstrained array and show that the normalized capacity converges to a limit. The limit is calculable analytically and matches that of the ideal identity MIMO system considered in [1]. Then we investigate the case of spatially constrained arrays. 3.1. Normalized capacity In order to study capacity growth independently of array gain effects we use the following normalized expression for the Shannon capacity (also used in e.g. [7, 12]) : h C = log2 det IM +
i ρ H H∗ , NM
(7)
where ∗ refers to transpose conjugate, and ρ is the receive signal to noise ratio (SNR) per receive antenna. The normalization is equivalent to assuming that not just the total transmit power is limited but also the total receive power across antennas. Note that a similar effect on capacity is obtained if considering that the noise samples become correlated across antennas. Beyond providing interesting insights into the properties of large arrays, one additional benefit of the metric used in (7) is that exact analytical expressions can be obtained for capacity limits while analysis of the regular capacity, be it in the large M, N case, has been done using lower and upper bounds only [1]. Another approach to this problem is to use the capacity per dimension (normalized by min(M, N ) outside the log function) as a metric. However that metric does not lead to tractable expression either, except in the compact array case where it simply converges to 0 (since the linear growth of capacity with M, N is of course not possible in that case). 3.2. The unlimited array To provide a reference for comparison with the RXcompact MIMO system of (4),(6), we first look at the normalized capacity of an unlimited, i.i.d. type, array. In this case, neighboring antennas are maintained sufficiently far apart so that: Rtx = Rrx = IM and H = H0
(8)
It is well known that as M, N become large, H0 ’behaves like’ a unitary matrix so we define: ΩM =
H0 H∗0 − IM N
The normalized capacity becomes: i h ρ ρ Ciid = log2 det IM + IM + ΩM M M
(9)
(10)
Note that although the central limit theorem suggests that the elements of ΩM converge to zeros for large M, N , the size of ΩM is also growing with the number of antennas, rendering the determinant analysis in (10) a non trivial one. In particular, in the usual case when the extra normalization by M is not used, ΩM cannot be removed from the expression (10) and bounds must be resorted to [1]. Here however, the following result can be proved: Lemma 1: In the large number of antennas case, i.e. N, M → ∞, we have Ciid → C0 =
ρ log(2)
(11)
Proof: Let us denote the eigenvalue decomposition (EVD) of ΩM by ΩM = UΣM U∗ where ΣM =
diag[σ1,M , ..., σM,M ] contains positive eigenvalues (ΩM is hermitian positive). We can re-write (10) into: Ciid
= = =
= =
i h ρ ρ IM + ΩM U log2 det U∗ IM + M Mi h ρ ρ IM + ΣM log2 det IM + M M M X ρσi,M ρ + ) log2 (1 + M M i=1 M ρσi,M 1 X ρ + + o(1/M ) log(2) i=1 M M P σi,M ρ +ρ i + o(1) log(2) M
(12)
Here o(x) denotes a quantity such that Po(x)/x tends to 0 as M, N → ∞. The key here is that i σi,M is also the trace of ΩM and can be shown to be a bounded quantity in variance. Indeed, from the central limit theorem, we have E|ΩMi,i |2 = K/M for large M , where K is a constant. Since diagonal elements of ΩM are uncorrelated, we have X E| σi,M |2 = E|trace(ΩM )|2 → K for large M (13) i
It follows from (12) and (13) that: Ciid =
ρ + o(1) log(2)
(14)
A few remarks are in order at this point. First the result above proves that the array-gain normalized capacity does converge to a finite limit even for ideal MIMO arrays (unlike with the usual capacity definition). This suggests that the array gain is a key driver of capacity growth in practical MIMO arrays. Second the normalized capacity limit is exactly proportional to SNR in the large array case. Interestingly, this same limit was shown to be realized by the identity channel (H = I) by Foschini in [1] and also among others in [7]. Therefore our results show that the large i.i.d. MIMO array behaves exactly like the identity system in the normalized capacity sense. Note that, in the usual capacity sense, it is well known that the limit for the i.i.d. array can only be lower and upper bounded analytically [1]. In the light of the earlier developments, the reason for this is that, in the absence of normalization by M in (12), the Taylor expansion of the log2 (1+x) function cannot be used as its argument x does not converge to zero, rendering the expression impossible to simplify. Worth noting also is that this limit coincides with the capacity limit obtained in a AWGN channel with infinite bandwidth and fixed power budget. The difference here is that the power is spread across an infinite spatial dimension instead of the spectral dimension.
3.3. RX-compact array We now turn to the case of the RX-compact array discussed in 2.1. Because the normalized capacity of the unlimited MIMO array levels off for a large number of antennas, one expects that the compact array will also converge to a value 0 < Cc < Ciid < ∞. This section introduces this result and describes the limit in terms of properties of the correlation model. The ability to compare between different finite limiting values for different (large) arrays further motivates the use of the normalized capacity as metric here. In view of (4) and (7), the normalized capacity of the RX-compact array is: ρ 1/2 H0 H∗0 1/2 R R , log2 det IM + M N i h ρ 1/2 ρ R+ R ΩM R1/2 , log2 det IM + M M
Cc
=
Cc
=
where the subscript rx has been dropped for simplicity. Lemma 2: Let us denote the EVD of R by R = VΛM V∗ where ΛM = diag[λ1,M , ..., λM,M ] contains positive eigenvalues. Then we have for large M, N :
correlation eigenvalues. To the difference of previous similar works, the eigenvalues here do not depend on the channel realization. The following result can be proved to help characterize the limit of the normalized capacity: Lemma 3: Assuming r(x), x ∈ [0, d0 ] is smooth enough1 , we have the following results λi,M M
=
αi with αi ≥ 0 constant(17)
r(τ − t)v(t)dt
=
αv(τ ) ,
lim
Z
M→∞ d0
0
∞ X
αi = 1
(18)
i=1
where (18) refers to a linear eigenfunction problem whose eigenvalue solution α is the series {αi }, i = 1, 2..∞. (17), (18) come from re-writing the eigenvalue of R problem into the Riemann approximation of integral in (18). Details are left out due to lack of space here. Note that in the i.i.d. case, the result above is not true as λi,M = 1 for all i and M . Using the result above, the limit of capacity in the large M, N case can be written as: ∞ X Cc = log2 (1 + ραi ) (19) M,N →∞
i=1
Note that the fact that the limit in (19) does exist and satisfies 0 < Cc < Ciid is trivial. This comes straight ρλi,M ) (15) from upper bounding log2 (1 + x) by x/log(2) in (19). log2 (1 + Cc ≈ M i=1 In the fully correlated case (d0 → 0) we have clearly α i = δ1,i , giving as expected the capacity of a 1 × 1 with exact equality for M, N = ∞ For a sketch of this channel C0 = log(1 + ρ). The analytical computation ∗ 1/2 1/2 proof, let us define ΨM = (ρ/M )V R ΩM R V, and ρ of the {α i } sequence is difficult (other than from solving DM = IM + M ΛM , we can write using a Taylor expanthe eigenvalue problem of R). Unfortunately the known sion: properties linking the distribution of eigenvalues of large i h ρ Toeplitz matrices to the Fourier transform of the maΛM + ΨM , Cc = log2 det IM + M trix elements [13] do not apply here because R does not X ∂ det include the whole correlation sequence but instead a con ΨMij + o(||ΨM ||) tinuously sampled fixed interval ([0, d0 ]) of it. Cc = log2 det(DM ) + ∂DMij i,j The simplest analytical characterization in the limit DM M, N → ∞ is shown in (18). Specifically, the {αi } sequence is given by the set of eigenvalues α correspondM X Πi6=j DMii ΨMii + o(||ΨM ||) (16) ing to eigenfunctions v(τ ) of a continuous linear operator = log2 det(DM ) + with kernel r(·). j=1 The numerical behavior of the distribution of the seWhere we exploit the standard co-matrix expansion of quence {αi } and the impact of the correlation model is the determinant and make use the fact P that DM is di- investigated in Sec. 4. agonal. Since trace(R) = M , we have j λj,M = M , from this we can upper bound Πi6=j DMii by eρ/ log(2) for 4. NUMERICAL EVALUATION all j = 1..M . Finally, the diagonal terms ΨMij are un√ correlated and behave as 1/(M M ) as M grows large. We examine the performance of a RX-compact uniform Therefore both the second and third terms approach zero array (1 wavelength wide) with a SNR of 10dB. We use as M grows large in the r.h.s of (16). The first term is the model defined in (3). First we illustrate the behavior of normalized eigenvalues λi,M /M of the correlation identical to that of (15). 1 the integral of r(x) is perfectly approximable by a Riemann The normalized capacity is the sum capacities of individual channels with SNR weighted by the normalized sum, as is usually the case. M X
0
10
Average normalized capacity vs. MIMO order. SNR=10 dB 16
14
Limit for uncorrelated
12
Average Capacity
matrix R in Fig. 2. The progression is negative exponential (up to the limit of matlab’s precision, bottom of figure), with most of the energy included in the 6 or 7 first eigenvalues. In Fig. 3, the normalized capacity is plotted for the unlimited (i.i.d.) array and the RX-compact array. We also show bounds for a fully correlated array and for the i.i.d array. The compact array with an infinite number of antennas give the same performance as an i.i.d. array with just 5 antennas. This i.i.d array will spread over much more than one wavelength. With 6 antennas the compact array achieves most of its limiting performance, after which there is no reason adding more antennas other than for array gain purposes. At that point the compact array will perform as well as a wider i.i.d array with 4 antennas (just two antenna less!).
10 Unlimited array 8
6 1 lambda array 4 Limit for fully correlated 2
0
2
4
6
8 10 12 14 Number of MIMO TXers and RXers
16
18
20
Figure 3: Normalized capacity versus MIMO order for i.i.d. array and RX-compact array (1 wavelength). Dotted and dash-dotted curves give the theoretical limits, only relevant in the large M, N case.
−1
10
−2
10
−3
eigenvalue
10
−4
[3] D. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multi-element antenna systems,” IEEE Trans. Comm., march 2000.
10
−5
10
[4] S. Pitchaiah, V. Erceg, D. Baum, R. Krishnamoorthy, and A. Paulraj, “Modeling of multiple-input multiple-output (mimo) radio channel based on outdoor measurements conducted at 2.5ghz for fixed bwa applications,” in Proc. International Conference on Communications, 2002.
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8 10 12 number of receive antennas
14
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Figure 2: Convergence of the top 8 normalized eigenvalues of correlation matrix R toward αi , i = 1..8. Total aperture is one wavelength (Jakes model). 5.
DISCUSSION
[5] K. Boyle, “The correlation limits of linear antennas,” in Proceedings of COST 273, (Guilford, UK), January 2002. [6] D. Gesbert, H. Bolcskei, D. Gore, and A. Paulraj, “Outdoor mimo wireless channels: Models and performance prediction,” IEEE Trans. Communications, 2002. [7] S. L. Loyka, “Channel capacity of mimo architecture using the exponential correlation matrix,” IEEE Communications Letters, vol. 5, pp. 369–371, Sept. 2001. [8] C. N. Chuah, D. Tse, J. M. Kahn, and R. Valenzuela, “Capacity scaling in mimo wireless systems under correlated fading,” IEEE Trans. Inf. Theory, vol. 48, pp. 637–650, March 2002.
This paper introduces the problem of achievable performance over compact MIMO arrays, constrained in space but not in the number of antenna elements and compared with traditional unconstrained arrays. We consider a normalized capacity measure and show the existence of finite limits for both compact and i.i.d. cases. The performance of small wavelength-like MIMO arrays is limited but can yield performance similar to that of much wider arrays, with a reasonable number of antennas.
[10] H. B¨ olcskei, D. Gesbert, and A. J. Paulraj, “On the capacity OFDM-based spatial multiplexing systems,” IEEE Trans. Comm., Feb. 2002.
REFERENCES
[12] A. Burr, “Channel capacity evaluation of multi-element antenna systems using a spatial channel model,” in Millenium Conference on Antennas and Propagation (AP2000), april 2000.
[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, March 1998. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Bell Labs Technical Memorandum, 1995.
[9] S. Wei, D. Goeckel, and R. Janaswami, “On the capacity of fixed length linear arrays under bandlimited correlated fading,” in CISS, Princeton, April 2002.
[11] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H. Reed, “Overview of spatial channel models for antenna array communication systems,” IEEE Personal Communications, pp. 10–22, Feb. 1998.
[13] R. M. Gray, “On the asymptotic eigenvalue distribution of toeplitz matrices,” IEEE Trans. Inf. Theory, vol. 18, pp. 725–730, Nov. 1972.
