Mimo Channel Capacity

Progress In Electromagnetics Research Symposium, Hangzhou, China, March 24-28, 2008

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MIMO Channel Model and Its Impact on the Channel Capacity Jun Wang, Quan Zhou, Wei Ma, and Lede Qiu National Key Laboratory of Space Microwave Technology Xi’an Institute of Space Radio Technology, Xi’an 710100, China

Abstract— In order to study the impacts of array configuration and channel model parameters including antenna spacing and scattering angle on the channel capacity of an S-MIMO (Satellite Multiple Input Multiple Output) system, a novel method is proposed to explore the channel capacity under flat fading. A novel channel model is constructed based on the fading correlation matrix which depends on the array configuration. And then using the properties of Wishart distribution, closed-form expressions for the upper and lower bounds on the ergodic capacity of N by M MIMO system are presented in detail. The novel method also could be generalized to MIMO-OFDM systems with any number of transmit and receive antennas. Computer simulation results show that for small spacing the UCA yields higher channel capacity than ULA. The channel capacity is maximized when the antenna spacing increases to a certain point, and further more, the larger the scattering angle, the quicker the channel capacity converges to its maximum. And at high SNR, the upper and lower bounds on the ergodic capacity are close to its true value. 1. INTRODUCTION

Multiple Input Multiple Output (MIMO) using multiple antennae simultaneously at both end in a wireless system has been shown to significantly improve spectrum efficiency of communication systems over traditional Single Input Single Output (SISO) systems. Theoretical work of [1] and [2] proved that if the fades between different pairs of transmit-receive antenna elements are independent and identical Rayleigh and receiver knows the channel perfectly, the channel capacity increases linearly with the minimum of the number of transmit and receive antennas. In satellite systems, how to use the limited resource to increase the system performance is the important problem. Many authors [3, 4] proposed that future land mobile satellite systems can take advantage of MIMO technology to boost data-rates in resource limited allocated spectrum. Although the land mobile satellite channel at low elevation is harsh, a significant increase in capacity can be achieved using space time coding from a single satellite MIMO system. In real propagation environments, correlation exists between the fades of deferent antenna elements due to the antenna spacing and the surroundings around antenna arrays are insufficient. This can cause the channel capacity of MIMO systems reduce significantly. In [5] the statistical model of the fading correlation is constructed and the asymptotic channel capacity is investigated when the number of antennas is infinite using random matrices theory. According to [6], the closed-form expression of MIMO systems can be derived using the probability density function of eigenvalues of Wishart matrix. But the computational complexity is very high. In [6] and [7] the upper and lower bounds on the ergodic capacity of MIMO system are presented, and the results shown that the upper and lower bounds are close to its real value. In this paper, we develop a method based on the receive UCA to investigate the channel capacity and analyze the impacts of antenna spacing and scattering angel on the capacity of S-MIMO systems. In addition, we introduce the properties of Wishart distribution to derive the upper and lower bounds in detail, which can reduce the computational complexity significantly. We also investigate the impacts of array configuration and channel model parameters including antenna spacing and scattering angle on the channel capacity of S-MIMO systems. 2. THE CHANNEL MODEL AND CAPACITY OF S-MIMO SYSTEMS

In the following, we assume: (1) M and N denote the number of transmit and receive antennae, respectively. (2) a single user in the system and employ the narrowband Rayleigh MIMO channel. (3) uniform circular antenna or uniform linear antenna is used at the receiver. (4) H is the matrix channel impulse, n is the M × 1 zero mean and unit variance additive white Gaussian noise vector. (5) AOA distribution is uniform and the central AOA is denoted as Θ, and the angle of arrival is uniform distribution in [Θ − ∆, Θ + ∆], the radius of UCA is r, the angle that each element location makes with the horizontal axis is ϕi , the angle spread is ∆.

PIERS Proceedings, Hangzhou, China, March 24-28, 2008

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The spatial fading correlation between the mth and nth antenna element is defined as µ µ ¶· µ ¶ Z ∆ 1 j4πr ϕn − ϕm ϕn + ϕm R(m, n) = exp sin sin z cos Θ − 2∆ −∆ λ 2 2 µ ¶¸¶ ϕn + ϕm + cos z sin Θ − dz 2 For the uplink, ∆ is usually small. We can approximate cos z ≈ 1 and sin z ≈ z which gives R(m, n) = exp (j4πrα/λ) sin c (4rβ∆/λ)

(1)

where α = sin ((ϕn − ϕm )/2) sin (Θ − (ϕn + ϕm )/2), β = sin ((ϕn − ϕm )/2) cos (Θ − (ϕn + ϕm )/2). When ULA is used at the receiver, the spatial correlation between two elements a distance d apart can be determined as ¶ µ ¶ µ ¶ µ Z Θ+∆ 1 2πd 2πd 2πd R(d) = sin(θ) dθ = exp j sin(Θ) sin c cos(Θ)∆ (2) exp j 2∆ Θ−∆ λ λ λ The spatial fading correlation matrix R is constructed based on (1) or (2). And then the channel matrix can be written as H = R1/2 U

(3)

We assume the channel is perfectly known at the receiver but unknown at the transmitter. The total power of the signal is constrained to P , regardless of the number of transmit antennas. The channel capacity is given by C = log2 (det(IM + ρHHT )) (4) where IM is M × M identity matrix, T is denote conjugate transpose, ρ = P/N . Using (3) and factorize R using the singular value decomposition, we get ³ ³ ´´ 1/2 1/2 C = log2 det I + ρΛR WΛR

(5)

For investigating the mean capacity of MIMO system, it is important to analyze the statistical 1/2 1/2 property of the eigenvalues of V = ΛR WΛR . It is known to all that W∼WM (N, I) and using the properties of Wishart distribution, we get V∼WM (N, ΛR ). There exists the closedform expression for the joint probability density of the eigenvalues of V, but it is very difficult to calculate the marginal pdf of the eigenvalues to derive the closed-form expression of the channel capacity. So we will give the closed-form expressions for the upper and lower bounds on the mean capacity and get some useful conclusions. Noting logx is concave function in x, so E log x ≤ log Ex. Using this property, we can get E(C) ≤ log2 E (det (I + ρV)) ≈ M log2 (ρ) + log2 (E (det V))

(6)

Wishart distribution has the property: if V is WM (N, ΛR ), then det (V)/det (ΛR ) has the M Q χ2N −k+1 , where the χ2N −k+1 for k = 1, . . . , M , denote independent χ2 same distribution as k=1

random variables. Using this property, we can get µ M

¶Á µ ¶ N N +1 ΓM 2 2 !,

E (det V) = det ΛR 2 ΓM ÃM X E (log2 (det V)) = (ln (λi ) + ψ (N − i + 1))

ln 2

(7) (8)

i=1

Using (7), then (6) can be written as ¶ X µ µ ¶Á µ ¶¶ µ µ ¶ M 1 1 P +1 + log2 (λi ) + log2 ΓM N +1 ΓM N E(C) ≤ M log2 N 2 2 i=1

(9)

Progress In Electromagnetics Research Symposium, Hangzhou, China, March 24-28, 2008

where ΓM (a) = π M (M −1)/2

M Q

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Γ (a − i + 1).

i=1

For a Hermitian matrix A, it is true that: det (I + A) ≥ det (A). From this property, we get E (C) ≥ E (log2 (det (ρV))) using (8), we can lower bound E(C) as !!, Ã ÃM X E (C) ≥ M ln (ρ) + (ln (λi ) + ψ (N − i + 1)) ln 2

(10)

(11)

i=1

where ψ(x) = Γ0 (x)/Γ(x). 3. SIMULATION

Simulation considers the narrowband single user S-MIMO system with M = 4 receive and N = 6 transmit antennae. The central angle of arrival is Θ = π/3, the radius of UCA is r, the receive SNR is 20 dB.

Figure 1: Mean capacity of MIMO for various scattering angels at Θ = π/3.

Figure 2: Mean capacity of MIMO for various scattering angels at Θ = 0.

Figure 3: Channel capacity and its upper and lower bounds.

Figure 1 and Fig. 2 show the mean capacity of spatially correlated Rayleigh-fading MIMO channels for different scattering angels. Note that for a four-element linear array, the√element spacing is d = (2/3)r, while for the UCA, the minimum spacing between elements is 2r. As shown, the channel capacity is maximized when the antenna spacing increases to a certain point,

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PIERS Proceedings, Hangzhou, China, March 24-28, 2008

and further more, the larger the scattering angle is, the quicker the channel capacity converges to its maximum is. And we also see that for small element spacing, when the scattering angle is determined, the UCA outperforms the ULA. Figure 3 shows the mean capacity and its upper and lower bounds of correlated Rayleigh-fading MIMO channels. We can see that, at high SNR, the upper and lower bounds on the mean capacity are close to its actual value. 4. CONCLUSIONS

We have constructed the statistical channel model based on fading correlation and developed a method to investigate the capacity of S-MIMO systems. The method using the properties of Wishart distribution to derive the closed-form expressions for the upper and lower bounds on the ergodic capacity based on receive UCA. And we also investigate the impacts of array configuration and channel model parameters including antenna spacing and scattering angle on the channel capacity of a MIMO systems. Simulation results show that for small element spacing, when the scattering angle is determined, the UCA outperforms the ULA. At high SNR, the upper and lower bounds on the mean capacity are close to its actual value. ACKNOWLEDGMENT

This work was supported by the National Key Laboratory Foundation of China under Grant 9140C5303010703. REFERENCES

1. Foschini, G. J., “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Technical Journal, Vol. 1, No. 2, 41–59, 1996. 2. Foschini, G. J. and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, Vol. 6, No. 3, 311–335, 1998. 3. King, P. R. and S. Stavrou, “Land mobile-satellite MIMO capacity predictions,” IEE Electronics Letters, Vol. 41, No. 13, 749–750, 2005. 4. King, P. R. and S. Stavrou, “Capacity improvement for a land mobile single satellite MIMO system,” IEEE Antennas and Wireless Propagation Letters, Vol. 5, No. 2, 98–100, 2006. 5. Chuah, C. N., D. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Trans. on Information Theory, Vol. 48, No. 3, 637–651, 2002. 6. Shiu, D.-S., G. J. Foschini, M. J. Gans, et al., “Fading correlation and its effect on the capacity of multi-element antenna systems,” IEEE Trans. on Communications, Vol. 48, No. 3, 502–512, 2000. 7. Grant, A., “Rayleigh fading multiple-antenna channels,” EURASIP Journal on Applied Signal Processing (Special Issue on Space-Time Coding (Part I)), Vol. 2002, No. 3, 316–329, 2002. 8. Muirhead, R. J., Aspects of Multivariate Statistical Theory, John Wiley, New York, 1982.