Channel Capacity Estimation for MIMO Systems with Correlated Noise

Transcription

1 Channel Capacity Estimation for MIMO Systems with Correlated Noise Snezana Krusevac RSISE, The Australian National University National ICT Australia ACT 0200, Australia Predrag Rapajic University of Greenwich Chatham Maritime United Kingdom Rodney A. Kennedy RSISE, The Australian National University National ICT Australia ACT 0200, Australia Abstract In this paper we determine the MIMO channel capacity in the presence of correlated noise. We employ a normalized noise correlation matrix in the channel capacity formula, in order to identify the noise correlation contribution to the MIMO subchannel decorrelation. Then, we analyze the thermal noise correlation due to mutual coupling. We calculate thermal noise correlation matrix in the multi-antenna system with closely spaced antennae by applying the Nyquist s thermal noise theorem. Simulation results shows that mean and outage MIMO channel capacity is underestimated if the noise correlation due to mutual coupling is not accounted for. We present results for cases when the transmitter does and does not know the channel realization. I. INTRODUCTION MULTIPLE-INPUT multiple output (MIMO) wireless systems, characterized by multiple antennas at the transmitter and receiver, have demonstrated the potential for increased capacity by exploiting the spatial properties of the multi-path channel [1]. If the channel matrix coefficients are i.i.d. (independent identically distributed) complex Gaussian variables, a linear increase in capacity with the number of antenna is possible. Such mutual independence of channel coefficients could be achieved by wide inter-element spacing in the multiantenna system. However, for typical small subscriber units this limit is not practically meaningful. Close antenna spacing results in antenna mutual coupling which significantly affects the performance of wireless communication systems [2]. The general approach in evaluating the impact of antenna mutual coupling on MIMO system performance is to examine how it affects the correlation of channel coefficients and consequently the channel capacity [3]. In [4], the MIMO channel capacity estimation is based on the rigorous networktheory framework analyzing the transmit power constraint due to mutual coupling and optimizing the receiving loads. Additionally, the channel and receiver noise model was analyzed, but the noise is taken to be i.i.d. complex Gaussian. While above studies present important contributions concerning the channel capacity estimation of MIMO systems with coupled antennae, they presume i.i.d. complex Gaussian noise at the receiver front-end. The assumption holds true for wide antenna spacing - uncoupled antennae. However for closely spaced antenna elements, the mutual coupling correlates thermal noise in the multi-antenna systems. The thermal noise correlation for two closely spaced antenna is discussed in [5]. Then, the thermal noise correlation due to mutual coupling in the multi-antenna system with a small inter-element spacing is investigated in [6]. In this paper, we analyze the capacity performance of MIMO systems in the presence of correlated noise. 1. We develop a MIMO channel capacity formula incorporating correlated noise. We use the normalized noise correlation matrix approach to estimate the amplification of each subchannel due to correlated noise. We consider the capacity performance in two scenarios: 1) the transmitter knows the channel, so that optimal transmit power allocation (also know as water filling) can be used; 2) transmitter does not know the channel, so the equal power is allocated to each of the transmit antenna elements. In both cases it is assumed that the receiver knows the channel perfectly. 2. We analyze the thermal noise correlation due to the mutual coupling effect. We derive the thermal noise correlation matrix in the multi-antenna system for, n R 2 antenna elements by using Nyquist s thermal noise theorem [7]. The derivation represents the further generalization of the work presented in [8] for two-antenna array. 3. Our simulation results show that ergodic and outage channel capacity is underestimated, if the noise correlation, due to the mutual coupling effect, is not accounted for. II. MIMO SYSTEM MODEL For a MIMO system with transmit and n R receive antennas, the transmitted and received signals are related by: y = H x + n (1) where x and y are respectively 1 transmitted and n R 1 received signal vector, H is n R channel matrix and n is n R 1 noise vector. A. Mutual Information with Equal-Power Allocation In this subsection, we analyze the channel capacity of MIMO system in the presence of correlated noise when transmitter does not know the channel characteristics. In this case, one way to distribute the transmit power is to share it equally on all transmit antenna elements. Additionally, IEEE Globecom /05/$ IEEE

2 we assume that transmitted signals are independent, their covariance matrix is then: Q = P T X I nt where P T X is the total transmitted power and I nt is a identity matrix. Defining N c as noise correlation matrix of the noise samples at the receiver and H c as the channel matrix that includes spatial (scattering [9] and electromagnetic [3]) correlation, the mutual information for n R MIMO system based on [1] becomes [10] I n = log 2 I nr + N 1 c H c QH H c (2) In order to estimate the noise correlation effect on the MIMO channel capacity, we multiply noise correlation matrix N c by identity matrix I nr = NN 1 in formula (2). Then, the mutual information is I n = log 2 I nr + ρ N cn H c H H c (3) where N = NI nr is the uncorrelated (AWGN) noise matrix, N is the noise power of isolated antenna element. Also ρ is signal-to-noise ratio of one subchannel ρ = P T X /N and N cn is the normalized noise correlation matrix defined as N cn = N 1 c N Applying a singular value decomposition to H c and N cn, the mutual information becomes I n = log 2 = n (1 + ρ λ i ν i ) (log 2 (1 + ρ λ i ν i )) where λ i is eigenvalue of matrix H c H H c and ν i is eigenvalue of matrix N cn and n = min(, n R ). The eigenvalues ν i of the normalized correlated noise matrix N cn represent the noise correlation contribution to decorrelation of MIMO subchannels. Furthermore, eigenvalues ν i of matrix N cn = N 1 c N, are given by ν i = N N c (i) (N 1 c and N are diagonal matrices, and inverse of diagonal matrix is diagonal matrix of its inverse elements A = diag(a 1,..., a n ) A 1 = diag(a 1 1,..., a 1 n )) In [6], it is shown that correlated noise power N c (i) of i th antenna element (20) of the multi-antenna system is lower then corresponding uncorrelated noise power N(i), when the mutual coupling is not considered. Thus, it means that (4) ν i 1 (5) Now based on (4) and (5), one can conclude that as eigenvalues of channel matrix represent virtual channel gains, eigenvalues of the normalized noise correlation matrix appear as increasing factors of channel gains due to the noise correlations. In fact, they appear as decreasing factors of subchannel correlations in the MIMO system, increasing the MIMO channel capacity. B. Capacity with Water-filling Power Allocation In this subsection, we derive the MIMO capacity C n assuming the transmitter has perfect knowledge about the channel. With this knowledge of channel, the total transmit power can be allocated in the most efficient way over the different transmitters to achieve the highest possible bit rate. Based on the system model and definitions (2), the MIMO capacity with optimal power allocation is C n = max log 2 I nr + N 1 c H c QH H c (6) Q where Q is the n n covariance matrix of x and Q must satisfy the average power constraint. tr(q) = E[ x i 2 ] P T X (7) The optimal solution is C n = log 2 ((λ i ν i )µ) + (8) where n = min(, n R ) and µ satisfies (µ 1 λ i ν i ) + = ρ (9) and λ i and ν i are the eigenvalues of H c Hc H and N cn, respectively. The optimal solution given in (8) and (9) are analogues to the optimal power allocation calculated through the waterfilling algorithm for parallel Gaussian channels [11]. Intuitively, (8) and (9) suggest that the original MIMO channel can be decomposed into n parallel independent subchannels, and we allocate more power to the subchannels with higher SNR ρλ i. Here, µ is the water level that marks the height of the power that is poured into the water vessel formed by the function {1/λ i ν i, i = 1, 2,..., n}. Each of these subchannels contributes to the total capacity through log 2 ((λ i ν i )µ) +. If λ i ν i µ 1, we say that this subchannel provides an effective mode of transmission and is called a strong eigenmode. Now, the eigenvalues of normalized noise correlation matrix ν i represent the additional subchannel capacity gain due to mutual coupling effect on thermal noise. In that way, the noise correlation due to the mutual coupling decrease the correlation between the subchannels, improving the MIMO channel capacity. III. NOISE CORRELATION MATRIX To explore the MIMO channel capacity in the presence of correlated noise, we ought to calculate the noise correlation matrix N c which exists due to the mutual coupling. In this section, we apply the generalized Nyquist s thermal noise theorem [7] to derive the noise correlation matrix. The generalized Nyquist s thermal noise theorem determines correlated thermal noise power in the multi-antenna systems with closely spaced antennae. IEEE Globecom /05/$ IEEE

3 J1 Then, the spectral density of thermal noise voltage of the multi-antenna system is: V1 j1 YL1 jl1 V = (Y + Y L ) 1 J = (Y + Y L ) 1 (j + j L ) (12) y11 y12. ynr,1 y21 y22. y2,nr J2 V2 j2 jl2 YL2 The power spectral density matrix of thermal noise N c (f) is diagonal matrix compose of diagonal elements of matrix N = 1 2 (Y L + Y L ) VVH, i.e.: ynr,1 ynr,2. ynr,nr N c (f) = dg(n) (13) = dg( 1 2 (Y L + Y L) VV H ) (14) JnR VnR jnr jlnr YLnR where dg( ) denotes the diagonal operator. In order to calculate (13), we derive the spectral squared noise voltage matrix VV H, in the following form Fig. 1. Nodal network repersention for multi-antenna system VV H = (Y + Y L ) 1 JJ H ((Y + Y L ) 1 ) H (15) Based on (10), the following relations are valid: A. Nyquist s Thermal Noise Theorem for Coupled Antennae The generalized Nyquist s thermal noise theorem [7] allows us to determine thermal noise power of coupled antennae in the multi-antenna system. The magnitude of correlation between two closely spaced antennae with isolated receivers is given in [5]. The theorem states that for passive network in thermal equilibrium it is possible to represent the complete thermal-noise behavior by applying Nyquist s theorem independently to each element of the network. In the case of the multi-antenna system these elements are self-impedances and mutual-impedances. The general, even nonreciprocal, network with a system of internal thermal generators all at absolute temperature T is equivalent to the source-free network together with a system of noise current generators I r and I s with infinite internal impedance [5]. Noise currents are correlated and their crosscorrelation is given by I s I r df = 2kT (Y sr + Y sr) df (10) where Y rs and Y sr are the mutual admittances and k is Boltzmann s constant and T is absolute temperature. Correlation is zero when the mutual coupling is purely reactive. Alternatively the internal noise sources can be represented by a system of nodal voltage generators. B. Noise correlation matrix The derivation of the noise correlation matrix is based on the nodal network for the multi-antenna system with n R antenna elements, as shown in Fig. 1. The spectral density of total thermal noise current for the multi-antenna system (Fig. 1) is given by: J = j + j L = (Y + Y L ) V (11) where J column vector of spectral density of total noise current and V column vector of spectral density of noise voltage. Additionally, Y is mutual admittance matrix of the multi-antenna system, Y L is diagonal matrix of receiver load admittances. 1. j j j k df = 2kT (y jk + y jk); 2. j Lj j k = 0; 3. j Lj j Lk = 0, j k; where (.) denotes complex conjugate. Now, the spectral squared current matrix of thermal noise becomes: JJ H = (j + j L ) (j + j L ) H (16) = 2kT (Y + Y H + (Y L + Y H L )) (17) Substituting (16) in (15), the spectral noise voltage matrix of thermal noise can be expressed as: VV H = 2kT (Y A ) 1 [Y A + (Y A ) ]((Y A ) 1 ) H (18) where Y A = Y + Y L Applying (18) in (13), one can finally write the power spectral density matrix of thermal noise for the multi-antenna system with closely spaced antenna as: N c (f) = 2kT dg[(y L + Y L)(Y A ) 1 [Y A + (Y A ) ]((Y A ) 1 ) H ] (19) For example, element N c (f)(1, 1) of the power spectral density matrix of thermal noise N c (f) for a two-antenna array is given: N c (f)(1, 1) = 2kT (Y L1 + Y L1 ) 2 D D ((y 22 + Y L2 )(y 22 + Y L2)((Y L1 + Y L1) + (y 11 + y 11)) y 21 (y 22 + Y L2)(y 12 + y 12) y 21(y 22 + Y L2 )(y 12 + y 12) +y 21 y 21((Y L2 + Y L2) + (y 22 + y 22)) (20) where y 11 and y 22 are self-admittance of antenna 1 and 2, and y 12 and y 21 are mutual admittances. The mutual admittance y 12 quantifies the contribution from the antenna 2 to total noise power of antenna 1. Therefore, the mutual admittance y 12 is a measure of the noise correlation IEEE Globecom /05/$ IEEE

4 level. The previous simple example of a two-antenna array is used to provide a better understanding of noise correlation effect. Based on (19), the noise correlation matrix of the multiantenna system with closely spaced antenna is given by: N c = N c (f)df (21) B where B is the frequency band of receiver bandpass filter. For narrow-band system, the noise correlation matrix (21) becomes: N c = 4kT Bdg[(Y L + Y L)(Y A ) 1 [Y A + (Y A ) ]((Y A ) 1 ) H ] (22) In (22), we assume that the mutual admittances Y and load admittances matrix Y L, are constant for the narrow frequency band B. IV. SIMULATION RESULTS In this section, we use simulations to demonstrate the effect of noise correlation upon the ergodic and outage capacity of MIMO systems. This effect exists due to the mutual coupling. Here, we consider the uniform linear arrays (ULA) with two and three half-wave dipoles. We use SONNET [12] software to calculate mutual admittances and impedances for different antenna spacings. Results are based on channel realizations. The spatial (scattering and electromagnetic) correlation is considered only at the receiver side, assuming wide antenna separation at the transmitter side. The spatial (scattering) correlation effect is included into channel capacity estimation through the expression H = R 1/2 rx H 0 [9] where R 1/2 rx [9] is the Hermitian square root of the n R n R receive antenna correlation matrix. Additionally, H 0 is an n R MIMO channel matrix such as one used in [1], whose elements are independent, unit variance random complex coefficients. The mutual coupling on signal is included into channel capacity estimation through a coupling matrix M rx and expression H c = M rx H [3]. The received coupling matrix is given by M rx = (Z L + Z A )(Z + Z T I nr ) 1 [3], based on circuit theory [2]. Here, Z A is the antenna impedance, Z L is the load impedance of each element and Z is the mutual impedance matrix. Fig. 2 depicts the 3 3 MIMO channel capacity as a function of antenna element spacing - the metric of correlation due to mutual coupling effect. We present how MIMO channel capacity evolves as the noise correlation level varies, by changing the antenna element spacings from 0 to 0.5λ. We studied two parameters of capacity distribution, the mutual information I n (transmitter does not have information about the channel - equal-power allocation is applied - EQ) and the capacity C n, see (6) (transmitter knows channel - water-filling power allocation - WF can be applied). In order to explore the noise correlation effect due to the mutual coupling on channel capacity, we estimate the capacity under three assumptions: 1) the mutual coupling affects both, signal and thermal noise Mutual Information and Capacity [bps/hz] nmc EQ mcs EQ mcstn EQ nmc WF mcs WF mcstn WF Antenna spacings (wavelengths) d/λ Fig. 2. Mean (ergodic) capacity with water-filling C n and equal-power allocation I n versus antenna element spacings, for 3x3 MIMO Systems Outage capacity C nmc 2 x 2 mcs 2 x 2 mcstn 2 x 2 nmc 3 x 3 mcs 3 x 3 mcstn 3 x Antenna spacings/wavelengths d/λ Fig. 3. C % Outage capacity versus antenna element spacings for 3x3 MIMO Systems (mcstn), 2) the mutual coupling effect is considered only for signal (mcs) and finally, 3) the mutual coupling effect is neglected either on signal or on noise (nmc). First, simulation results confirm the signal correlations due to mutual coupling effect increase the MIMO channel capacity [3] (mcs) in comparison with capacity value obtained without considering the mutual coupling effect (nmc). Also, we confirm that calculated mean capacity with water-filling power allocation is higher then mutual information based on the equal power allocation scheme in all three cases (nmc, mcs and mcstn). From Fig. 2, one can conclude that the correlated noise due to the mutual coupling acts as an increasing factor in the MIMO channel capacity estimation. We base our conclusion on both parameters of capacity distribution, the mutual information and water-filling capacity. We confirm our theoretical IEEE Globecom /05/$ IEEE

5 Probability [Capacity < apscissa] nmc mcs mcstn d=λ/6 2x2 MIMO d=λ/3 d=λ/6 3x3 MIMO d=λ/ Capacity [bps/hz] Fig. 4. Cumulative distribution function (cdf) of channel capacity for 2x2 and 3x3 MIMO systems for antenna spacing d = λ/6 and d = λ/3 analysis that correlated noise increases the eigenvalues of channel matrix, improving the strength of each subchannel. Therefore, the channel capacity calculated under assumption that mutual coupling correlates both, signal and noise, (mcstn) represents the better estimation. In fact, the more accurate estimation of channel capacity is obtained when we include all factors, scattering and electromagnetic signal correlation and electromagnetic noise correlation. Fig.3 depicts the 1% outage capacity C 0.01 in case when transmitter knows the channel characteristics. The outage capacity C 0.01 means that there is a probability of 0.01 that the capacity is less then what is displayed on the figure. We compute the outage capacity with no mutual coupling (nmc), mutual coupling only for signal (mcs) and mutual coupling on both, signal and thermal noise (mcstn). From Fig.3, we confirm that MIMO systems perform better in terms of capacity when the thermal noise correlation (due to the mutual coupling effect) is considered. Fig. 4 shows the cdf for three cases: with mutual coupling on both signal and thermal noise, with mutual coupling on signal and without mutual coupling either on signal or thermal noise. We present cdf:s for two antenna separation at receiver side d = λ/6 and d = λ/3 for 2x2 and 3x3 MIMO systems. If one ignores the effect of mutual coupling on both, signal and noise, it can be seen that one significantly underestimates the channel capacity. The channel capacity underestimation increases with decreasing the inter-element separation or with higher order of correlation both, noise and signal. Furthermore, the underestimation is worse in the multi-antenna system with the greater number of antenna elements. The underestimation becomes more noticeable if the noise correlation due to the mutual coupling is omitted. In Fig. 3 and 4, we also study how ergodic MIMO channel capacity evolve as the number of the transmit and receive antennae increases. We observe that the capacity underestimation due to the noise correlation increases as the number of the transmit and receive antennae increases. Figs. 2 and 3 verify that the mutual coupling which correlates both signal and thermal noise could be beneficial factor in order to estimate maximal possible information rate for the MIMO systems. The statement is valid for the case when the small inter-element separation is enforced due to limited size of subscriber unit of the multi-antenna communication systems. V. CONCLUSION In this paper, we have analyzed the channel capacity of MIMO systems in the presence of correlated noise. We presented the thermal noise correlation matrix of the multiantenna system due to mutual coupling effects. We show that the ergodic and outage channel capacity of MIMO systems is underestimated if the noise correlation due to mutual coupling effect on thermal noise is neglected. Based on the presented results, one can conclude that the channel capacity of MIMO system with closely spaced antennae can be estimated accurately only by considering both signal correlation and noise correlation. ACKNOWLEDGMENTS National ICT Australia is funded through the Australian Governments Backing Australia s Ability Initiative, in part through the Australian Research Council. REFERENCES [1] G. Foschini, On limits of wireless communication in fading environment when using multiple antennas, Wireless Personal Communication, no. 6, pp , Mar [2] I.J.Gupta and A.K.Ksienski, Effect of the mutual coupling on the performance of the adaptive arrays, IEEE Trans. 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