Abstract
Novel closed-form expressions are derived for the performance analysis of a multiple-input multiple-output (MIMO) system in Rayleigh fading using transmit antenna selection (TAS) at the transmitter and maximal ratio combining (MRC) at the receiver. Receive antennas are assumed to be arbitrarily correlated, as no restriction is imposed on the correlation matrix. General exact and asymptotic expressions to evaluate the bit error rate (BER) of different modulation schemes are presented for uncoded transmission, and a closed-form expression is presented for the channel capacity. It is demonstrated that channel capacity may improve due to correlation at the receive antennas if the transmit array size is large enough as a result of a higher signal variability and the antenna selection performed at the transmitter. Monte Carlo simulations have been carried out to validate the analysis, showing an excellent agreement with the theoretical results.
1. Introduction
Multiple-input multiple-output (MIMO) systems use multiple antennas at both the transmit and receive ends to improve the performance of wireless links. Among the different possible MIMO schemes, TAS/MRC, employing transmit antenna selection at the transmitter and maximal ratio combining at the receiver, has been proposed as a means to provide full diversity gain and reduce implementation complexity [1].
Several works have analyzed the performance of TAS/MRC systems in different fading conditions in the last decade; however, most previous works assume uncorrelated antennas at both the transmit and receive ends [1–5]. In practice, space correlation among antennas may occur, which can be significant in portable terminals due to space limitations. Few works have studied the performance of TAS/MRC systems in correlated fading. An error rate analysis for a semicorrelated scenario, assuming receive antenna correlation, was presented in [6] for Rayleigh fading; however, the analysis was simplified by assuming that all the eigenvalues of the correlation matrix are different, which may not be realistic in practice. In [7], the outage probability was evaluated for Nakagami fading, which includes Rayleigh fading as a special case, and the same system model was used in [8] to obtain an exact expression for the bit error rate (BER) of binary phase-shift keying (BPSK), but again considering that all eigenvalues are distinct. A TAS/MRC system with arbitrarily correlated receive antennas is considered in [9], but the analysis is approximated by considering that the sum of gamma variates with arbitrary parameters is a gamma variate. The error made with this approximation was extensively studied in [10], under the assumptions of constant or circular correlation, where it was shown that the approximated pdf tends to deviate in the lower tail from the exact pdf, and this lower tail is critical for error rate calculations. Lee’s spatiotemporal model was considered in TAS/MRC in [11], but only a uniform linear receive antenna array is studied. The analysis of ergodic capacity in TAS/MRC systems seems to have received less attention. Recently, ergodic capacity was studied in [12] for TAS/MRC in the context of spectrum sharing systems. However, the study was restricted to uncorrelated antennas.
In this work, we study a TAS/MRC system in semicorrelated Rayleigh fading channels where transmit antennas are assumed to be independent and receive antennas are considered to be arbitrarily correlated. No restriction is imposed on the correlation matrix of the receive array, as arbitrary correlation is considered between any pair of receive antennas, thereby allowing the multiplicity of the receive correlation matrix eigenvalues to be arbitrary. We derive closed-form expressions for the pdf and CDF of the signal-to-noise ratio (SNR) as well as for the BER of different modulation schemes for uncoded transmission. In order to provide insight into the impact of the different system parameters on performance, a simple and compact asymptotic expression of the BER for the high SNR regime is also derived and it is shown that correlation results in an increased BER for high values of the SNR. We also develop an expression for the ergodic capacity and, by studying the derivative of capacity with respect to correlation, we demonstrate that the presence of correlation may increase channel capacity if the number of transmit antennas is high enough. This beneficial impact of correlation on capacity has also been observed in multiuser MIMO schemes (see [13] and references therein) where the user with the best channel gain is selected for transmission. However, results in multiuser MIMO are usually based on bounds or, as in [13], asymptotic analysis on the number of users. On the other hand, our analysis yields closed-form expressions for any number of transmit antennas, which allow calculating the number of the required transmit antennas for the correlation to be beneficial, which will be a function of the number of receive antennas. Monte Carlo simulations have been carried out to validate our analysis, showing an excellent agreement with the theoretical results.
2. System Model and Output SNR Statistics
Let us consider a wireless system with transmit and receive antennas where the signal at every receive antenna is corrupted by additive white Gaussian noise (AWGN). At the receive end, MRC is used, whereas at the transmit end the antenna that maximizes the instantaneous SNR after combining is selected for transmission. A simple diagram of the system model is shown in Figure 1. We assume independent and identically distributed (i.i.d.) slowly varying (with respect to the symbol duration) flat Rayleigh fading between any pair of transmit and receive antennas. Thus, the instantaneous SNR at receive antenna due to the signal transmitted from transmit antenna , denoted as , follows an exponential distribution with probability density function (pdf): where is the average SNR per branch.
For MRC combining, it is known that, when the transmit antenna is selected for transmission, the instantaneous output SNR per symbol is . As the receive antennas are assumed to be arbitrarily correlated, the pdf of can be obtained from its moment generating function (MGF), which is given by [14] where are the distinct eigenvalues of the matrix , is a full rank signal correlation matrix of the receive array, is the number of distinct eigenvalues, and eigenvalue is assumed to have multiplicity . The elements of the SNR correlation matrix are denoted by , which are related to the elements of by [15]
Using partial fraction expansion, (2) can be written more conveniently as where the coefficients can be written in closed form as and is the set of -tuples such that
From (4), by the inverse Laplace transform, the pdf of can be calculated as and its cumulative distribution function (CDF) is obtained by integrating (7), yielding
The unconditional instantaneous output SNR per symbol will be , and its CDF can be calculated, assuming statistical independence among transmit antennas, as . Then, which, for subsequent manipulation, can be rewritten more conveniently, using the binomial expansion theorem, as a finite summation as where and where we have defined the function
Using the multinomial theorem [16], the jth power of function can be written as where is the multinomial coefficient and is the set of tuples and where we have defined the following coefficients: For the sake of compactness, in the following we will write . Note that and because .
The pdf of the output SNR can therefore be written in compact form by differentiating (10), yielding
3. Bit Error Probability
The average BER depends on the fading distribution and the modulation technique and can be obtained by averaging the conditional error probability (CEP), that is, the error rate under AWGN, over the output SNR. Alternatively, the average error rate can be expressed in terms of the CDF of the output SNR as where is the first order derivative of the CEP.
For the sake of compactness, in order to compute the BER of the considered modulations, we will use the following expression for the CEP [17, equation (8.100)]: where is the complementary incomplete gamma function and is the gamma function. The values of the constants in (17) are for BPSK, for BFSK, for DBPSK, and for NCBFSK. Moreover, introducing an additional multiplicative constant and considering the relation between the complementary incomplete gamma function and the Gaussian -function, it is possible to use (17) to obtain a good approximation of the symbol error rates for several M-ary modulation schemes [18, Table 6.1].
Taking the derivative of the CEP, we can write
An exact expression of the average BER for the considered modulations can be obtained by introducing (10) and (18) into (16) and solving the integral. The integration is reduced to a sum of terms that match with the gamma function definition. Therefore, we obtain the average BER as Note the simplicity of (19), which is actually a finite sum of elementary functions.
Although the error rate expression (19) is exact and can be easily computed, it does not provide a straightforward insight into the impact of the different system parameters on performance. In the appendix, we show that an asymptotic, yet accurate, simple expression of the error rate for the high SNR regime, from which the effect of system parameters on performance can be established in a simple way, is given by
It can be observed in (20) that the average BER decreases inversely with the power of the average SNR per antenna, which indicates that the proposed TAS/MRC scheme achieves full diversity gain. On the other hand, it is straightforward to show that the error rate increases in the high SNR regime as antenna correlation increases. Note that, for example, for two receive antennas with SNR correlation denoted as , we have that .
4. Channel Capacity
4.1. Closed-Form Expression of Channel Capacity
The ergodic capacity of the channel , also referred to as channel capacity, is given by [18] where is the channel bandwidth, is the natural logarithm, and is the pdf of the output SNR given in (15). In this expression, an optimal channel encoding is implicitly assumed. Introducing (15) into (21) and using [19] the channel capacity can be calculated as
4.2. Impact of Receive Antenna Correlation on Capacity
When MRC is performed at the receiver and there is a single transmit antenna, it is known that correlation has a detrimental effect on capacity [20]. In this section, we demonstrate that, when multiple independent antennas are considered at the transmitter performing transmit antenna selection, correlation at the receive antennas may result in an increased capacity.
In order to simplify our analysis, let us consider the case of constant correlation, denoted as , between any pair of receive antennas. For simplicity, we define , so that any function increasing or decreasing with will have the same behavior with . For this case, the elements of matrix become and only two distinct eigenvalues are obtained [10, equation ]: Introducing (25) into (5), the coefficients can be written as
A general treatment of (23) using (26) would be too long and difficult to manage. However, the simplest case with two receive antennas () can be easily studied. Then, the different parameters used in this section become Therefore, the capacity expression (23) can be written in this case as
In order to study the effect of correlation on capacity, we calculate the derivative of capacity with respect to . Using [21, equation (6.5.25)], we can obtain where
To study the monotonicity of capacity, the capacity derivative given in (29) is represented in Figure 2 for different number of transmit antennas as a function of the correlation coefficient for an average SNR per branch of 0 and 20 dB, and, in Figure 3, the capacity derivative is represented as a function of the average SNR per branch for 2 receive antennas. It can be observed that, for only one transmit antenna (), the derivative is always negative, which means that the capacity is monotonically decreasing with correlation; that is, the higher the correlation, the lower the capacity, which is consistent with the results in [14, 20]. Figure 2 also shows that the capacity will be quickly degraded, for , as correlation increases if the average SNR per branch is high. On the other hand, for , capacity is monotonically increasing with correlation; that is, the higher the correlation, the higher the capacity. An intuitive explanation of the reason of such behavior can be found in the fact that the higher the correlation the higher the variability of the output signal received from a given transmit antenna. Therefore, as the number of transmit antennas increases, the probability that a transmit antenna with good channel conditions is selected for transmission also increases. Ultimately, if the number of transmit antennas for selection is high enough, this will lead to an increase of capacity.
5. Numerical Results
In this section, numerical results are presented for the BER as well as for the channel capacity of the analyzed TAS/MRC scheme. Monte Carlo simulations have also been carried out showing an excellent agreement with the analytical results.
Figure 4 shows the average BER for BPSK as a function of the average SNR, for a TAS/MRC system with 4 receive antennas, high and low correlations ( and , resp.), and different number of transmit antennas. The asymptotic behavior is also shown. In all cases, it can be observed that a lower correlation results in a lower BER, except for very low values of the average SNR.
Figures 5 and 6 show the ergodic capacity for 3 and 4 receive antennas and an average SNR per branch of, respectively, 0 and 20 dB as a function of the correlation coefficient , which is assumed to be the same for any pair of receive antennas. It can be observed that when there is only one transmit antenna, , the channel capacity diminishes as correlation increases, which is consistent with the results in [14, 20]. However, for , highly correlated antennas result in an increased capacity. Note that these results do not contradict the detrimental impact that the correlation has on the BER (except for very low average SNR) in uncoded communications, as channel capacity implicitly assumes an error-free communication link. Figure 7 reveals how the width of the SNR pdf becomes larger when correlation increases for a given transmit antenna (note that the abscissa is represented in dB), which indicates a higher variance. This higher variability for high correlation is responsible for the increased capacity when transmit antenna selection is performed. Note that the probability of having a higher instantaneous SNR after combining increases as correlation increases.
Figure 8 shows the ergodic capacity as a function of the average SNR per branch for different number of transmit and receive antennas considering a constant correlation or linear-array correlation. The linear-array correlation matrix has been built such that if and if . Again, an increased capacity with correlation when can be observed, which is larger if the correlation is constant (as the receive antennas suffer a stronger correlation). For low correlation (), almost no difference can be appreciated between constant and linear array. For , there is no appreciable difference in any of the depicted cases.
6. Conclusion
We have presented novel closed-form expressions for the pdf and CDF of the output SNR of TAS/MRC systems in Rayleigh fading with arbitrarily correlated receive antennas. Additionally, a novel closed-form expression has been derived for the average BER under several modulation techniques, and the corresponding asymptotic expression is also presented. Our results show that antenna correlation has always a detrimental impact on BER except for very low average SNR per receive antenna. We have also studied the impact of correlation on ergodic capacity and we have shown that correlation may have a beneficial impact on ergodic capacity of TAS/MRC systems due to the antenna selection performed at the transmitter, as a higher variability of the output signal increases the probability that a transmit antenna in good fading conditions is selected for transmission. As a consequence, the capacity will increase as receive antenna correlation increases if the number of transmit antennas is high enough.
Appendix
Derivation of (20)
We will follow the framework provided in [22], where it is shown that the asymptotic error performance depends on the behavior of the SNR’s pdf at the origin or, equivalently, on the decaying order of its MGF.
First, let us consider that a single transmit antenna is used. From (2), we can write where
The pdf of the random variable near the origin can be approximated by calculating the inverse Laplace transform of (A.1). By noticing that the single-sided Laplace transform of , with integer, is , we can write By integrating (A.3), the CDF of at the origin is obtained as
When transmit antenna selection is performed, the CDF of the output SNR at the origin can thus be written as
Note that (A.5) is actually the first nonzero term of the Maclaurin series expansion of the CDF of , that is, the term with the lower exponent in . The remaining terms become lower as increases. The asymptotic BER can thus be calculated by introducing (18) and (A.5) into (16), yielding, after solving the integral, (20).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work has been supported by European FEDER funds and the Spanish Ministry of Economy and Competitiveness (Grant TEC2009-13763-C02-01).