Research Article | Open Access
Keerti Tiwari, Davinder S. Saini, Sunil V. Bhooshan, "On the Capacity of MIMO Weibull-Gamma Fading Channels in Low SNR Regime", Journal of Electrical and Computer Engineering, vol. 2016, Article ID 9304597, 8 pages, 2016. https://doi.org/10.1155/2016/9304597
On the Capacity of MIMO Weibull-Gamma Fading Channels in Low SNR Regime
We present capacity analysis for multiple-input multiple-output (MIMO) system under a low signal-to-noise ratio (SNR) regime. We have selected a composite fading channel that considers Weibull fading for multipath and gamma fading for shadowing. We have presented the analysis in a detailed form for three different techniques, namely, spatial multiplexing (SM) with optimal detection, SM with minimum mean square error (MMSE) detection, and orthogonal space-time block codes (OSTBC). Because capacity analysis at arbitrary signal-to-noise (SNR) is stringent, a low SNR regime is considered to achieve a positive rate and wideband slope. The improvement is the result of minimizing energy per information bit . For the first time, closed-form expressions are evaluated for the capacity of MIMO systems at low SNR under WG fading channels which facilitate the performance comparison for proposed techniques.
MIMO wireless systems have been employed to overcome the disruptive effects of multipath fading and shadowing. These systems gradually became mature which enabled high reliability and increased capacity in bandwidth limited channels. The varying nature of wireless radio environment is responsible for some well-known multipath fading such as Rayleigh , Rician , Weibull , and Nakagami-m . In addition, shadowing conditions can also be formed and determined in terms of gamma and lognormal distribution . The shadowing effects resulting from the slow variations are incorporated in the rapid fluctuations arising from multipath propagation. Subsequently, several models have been proposed for analyzing the performance of the wireless system. Table 1 presents the existing models which define both the multipath fading and shadowing conditions, collectively termed as composite channel models.
Several generic distributions have been proposed using gamma distribution instead of lognormal distribution. The problem is that the lognormal distribution is not mathematically tractable . In , author has proposed a composite Weibull-gamma (WG) distribution for modeling multipath along with shadowing. In the recent work [17–21], WG fading channel is used to analyze the MIMO system performance due to its less computational complexity. Consequently, under such fading channel conditions, the ergodic capacity has been evaluated in  and the error rate performance of MIMO systems has been evaluated using efficient detection technique in .
Since many locations in the geometry lie on the edge of their cells, apart from operating at high SNR, users frequently operate at low SNR. According to , 40% of geographical area receives signal as low as 0 dB and less than 10% area has a SNR greater than 10 dB. In such conditions, the system performance evaluation at low SNR is imperative to the designers. In MIMO systems, figure of merit depends on normalized energy per information bit () instead of SNR. In low SNR regime, the channel capacity analysis in terms of per symbol gives misguiding outcomes and hence capacity is defined in terms of minimum energy per information bit . Henceforth, channel bandwidth, power, and rate of transmission have been analyzed with arbitrary number of transmitting and receiving antennas by minimizing .
In [12, 13], the ergodic capacity of spatially multiplexed (SM) and OSTBC MIMO system has been computed over fading channel. This channel model decreases energy levels to a prominent range . MIMO systems, namely, SM with optimal detector, SM with MMSE detector, and OSTBC, have been developed to evaluate performance at low SNR over Weibull fading channel ; however, the shadowing effect is not considered.
To the best of the authors’ knowledge, statistical characterization of MIMO WG fading channel is not fully investigated in existing literature. Hence, the capacity performance is measured with this generic distribution in this paper. This generic channel model is analytically better than other composite channel models which depict the linear approximation of the multipath and shadow fading conditions. and wideband slope are important parameters to convey any positive rate of data reliably and to analyze the capacity of MIMO system at low SNR. The superiority of SM against OSTBC in giving better capacity is established elsewhere. However, SM scheme is not optimum at low SNRs and, apart from eigenstatistics, it permits working with channel matrix trace which is given in . Optimal detectors offer the maximum capacity of MIMO system; however, MMSE detectors are less complex than optimal detectors and offer improved capacity performance. In addition, OSTBC is diversity-oriented technique. Thus, we have explored these three techniques under WG fading in our work. Also, we have evaluated the capacity of MIMO system at low SNR in this scenario.
This paper is organized as follows. In Section 2, we present MIMO system models used in this work. In Section 3, closed form expressions are evaluated for capacity analysis of MIMO systems in low SNR regime using three models: (a) SM with the optimal detector, (b) SM with MMSE detector, and (c) OSTBC. Section 4 concludes the paper.
2. MIMO System Models
Consider a MIMO system with transmit and receive antennas and complex channel matrix . Each channel element is independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and variance. Let denote the total transmit energy attainable at each time interval. The received signal is a complex matrix with size over successive symbol durations and is given bywhere is the receiver noise having matrix size with constituents modeled as i.i.d. random variables and is transmit signal with the size .
The effects of both large-scale fading and small-scale fading are considered to model , which is determined as WG fading channel. The channel matrix is given by . Here, is path loss exponent, is the distance between transmitter and receiver; for average fading power , ; represents i.i.d. gamma random variable with probability density function (PDF), given byThe elements of matrix follow an i.i.d Weibull PDF, given in (3). Here, phase distribution is uniform in the interval and amplitude is Weibull distributed for each entry of :In (3), represents scaling parameter; and are gamma and Weibull fading parameters, respectively. It has been reported in  that a conditional PDF called the WG composite PDF is determined by combining (2) and (3). The WG distribution is approximated to Weibull distribution for , Rayleigh distribution for = 2, , and additive white Gaussian noise (AWGN) channel for , . It follows or Rayleigh-lognormal distribution for = 2 using [27, Equation (9.34/3)].
It is desirable to estimate and to compute , as both the Weibull and gamma random variables are independent of each other. Here, is Hermitian transposition and is the summation of diagonal elements of matrix or matrix trace. First and second moments of gamma random variable are computed as Similarly, the th moment is also calculated for Weibull random variables  and given bywhere , , is Frobenius norm of matrix, and denotes gamma function.
The proposed three configurations are as follows.
2.1. Optimal Detection for SM MIMO Systems
When all data vectors are identical, optimal detectors are used to minimize the probability of error. Although the implementation complexity is very high, power distribution is uniform across the transmit antennas with an average SNR . The SNR and ergodic capacity  are expressed, respectively, as
2.2. MMSE Signal Detection for SM MIMO Systems
Optimal detectors are complex in nature which makes them unemployable in cost-effective communication systems. Therefore, linear detectors like MMSE detectors are designed to reduce computational complexity . Thus, at the th receiver output, the postprocessing SNR when is represented aswhere returns the th diagonal element of a matrix. The achievable sum rate can be determined by assuming the independent decoding at the receiver side, which is given by
2.3. OSTBC MIMO Systems
OSTBC scheme is preferred due to its simplicity and reliability. This scheme is used to achieve the maximum diversity order of and is computationally efficient for per symbol detection. MIMO channel can be converted into identical scalar channel by taking the response similar to that of Frobenius norm of channel matrix . For the OSTBC MIMO systems, the SNR and Shannon’s capacity with rate can be expressed, respectively, as
3. Low SNR Analysis
According to , minimization of governs a trade-off between bandwidth and power of the communication channels in wideband regime which is desirable for efficient signal communication. Also, is preferred over per symbol SNR in the measurement of MIMO system performance precisely at low SNR over distinct fading channels. Therefore, the capacity is characterized in  aswhere the parameters and prompt the low SNR nature desired for efficient transmission of positive rate and wideband slope. First-order derivative and second-order derivative of ergodic capacity are derived from (7) at . They are used to determine the following two figures of merits:
Theorem 1. The following properties are represented by MIMO systems over i.i.d. WG fading channels in low SNR regime:
Proof. Refer to the Appendix.
Using (12), we evaluated the capacity performance of SM MIMO with the optimal detector and SM MIMO with MMSE detector and OSTBC MIMO systems at low SNR.
Proposition 2. Respective and for SM MIMO systems with optimal detectors using antennas are represented aswhere
Proof. In , the matrices for low SNR are rearranged to formSubstituting (14)-(15) into (20), we get (17) and (18) using simple mathematical formulation. Equation (17) is the increasing function of and the decreasing function of As is a function of in (19), increases with the increase in . In (17), do not depend on and . When extra receive antennas get additional power, decreases monotonically with the increase in [12, 13]. Equation (17) remains unchanged with the increase in ; however, capacity increases due to higher value of . The wideband slope is lower bounded and upper bounded () as Nevertheless, is reduced by increasing because a higher number of receive antennas need more power as mentioned in [12, 23]. The parameters chosen for -fading and Rayleigh fading are and , respectively. Equations (18)-(19) are simplified to form Rayleigh fading as
In Figure 1, the capacity results are evaluated at In this case, the effects of low and high multipath fading are considered for and , respectively, in the presence of light shadowing () and heavy shadowing (). In light shadowing environment, that is, for , Rayleigh fading is observed. In addition, the capacity for the special cases (i.e., Rayleigh and -fading) of GK fading in  approaches both the special cases of WG fading. Thus, it is observed that WG is an alternative to GK fading channel. The range of low SNR takes numerically negative values from −8 dB to −1 dB. Simulation results show that capacity increases in low multipath and light shadowing environment. As for light shadowing (when is large), the impact of shadowing on wideband slope is negligible.
The generation of WG MIMO fading channels occurs as the product of a gamma random variable and with i.i.d. Weibull entries. Moreover, the simulation results of capacity at low SNR for SM MIMO systems with optimal detectors follow analytical approximations of Proposition 2.
Proposition 3. and for SM MIMO systems with MMSE detectors using are given by
Proof. In , the following expressions are derived for the derivatives above: is replaced by after removal of th column. The elements of follow i.i.d WG fading. Using (14)-(15), the expectations in (25)-(26) can be evaluated. The desired results are obtained using (13). It is observed that (17) and (23) represent the same mathematical expression. Therefore, due to , optimal detection is realizable through MMSE detectors and gives suboptimal detection. It can be shown thatEquation (27) decreases with the increase in and increases with the increase in . Subsequently, MMSE detector has degraded interference cancellation capability for larger which increases number of data streams. In case , and, therefore, interfering data streams cancellation is not possible. For Rayleigh fading conditions, (23)-(24) are simplified and resemble [28, Eq. and Eq. ], which is given by
Also, decreases consistently due to the influence of shadowing on wideband slope, which reflects the divergence from small scale fading. Here, shows the severe shadowing effect and consequent reduction in which is depicted in Figure 2. For a fair comparison, we have compared our result with the results of  under a special case (i.e., Rayleigh fading) of WG fading except we considered shadowing effect. SM MMSE offers performance with a high data rate keeping the complexity low. However, it is disadvantageous to system reliability. Therefore, we have considered OSTBC in Proposition 4.
Proposition 4. The respective and for OSTBC MIMO systems using are expressed as
Proof. It can be seen from (11) thatIf we combine (14) and (16) with (30) and then substitute into (13), desired results (29) are obtained after some simple algebraic calculation:Since , and ; therefore ; this shows that SM MIMO systems with optimal/MMSE detectors have higher wideband slope than OSTBC system. For i.i.d. Rayleigh fading conditions, (29) is simplified asIn Proposition 4, it is discussed that is affected by code rate, transmission distance, and average channel gains. In Figure 3, spectral efficiency of OSTBC systems in MIMO WG fading channels for and 20 and and 70 is shown for , , and bps. It is shown that is reduced by 50% when instead of . increases with the increase in fading parameter , and an elevated wideband slope is observed which compensates the increased maximum energy per bit. OSTBC diversity leads to a reduction in capacity, which we examined by comparing it with the average capacity obtained from i.i.d. Gaussian inputs and by assigning equal transmit power.
After some simple mathematical formulation, it is observed that for all three configurations is identical, while is different. We have determined that the approximations are precise for the proposed cases at adequately low SNR values, particularly for high values of fading parameters. However, they become inappropriate for small values of and (; due to the unreliable nature of . As shown in Figure 3, the capacity does not show significant improvement for less fading and high shadowing under GK fading  compared to WG fading condition. In addition, for , MIMO system can achieve more capacity in WG fading than that of GK fading; however, WG fading degrades the capacity at high values of both parameters. Our results are approximately same as obtained and compared in  for special cases such as Rayleigh fading and Weibull fading. Also, analytical results are well suited to simulation results.
Figure 4 shows the comparison of optimal and MMSE detection with OSTBC diversity technique on the basis of capacity performance. At −3 dB SNR, approximately 0.5 bps/Hz and 0.24 bps/Hz improved capacity is achieved using optimal and MMSE detection, respectively, compared to that of OSTBC for . However, MMSE detector reduces 0.25 bps/Hz capacity compared to that of optimal detector for the same parameters. It is noted that optimal detector experiences more computation complexity than MMSE detector.
The WG fading model has shown a wide range of agreement with measured data under various environment conditions. The performance estimation of MIMO systems in WG fading channels is bounded, generally due to the adversity to deal with the non-Gaussian nature of the fading coefficients. A specified low SNR analysis of different MIMO systems operating in WG fading channels is presented in this paper. Novel tractable expressions for and are deduced. These expressions are verified by previous results of Rayleigh and K/GK-fading. The proposed analysis of MIMO system deals with three techniques, namely, SM with optimal detection, SM with MMSE detection, and OSTBC. The average capacity of OSTBC systems is usually subsidiary to that of capacity-oriented SM MIMO techniques, whereas OSTBC system is a diversity-oriented technique. Further, this work can be extended using the same channel model with other performance measures.
The proof depends on the moments of WG variates, which can be evaluated by combining (4)-(5). The procedure to evaluate (15) is given by (A.1). Firstly, th () diagonal element of is augmented, which is computed by After some simple algebraic calculation, the expected value of (A.1) can be obtained asAll diagonal elements of (A.2) are summed up to get (15). Concerning (16), it is noted that all the elements of are i.i.d. random variables to obtain Equation (16) is obtained by employing (4) and (5) in (A.3) after some simplifications.
The authors declare that they have no competing interests.
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