Adaptive Transmission with Multiuser TAS/MRC Systems over Exponentially Correlated Rayleigh Fading Channels

—This paper analyzes the channel capacity for multiuser Transmit antenna selection (TAS) / Maximal ratio combining (MRC) system over exponentially correlated Rayleigh fading channels. TAS scheme reduces the hardware complexity and the number of RF chains in MIMO systems and MRC is a diversity technique which provides optimal performance. The correlation is almost unavoidable in spatial diversity combining scheme due to reduced size of the wireless portable receivers. Different types of correlations exist based on their antenna placing. Exponential correlation is observed when antennas are put in a linear array in a communication receiver. Expressions for channel capacities with different power and rate adaptation techniques have been found. The influence of diversity order and correlation on the channel capacity has been examined for different adaptive transmission techniques.


II. SYSTEM AND CHANNEL MODEL DESCRIPTION
We consider a multi user, multiple input and multiple output (MIMO) wireless communication system with a base station communicating with K users. A block diagram of the system is shown in Fig. 1 for analysis of downlink data transmission. The base station has N transmit antennas and each user has L receive antennas. The base station is serving K users. Each user is served by a single antenna of base station. In Fig. 1 The channels between the transmit antennas and the users are modelled as a slow flat fading Rayleigh channel. Using channel state information (CSI) received through an error free feedback path, the base station selects the best transmit antenna to transmit data for each user. The complex low pass equivalent of the signal received at the  and , k l  is the Rayleigh distributed fading amplitude with PDF given by [17]. , , , The receiving antennas of each user are placed in a linear array, thus they are affected by exponential correlation. The correlation coefficient between i th and j th received fading signals is defined as [17,18]. , , 1,2,., ) ;( In the system, MRC is performed by each user to improve the quality of the downlink information. In the MRC receiver, the received signals are obtained by summing the signal of all branches after performing cophased and scaling by a component proportional to the branch SNR. The instantaneous output SNR of the MRC receiver is given by [17] 1 , is the instantaneous SNR of l th branch and L is the total number of diversity branches in the combiner. By using the channel state information (CSI) of each user, the best transmit antennas out of all N transmit candidates, which maximizes the SNR at the MRC output of L receive antennas, are selected to transmit data.
III. PDF OF OUTPUT SNR From the formula given for the exponentially correlated sum of gamma RVs in [19], sum of the Rayleigh square distribution can be very closely approximated. From which performing RV transformation the PDF of SNR between t th transmitting antenna and k th user can be given as and  is the average input SNR in each branch of an MRC receiver.
We assume equal average SNR in each branch i.e , Simplifying using [20, (3. is the lower incomplete gamma function. For N number of transmit antennas and K number of users the total number of communication link will be NK . Each of link will have PDF and CDF as given in (5) and (7) respectively. The joint CDF of all links can be expressed as Differentiating the expression (9), the PDF of SNR for the TAS/MRC system over Rayleigh Fading Channel can be expressed as: Writing the lower incomplete gamma function in infinite series from [21, (1.7], the PDF of output SNR for the TAS/MRC system over Rayleigh Fading Channel can be obtained as: (11)   1 2

IV. CAPACITY ANALYSIS OF ADAPTIVE TRANSMISSION SCHEMES
The various adaptive transmission schemes are highlighted in [6] and [14] along with the general expression of channels capacities. We have considered the adaptive transmission system to analyze the capacities of the TAS/MRC scheme along with antenna correlation. For finding out the capacity expression the PDF of system SNR (11) has been used and is explained in the following subsections.

Optimal Power and Rate Adaptation at the Transmitter
This technique is suited for power limiting scenario. With a constraint on the average transmitting power this method adaptively optimizes the transmission rate to enhance the capacity of the system. The analytical expression of capacity with this technique is given as [6]   where B is the channel bandwidth,   f   is the PDF of the combiner output SNR and 0  is the optimal cutoff SNR, below which no transmission is allowed. The optimal cutoff SNR 0  has to satisfy the condition Putting the expression of (11) into (12) and arranging the integral, the capacity for OPRA scheme can be given as    (14), the optimal cutoff SNR, 0  should satisfy (13). Substituting   f   from (11) in (13)

Constant Transmitting Power
In a scenario when power is sufficient a constant transmitting power can be maintained in the transmitter with variable rate to reduce the complexity of the system. The channel capacity for this technique (bits/sec) can be given as [6].
Substituting (11) into (16) and solving the integral following an approach similar to OPRA scheme, the capacity for constant transmitting power techniques can be obtained as   where    

Channel Inversion with Fixed Rate
Channel inversion is a technique in which the received SNR is maintained constant by changing the transmitting power. In channel inversion technique the same rate is maintained as the effect of channel is nullified by inverting the channel effect. The channel capacity formula for this scheme is given as [6] The capacity for this scheme requires initially a solution to the integral of cifr R in (18). Putting (11) Thus, the capacity of this scheme can be calculated by substituting (19) into (18).

Truncated Channel Inversion with Fixed Rate
The major drawback of the channel inversion technique is that in case of worst channel to invert the effect of the channel it requires large amount power which is difficult to realize in real system. To avoid this problem truncated channel inversion with fixed rate (TIFR) is developed which is a modified version of CIFR. In TIFR the transmission is suspended when the received SNR is below a predefined threshold 0  . The capacity formula for TIFR can be given by [14]   2 0 1 log 1 1 where   An expression for   0 out P  can be obtained from (9) by putting 0    only. Thus a final expression for the capacity of this scheme can be obtained by placing (21) and (20).

V. NUMERICAL RESULTS AND DISCUSSION
In this section we have discussed the results obtained from the derived expressions for capacity with different power and rate adaptation techniques assuming exponential correlation among received antennas. We consider a TAS/MRC system having a number of transmitting antennas 2 N  and the number of users 2 K  with different diversity order for the purpose of illustration.   Fig. 3 and 4 present the capacity vs. average SNR per branch in dB for CIFR and TIFR schemes respectively. It is seen that the increase in correlation coefficient results in an improvement of the capacity of the channels for all the schemes. This is due to the reason that these formulas give the maximum possible capacities through the channels. Therefore, when all the channels are good (means they are highly correlated) one can expect maximum capacity through the channels. The similar kind of result is also reported in [22]. It can also be observed that the larger the value of diversity order L , the capacity of the system increases considerably for a constant average SNR per branch. For TIFR scheme we have considered the threshold SNR, 0 2dB   . The observations of CIFR and TIFR schemes are similar.

VI. CONCLUSIONS
In this work we have analyzed the channel capacity of multiuser TAS/MRC wireless systems under Rayleigh fading with exponentially correlated receive antennas. The channel capacity has been analyzed for different adaptive transmission techniques available in the literature. Expressions have been obtained for the capacity of OPRA, ORA, CIFR and TIFR schemes. The obtained expressions are in the form of infinite series and incomplete gamma function. The capacity expressions are numerically evaluated and plotted for different parameters of interest. It has been observed that the channel capacities increases with higher correlation and diversity order.