L-MRC Receiver with Estimation Error over Hoyt Fading Channels

Abstract We present the performance of arbitrary branch maximal ratio combining (MRC) over Hoyt fading channel with estimation error. Among the diversity receiver MRC performs optimally provided the phase and SNR estimation are perfect. Yet, in real life the perfect channel estimation is a challenging job. There may be error present in both phase and the envelope estimation, and the amount of error will be depend upon channel conditions as well as the techniques used for the detection. In this paper performance degradation of the MRC receiver by phase and envelop estimation error have been analyzed in terms of ABER and channel capacity. From the analysis, it can resolve that the receiver operation is affected severely by the estimation error in the envelop than the phase. Keywords-Imperfect channel estimation, Error estimation, MRC, Hoyt Fading, ABER, Channel capacity.

branches MRC receiver is derived, which has been used to obtain the ABER expression for coherent BPSK and the capacity analysis. The paper is organized as follows: In section II the system model considered for analysis is explained. In section III PDF of MRC receiver effective output SNR with ICE is discussed. In IV ABER of MRC receiver over Hoyt Fading with ICE is given. In section V Capacity analysis for a L-MRC receiver is done for ICE. In VI the results and analysis is presented. Conclusions are presented in section VII.

II. SYSTEM MODEL
The wireless communication system receiving multipath fading signals in the presence of additive white Gaussian noise (AWGN) is considered over here. To receive multipath signals the receiver is equipped with L antennas and perform MRC to improve the quality of the signals. The channel is considered to be slow and frequency non selective. For the assumed channel the received signal over 'i th ' symbol interval is given as, where, d (i) is the transmitted symbol in 'i th ' interval, with energy E S = E [|d (i)| 2 ] and noise vector n (i) =[n 1 (i) ,……, n L (i)] T , is the complex Gaussian noise having zero mean and two sided power spectral density 2N 0 .In  is the channel co-efficient vector for L branches and its elements are Hoyt distributed. The envelop pdf of a Hoyt distributed receive signal is given by [1], where, where, l = 1, 2, 3,….., L , 'f' is the diffused component and the errors  [15]. Applying the half plane decision method [21] the complex DV D  will be rotated with a plane angle  to obtain a new DV as, where, M is the constellation size and 0   for BPSK (M=2). So considering the half plane decision method for a DV D , the effective output SNR of a MRC receiver is given as [6], where, B is a function of  and given by,       It is already described that the principle of MRC is based on the channel estimation at the receiver. Hence, it is advantageous to use the adaptive transmission technique in case of MRC receiver. An adaptive technique very the power and rate of transmission based on the channel condition between the transmitter and receiver which is available to the transmitter through a lossless channel from the receiver [7], [16].

III. PDF OF MRC RECEIVER EFFECTIVE OUTPUT SNR WITH ICE
Using the Hoyt fading model given in [22] the square of the Hoyt distribution can be written in terms of square Gaussian distribution as, where l X and l Y are independent zero mean Gaussian RVs with variances 2 x  and 2 y  respectively. In this representation the Hoyt RV l  has the PDF given in (2) where the fading parameter For the convenience of presentation, but without loss of generality, we assume This channel model is suitable to obtain the PDF of the output SNR of the MRC receiver. From this model doing some mathematical calculation followed by a random variable transformation a closed-form expression for the SNR PDF of MRC receiver is given as [23], Since  and SNR with ICE  is identical for 1 l   , so eq. (7) can be reproduced for  as,

IV. ABER OF MRC RECEIVE OVER HOYT FADING WITH ICE
In PDF based approach the ABER can be derived by averaging the PDF of the system SNR as [1], where,   P e  is a conditional BER. In our analysis the coherent BPSK modulation has been considered and for this the coherent conditional BER is given by [24], where, a=1 for Coherent BPSK. According to the eq.(5) multiplying eq.(8) with the function 'B' followed by a RV transformation the expression can be expressed as, substituting eq.(11) and eq.(10) for  in eq.(9) and solving the integral using [ [26] where, 2 1 F is a Hypergeometric function.

V. CAPACITY ANALYSIS OF ADAPTIVE MRC RECEIVER WITH ICE
The adaptive transmission scheme (both power and rate) is employed to increase the receiver performance. In this section we have presented the upper limit of the channel capacities considering the adaptive transmission schemes along with estimation error. The effect of estimation errors on the channel capacities has been discussed in details in the results and discussion section.

A. Optimum Rate Adaptation (ORA):
Optimal rate adaptation is the simplest adaptive transmission techniques. This method have lot of more practical value due to its simplicity. In this technique adaptation is implemented to the rate of transmission depending on the channel condition while transmitter power remains constant [16], where, 's' is the bandwidth of the channel. Substituting (8) in (13) where, ora C s the spectral efficiency (Bit/sec/Hz) and 1 1 F is a Hypergeometric function. Solving (14)

B. Channel inversion with Fixed Rate (CIFR):
In case of CIFR, the transmitter adapts its power to maintain constant received SNR, so that the inversion of the channel fading effect is possible. Then the channel appears to the encoder and decoder as a time-invariant AWGN channel. The Channel capacity for CIFR is given by [16], where, Putting (8) in (17), calculating for the SNR  the equation can be written as, Solving (18) (19) substituting, (19) in (16) the final expression for spectral density of CIFR can be obtain.

C. Truncated Channel inversion with Fixed Rate (TIFR):
The major drawback of the CIFR system arises when the channel goes to deep fades. Under this situation, it needs very large amount of power to maintain constant SNR at the receiver. Hence, system may not be practically possible. This leads to a modified version of it with a modification that the transmission suspended if the SNR goes below a threshold value 0  . The channel capacity of this scheme can be obtained as [16], where, and Putting, (8) in (21) and solving the equation, using [26,351(2)], the expression for tifr R can be expressed as,   (23) and (24) in (20) the final expression for TIFR over Hoyt fading channel for ICE can be obtain.

VI. RESULTS AND DISCUSSION
The analytical results of the above expression of an MRC system, has been presented in this section with proper analysis. ABER vs   has been shown for various channel conditions (for different fading parameter 'q'), diversity order ( L), phase and envelope estimation errors from Fig. 1 to Fig. 4. We have observed similar observation presented in the earlier literature for parameters 'L' and 'q' when there is no estimation error. However, the performance completely changes with the amount of estimation error. In Fig.1 the amount of envelope estimation error '  ' is considered as 0.999 and the phase 32 As expected with the decrease of 'q' parameters of Hoyt distribution the ABER is increasing. But if we increase the antenna number at the receiver side (for L=4) the BER decrease considerably. The same thing can be seen in Fig. 2  10  bit/sec).
In Fig. 3, we have considered only the phase error and have found that, ABER is comparatively low than the first two cases (Ex. For L=2 , when SNR is 20 dB, ABER is 4.6 10  bit/sec). From this we can conclude that the accuracy in the estimation of envelope is much more essential than phase for an MRC receiver. A comparative study of perfect CSI with ICE has been shown in Fig. 4. . From here it is clear that, the channel capacity is reduced remarkably for imperfect estimation irrespective of the diversity order. In Fig: 6 the effect of '  ' (Envelop error ) and   (Phase error ) on channel capacity is studied. As expected for channel capacity also the effect of enveloped error is comparatively more sensitive than the phase error irrespective of order of diversity for a MRC system.

VII. CONCLUSION
This paper analyzes the performance of L-MRC receiver with estimation error over Hoyt fading channels. The expression for ABER and capacity has been derived considering both envelope and phase estimation error. Form the analysis it can be concluded that the performance of MRC receiver degrades considerably with improper estimation. Infect, from the result we found that if the envelope estimator is not accurate enough it is better not go for the MRC receiver. Inaccurate estimator may completely nullify the advantages expected from the MRC system.