www.banglajol.info/index.php/JSR Symbol Error Performance Analysis and Comparison between Alamouti and MRRC Diversity Combining Scheme for Wireless Communication

The symbol error performance of a wireless communication system is analyzed using diversity combining scheme in Rayleigh fading channels . The performance of the system is compared using Alamouti scheme and maximal-ratio receiver combining (MRRC) scheme. The performance results of the system are evaluated by numerical computation and by simulation using MATLAB. A detailed analysis and exact (closed-form) expressions of the probability of error of the wireless communication systems in Rayleigh fading channels are provided for both M-ary phase-shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM) schemes. Monte Carlo simulation shows exact match with the theoretical results. Keywords: Alamouti scheme; M-PSK; M-QAM; MRRC; Symbol error probability (SEP). © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i1.3000 J. Sci. Res. 2 (1), 54-66 (2010)


Introduction
Diversity techniques have almost exclusively been applied to base stations to improve their reception quality. This is because it is more economical to add equipment to base stations rather than the remote units. For this reason, transmit diversity schemes are very attractive. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. Numerous research works have been carried out during the last few years on the performance evaluation of wireless communication system using diversity technique. Wireless Communications and Networking including coding, different multiplexing and multiple accesses, forms of diversity, generations of communications, WiMAX and many topics related to wireless communications are well discussed [1][2][3][4]. Discussions on diversity technique, multiple accesses and various information on wireless communication system can also be found in the Wikipedia [5]. A simple two-branch transmit diversity scheme is presented in ref. [6]. Using two transmit antennas and one receive antenna the scheme provides the same diversity order as maximal-ratio receiver combining (MRRC) with one transmit antenna, and two receive antennas. It is also shown that the scheme may easily be generalized to two transmit antennas and M receive antennas to provide a diversity order of 2M. Based on Alamouti's work, Tarokh et al. [7] proposed orthogonal space-time block codes (OSTBCs) combining the orthogonal coding method and the simple diversity technique. Tarokh [8] studies the encoding and decoding algorithms for various codes and provides simulation results demonstrating their performance. A squaring method to simplify the decoding of orthogonal space-time block codes in a wire-less system with an arbitrary number of transmit and receive antennas is reported in ref. [9].
In this paper, we present the theoretical analysis to evaluate the symbol error performance of a wireless communication system using Alamouti-scheme and MRRC diversity combining scheme. Closed-form expressions of signal-to-noise-ratio (SNR) are analytically derived for both Alamouti scheme and MRRC scheme. Using these SNR equations Exact expressions of symbol error probability (SEP)are derived analytically for up to 2Tx-2Rx transmit diversity scheme and as well as 4 branch receiver diversity scheme for binary phase-shift keying (BPSK), quadrature phase shift-keying( QPSK) and M-ary quadrature amplitude modulation (M-QAM) schemes. The symbol error performance results are numerically evaluated and simulated using MATLAB software.

Two-branch transmit diversity with one receiver
The combiner builds the following combined signal that is sent to the maximum likelihood detector. We can write as [6] The signal power where, 2 w σ is the noise power.

Two-branch transmit diversity with two receiver
The received signal can then be expressed as [6] where n 0 , n 1 , n 2 and n 3 are complex random variables representing receiver thermal noise and interference. The combiner builds the following signal that is sent to the maximum likelihood detector: x h r h r h r h r = + + + The signal-to-noise-ratio (SNR) becomes 3. Receiver Diversity: MRRC Scheme

Two branch maximal-ratio receive combining (MRRC) scheme
The receiver combining scheme for two-branch MRRC is as follows: The signal-to-noise-ratio (SNR) becomes

Four branch maximal-ratio receive combining (MRRC) scheme
The resulting received baseband signals are The receiver combining scheme for the four-branch MRRC is as follows: The signal-to-noise-ratio (SNR) becomes

One transmitter -one receiver
From (3) The PDF of η becomes

M-ary PSK modulation
The symbol-error probability ( ) e P η for M-ary PSK modulation can be given as [12]

Two transmitters and one receiver
From (3) the instantaneous SNR per symbol λ becomes, The average probability of error for the system can be calculated as

M-ary PSK modulation
The symbol-error probability ( ) e P λ for M-ary PSK modulation can be given as . Substitution of (22) and (24) in (23) and by performing the integral the equation becomes

M-QAM modulation
The symbol-error probability ( ) e P λ for square M-QAM modulation can be given as Let us define the function 2 ( , , ) a ψ β Δ as ( ) is given by the right-hand-side expression in (25). The average symbolerror probability for M-QAM modulation can be given as and .

One transmitter and two receivers
The PDF of instantaneous SNR per symbol 1 λ becomes The average probability of error 1

M-ary PSK modulation
The symbol-error probability 1 ( ) e P λ for M-ary PSK modulation can be given as Substitution of (30) and (32) in (31) gives and by performing the integral the equation becomes

M-QAM modulation
The symbol-error probability 1 ( ) e P λ for square M-QAM modulation can be given as The average probability of error for the system can be calculated as 1 4 x x T R e P −

M-ary PSK modulation
The symbol-error probability 1 ( ) e P ε for M-ary PSK modulation can be written as Substitution of (37) and (39)

M-QAM modulation
The symbol-error probability where c and

Two transmitters-two receivers
The PDF of instantaneous SNR per symbol 3 ε becomes The average probability of error for the system can be calculate as

M-ary PSK modulation
The symbol-error probability 3 ( ) e P ε for M-ary PSK modulation can be given as Substitution of (45) and (47)  2)

M-QAM modulation
The symbol-error probability 3 ( ) e P ε for square M-QAM modulation can be given as and for 4 π β =

Performance Results
Following the analytical approach we compute the symbol error rate performance of wireless communication system using Alamouti scheme and MRRC scheme. The symbol error rates using two transmitter-one receiver antenna and two transmitter-two receiver antenna for BPSK and QPSK modulation depicted in Figs. 1 and 2, respectively.       It is evident from Fig. 1 and Fig. 2 that significant improvement of the SEP performance is achieved by applying diversity scheme and as the number of transmitting antenna increases probability of error decreases. The symbol error probability of Alamouti scheme using one and two transmit antenna with single receive antenna for 16-QAM modulation in Rayleigh fading is shown in Fig. 3. Fig. 4 shows the symbol error probability for two transmitter-two receiver for 16-QAM modulation. The symbol error probability of MRRC scheme using one transmitter-two receiver and one transmitter-four receiving antenna for BPSK modulation in Rayleigh fading shown in Fig. 5. The comparison between Alamouti scheme and MRRC scheme is shown in Fig. 6. The maximum ratio receive combining scheme (MRRC) technique that was used for Fig. 6 employed one transmit antenna and two receive antennas. It is evident from the Fig. 6 that adding another transmit-antenna and employing temporal diversity leads to a significant boost in symbol error performance. The Alamouti scheme gradually outperforms the MRRC technique as the SNR values increase. The MRRC technique outperforms the Alamouti scheme with just one receive antenna. The graph shows about a 3-dB penalty in error rate performance. Alamouti explains this penalty by noting a constraint that he imposes on his code; namely, he has each transmit antenna radiating half of the energy that the one antenna for the MRRC scheme radiates. The Alamouti scheme is practical because it allows a boost in performance through the addition of receiver antennas. The Monte Carlo simulation of symbol error rate performance of Alamouti coded system for BPSK modulation in Rayleigh fading for 2Tx-1Rx and 2Tx-2Rx are depicted in Figs. 7 and 8, respectively. It is seen form these figures that, Monte Carlo simulation showed an exact match with the theoretical results.

Conclusion
We investigate the probability of error of single-input-single-output (SISO) system and Alamouti scheme using 2 Tx-1Rx, 2Tx-2Rx and MRRC scheme using 1Tx-2Rx, 1Tx-4Rx in Rayleigh fading channels. A detailed analysis for finding the symbol error probability of the wireless communication systems in Rayleigh fading channels is provided. Exact closed-form expressions are derived for average symbol-error-probability (SEP) for both M-PSK and M-QAM modulation schemes. The numerical results show exact match with Monte Carlo simulation.