COMPARISON OF CODE RATE AND TRANSMIT DIVERSITY IN 2 2 MIMO SYSTEMS

Transcription

1 126 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.107 (3) September 2016 COMPARISON OF CODE RATE AND TRANSMIT DIVERSITY IN 2 2 MIMO SYSTEMS D. Churms, O.O. Ogundile, and D.J.J. Versfeld School of Electrical and Information Engineering,University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. duane.churms@students.wits.ac.za, olayinka.ogundile@students.wits.ac.za, and jaco.versfeld@wits.ac.za. Abstract: This paper compares the Alamouti STBC and two BLAST (VBLAST and TBLAST) based 2 2 MIMO schemes using different channel code rates. The overall rate of the system is kept constant by making the product of the MIMO scheme s rate and the channel code rate constant. Reed-Solomon (RS ) soft and hard decision decoding algorithms are adopted as the forward error correction (FEC) scheme. The Berlekamp-Massey (B-M) algorithm is used as the hard-decision decoding FEC scheme. The Koetter-Vardy (KV) algorithm is employed as the soft-decision decoding FEC scheme to maximise the error rate performance of the MIMO systems. The performance of these MIMO schemes for different channel code rates are documented through computer simulation. From the computer simulation results, it is shown that given two systems with equal overall rate, the system with the lower MIMO rate exhibits better performance due to the increased diversity. In addition, the results show that diversity do not have a significant impact on the soft-decision gain. Finally, the two BLAST based MIMO systems are shown to have near identical performance for a 2 2 MIMO system. Key words: Alamouti STBC, code rates, diversity, MIMO, RS codes, TBLAST, VBLAST. 1. INTRODUCTION Multiple input multiple output (MIMO) systems are fast gaining popularity in wireless communication, most notably LTE [1] and WiFi [2, 3]. MIMO systems employ multiple antennas both at the transmitter and receiver to give an edge over single input single output (SISO) systems. Applying the MIMO technique in communication systems give advantage of two prime properties: Diversity and Multiplexing. The system transmission rate is improved by multiplexing and the link reliability of the system is also improved by taking advantage of space and time diversity [4]. MIMO systems provide other advantages such as combating multipath fading and higher throughput in rich scattering environments. There are a wide range of MIMO encoding schemes which are designed to prioritise different strengths of MIMO. Low rate schemes offer high diversity and combat multipath fading while high rate schemes place more emphasis on higher throughput and spectral efficiency. In some MIMO scheme, the additional transmit antennas are not necessarily used for diversity. These additional transmit antennas are utilized to send multiple symbols per time slot; thus, increasing the rate and spectral efficiency of the system. With respect to the MIMO encoding and decoding schemes, MIMO systems are combined with different forward error correction (FEC) codes (such as Turbo codes, LDPC codes Reed-Solomon (RS ) codes, etc) in order to improve the system s transmission reliability in different wireless communication channels. Therefore, this paper investigates the performance of RS codes using symbol level decision decoding over three different MIMO schemes namely the vertical bell laboratories layered space-time (VBLAST) scheme [5], the Turbo bell laboratories layered space-time (TBLAST) scheme [6], and the Alamouti space-time block code (STBC) scheme [7]. Of particular interest, the paper probe the impact of using a low rate MIMO scheme with a high rate RS code and vice versa. Reed-Solomon codes introduced in [8] are a class of non-binary error correcting codes that are maximum distance separable. For a given code rate, RS codes thus offer the largest possible minimum distance d min = n k + 1, where n is the codeword length and k is the information symbol length. This allows RS codes to offer a conventional error correcting capability of n k 2 for hard-decision decoding. Examples of such hard-decision decoding algorithm are found in [9 12]. In the case of soft-decision RS decoding, the error correcting capability goes beyond the hard-decision bound, allowing improved performance at the cost of increased decoding time complexity. Various bit level and symbol level soft-decision decoding algorithms have been proposed for RS codes. Examples of such decoding algorithm includes [13 15]. However, we focus our attention to symbol level soft-decision decoding algorithm proposed by Koetter-Vardy (KV) in [13] and the hard-decision decoding proposed by Berlekamp-Massey (B-M) in [9, 10] in order to maintain a fair comparison. The paper is organised as follows. Section 2 describes the system model. In particular, the RS encoding and decoding steps, the MIMO encoder and decoder schemes, and the channel model assumed in this paper are explained in detail. In section 3, the optimal RS code rate is investigated to ensure a fair comparison among the three MIMO schemes. More so, results are presented to analyse the gain achieved using soft-decision decoding compared to hard-decision decoding. The section also compares the

2 Vol.107 (3) September 2016 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 127 performance of the two MIMO BLAST based schemes. In addition, the effect of using a reduced RS code rate in conjunction with a high rate MIMO scheme is investigated in this section. Finally, the paper is summarized with concluding remarks in section SYSTEM MODEL Consider the simulation system set up of Fig. 1. The input data is encoded using an (n, k) RS code, where n and k are defined as above. The encoded data is mapped to a rectangular M-ary quadrature amplitude modulation (M-QAM, where M = 16) complex data symbol. The mapped data is interleaved to phase out burst errors at the decoder. Subsequently, the mapped data is encoded to the desired MIMO scheme. The MIMO encoded data is therefore transmitted over a block Rayleigh fading channel with normalized Doppler frequency F D T (where F D is the Doppler frequency and T is the symbol period), and distorted with additive white Gaussian noise (AWGN). Note that the paper assumes Jakes isotropic scattering model [16] for the complex Rayleigh fading. At the receiving end, the reverse of the transmitter stages occur. The receiver stages start with the MIMO decoding, followed by the deinterleaving, demodulation and RS decoding stages. This paper focuses on the RS encoding and decoding blocks, the MIMO encoding and decoding blocks, and the channel model as described in sections 2.1, 2.2 and 2.3 respectively. Input Data Reed-Solomon Encoder 16QAM Modulator Interleaver MIMO Encoding Rayleigh Channel AWGN Data k n Parity n k Figure 2: RS codeword is used in conjunction with the Guruswami-Sudan (GS ) algorithm [17] for the symbol level SDD. The GS algorithm is a hard-decision list decoding algorithm which can correct up to n nk errors [17]. The algorithm interpolates a polynomial with roots corresponding to the received symbols, which is then factorised to create the decoded message [17]. The KV algorithm transforms the GS algorithm into a soft input list decoder [13]. The KV algorithm operates on the symbol reliability matrix generated from the received symbols by assigning higher multiplicities to roots with high reliability [13]. The symbol reliability matrix is obtained from the probability of the received symbols being at a given distance from the constellation point as described in [13, 18]. The maximum output list length L s for the KV algorithm is set to 4 (L s = 4), which is sufficient to significantly outperform the GS bound. This L s also determines the decoding performance of the KV algorithm, the higher the value of L s, the better the performance of the algorithm. Although, choosing a longer list length causes the decoding algorithm to become prohibitively computationally intensive. All error analysis is performed using codeword error rate (CER), i.e. the fraction of messages that do not appear in the decoded list of potential messages. Simulations for HDD are run until 100 codeword errors are detected, while the SDD simulations are run for only 30 codeword errors due to limited computational time. Output Data Reed-Solomon Decoder Demodulator Deinterleaver MIMO Decoding 2.2 MIMO Scheme Figure 1: Simulation system set up 2.1 Reed-Solomon Coding To align RS symbols to modulation symbols, RS symbols are chosen to have a size of 4 bits. The RS codes of interest thus have a codeword length of n = 15. The message length k is adjusted depending on the simulation, with a (15,9) code being used for comparing hard and soft decision decoding. In addition, to evaluate the diversity gain from RS codes, (15,5) RS codes over a rate 2 MIMO system are compared to (15,10) RS codes over a rate 1 MIMO system. Similar to other error correcting codes, the encoded RS codeword is also divided into separate block of data as shown in Figure 2. The RS decoder can be set up to either perform hard-decision decoding (HDD) or soft-decision decoding (SDD). As earlier said in section 1, B-M algorithm is used for HDD. On the other hand, the KV algorithm [13] The physical MIMO design for this simulation set up is a 2 2 antenna system, representing two transmitting (TX) and two receiving antennas (RX) as shown in figure 3. This type of antenna configuration along with the 4 4 configuration are proving to be the most common, largely due to space limitations in mobile equipment [1, 2]. MIMO systems generally offer benefits in three categories: diversity, spectral efficiency, and beamforming. Increased diversity improves the robustness of the system by transmitting each symbol over more than one antenna in separate time slots, mitigating the effect of multipath fading and noise. Higher spectral efficiency, on the other hand, improves overall throughput of the system without increasing the required bandwidth. This is done by transmitting unique symbols over each antenna in all time slots. A third technique, beamforming [19], differs from the previous two categories in that the same symbol is transmitted over multiple antennas in a single time slot. The phases at each antenna are adjusted so that the signals interfere constructively in the intended direction of

3 128 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.107 (3) September 2016 transmission. This increases the received signal-to-noise ratio (SNR), and is typically used when signal strength is low due to shadowing or the transmitter and receiver being far apart. TX1 TX2 h 21 h 12 h 11 h 22 RX1 RX2 Figure 3: Simplified signal paths for a 2 2 MIMO system In MIMO schemes that offer diversity or increased spectral efficiency, symbols are transmitted coherently, while in beamforming systems a phase shift is introduced. Beamforming systems will not be considered in this paper. Unfortunately it is not possible to maximise both diversity and spectral efficiency within a single MIMO scheme. A high rate scheme, which transmits many symbols per time slot, will provide very good spectral efficiency at the cost of diversity. A low rate scheme on the other hand will require more time slots to transmit the same amount of data, lowering its spectral efficiency but providing very good diversity. Many different MIMO schemes exist which are designed to achieve one of three things: maximum diversity, maximum spectral efficiency or a trade-off between the two [5, 7, 20, 21]. This paper considers only three MIMO encoding schemes as mentioned in section 1. The VBLAST and TBLAST are examples of maximum rate MIMO schemes considered. The Alamouti scheme is considered because it is an orthogonal STBC which offers maximum diversity. The VBLAST scheme transmits independent symbols over each antenna during each time slot. For a 2 2 antenna system, VBLAST thus transmits two symbols per time slot, which means it is a rate 2 scheme [5]. Decoding is performed by decoding the transmitted symbol with the greatest contribution to the received vector first, based on the estimated channel matrix. The other symbol is assumed to be equal to zero in the first step. The first symbol is quantised to the nearest value from the modulation constellation, following which its contribution is cancelled from the received vector. The second symbol is then decoded. The VBLAST scheme does not offer any transmit diversity but contributes receive diversity only in the form of two receive antennas. The diversity order for the VBLAST scheme is thus 2. Turbo-BLAST, or TBLAST, was developed in [6] to minimise the effects of co-antenna interference (CAI). The transmission works identically to VBLAST, so it is also a rate 2 scheme for a 2 2 antenna system. Decoding is performed by iteratively estimating the values for all symbols in the received vector based on the previously estimated values of the other symbols. The expected value of the CAI is removed from the received vector before generating the new estimate. As the number of iterations becomes large, the symbol estimates approach the actual values. Since each symbol s estimate is generated using knowledge of other symbol s estimates, the effect of CAI is mitigated. Neither of the BLAST schemes offer any transmit diversity. Thus, the diversity order for TBLAST schemes is also 2. The Alamouti scheme is an orthogonal STBC. It utilises two time slots for every two symbols that are transmitted, so it is a rate 1 scheme [7]. Table 1 shows the structure of the Alamouti STBC. The decoder uses maximal ratio combining (MRC) to recover the transmitted symbols. The diversity order achieved by the Alamouti STBC scheme on a2 2 antenna system is Channel Model Table 1: Alamouti STBC structure t 0 t 1 TX1 s 0 TX2 s 1 s 1 s 0 The channel is modelled as a block Rayleigh fading channel. The entries h ij of the 2 2 channel transfer matrix H are independent and identically distributed Rayleigh random variables. These entries have normally distributed real and imaginary components that have zero mean and 1 variance. The channel matrix is also normalised so 2 that E H ij 2 = 1 [22]. Following every fading block, a new channel matrix is generated which is independent of all previous channel matrices. The H matrix in a real system is estimated at the receiver by means of a training sequence. For the purposes of these simulations, perfect channel knowledge is assumed at the receiver in order to avoid estimating the channel state information (CSI). Since channel estimation is not the focus of this paper, assuming a perfect CSI will make the simulation design less computationally intensive. More so, the transmitter do not have any knowledge of the CSI. This means that all transmitting antennas operate at the same power level. Besides, using a block Rayleigh fading channel, a correlated channel matrix causes many errors to occur (an entire fading block can solely consist of errors). The burst of errors will be concentrated within a few error correcting codewords if no interleaver is used. This will result in these codewords containing more errors than the error correction code can correct, resulting in decoding failures. In order to avoid having large bursts of errors within a single codeword, a block interleaver is used to spread the codewords across multiple fading blocks. Figure 4 shows an example of the interleaver. The size of the interleaver is chosen to be sufficiently large that no two symbols in a codeword occur within the same fading block. For the purposes of these paper, the performance of the interleaver is assumed to be close to optimal.

4 Vol.107 (3) September 2016 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 129 Block fade length Codeword Length Codeword 1 Codeword 2 Codeword 3 Transmission order Figure 4: Interleaver structure showing ordering of transmitted symbols The 2 1 received signal from the channel output (received vector r) is given as: r = Hx + n, (1) where H is the 2 2 channel matrix, x is the 2 1 transmitted vector, and n is the 2 1 noise vector. The channel matrix H represents the individual paths between antenna pairs such that: [ ] h11 h H = 12, (2) h 21 h 22 where h ij is a complex value representing the magnitude and phase shift of the path from transmitting antenna j to receiving antenna i. Objects such as walls reflect the radio waves travelling from the transmitter to the receiver, causing changes in phase and angle of arrival of the waves. The signal propagates along a slightly different path for each antenna pair. Channel paths are thus of different lengths and experience different levels of attenuation. The channel entries h ij are complex numbers indicating the phase and magnitude change introduced by each of the channel paths. The exact values of the channel entries used in simulations are determined by the channel model used. 3. RESULTS The results are presented using simulations run in MATLAB. Feasibility of systems is determined by comparing the symbol error rate of the systems across a range of signal to noise ratios. The MIMO configuration used has two transmitting and two receiving antennas (2 2). A Rayleigh fading channel is used as this models a rich scattering environment, as encountered in indoor and heavily built up environments [16]. Three different MIMO schemes are simulated: one system with rate 1 and two systems with rate 2. The rate 1 scheme is the Alamouti scheme, which is an orthogonal STBC specifically designed for 2 2 systems. The rate 2 schemes are the VBLAST and TBLAST schemes, which have identical transmission structures but differ in the decoding algorithm as explained in section 2.2. The system uses 16-QAM modulation. Reed-Solomon channel coding [8] is used with a symbol size of four bits so that each modulation symbol maps to a single code symbol. To achieve the longest possible codewords given the size of the symbol space, the codeword length is n = 15. The message k assumes different sizes in order to perform high and low rate simulations. Both hard and soft decision decoding are implemented. The hard decision decoding is implemented using the Berlekamp-Massey algorithm [9, 10]. Soft decision decoding is used since it can decode beyond the conventional error correcting ability of the code and is particularly suited to low rate codes [17]. The Guruswami-Sudan algorithm [17] is used along with the Koetter-Vardy algorithm [13] for soft decision decoding. The primary objective of this paper is to compare three 2 2 MIMO systems with equal overall rate but with different MIMO and RS error correcting code rates. The data generated in order to perform this comparison can also be used to draw several other conclusions. This section is therefore structured as follows. In order to justify the selection of code rates, section 3.1 evaluates a range of rates using each of the MIMO schemes. The preferred high rate MIMO scheme is then selected by comparing VBLAST and TBLAST in section 3.2. Section 3.3 analyses the impact of using soft decision decoding with various code rates over all three MIMO schemes. Finally, systems with equal overall rates are compared in section 3.4, which addresses the aim of the paper. 3.1 Optimal code rates When performing comparisons between different rate codes, it is important to keep the total energy transmitted per message symbol constant. Suppose the reference value of the energy per symbol E s is based on an uncoded data stream. If an (n,k) code were used and each of the n symbols were transmitted using E s energy, the total energy used to transmit the stream should be n k times higher than the reference data stream. It should thus achieve lower error rates than the reference stream not solely due to the error correcting code, but also due to the higher SNR. Therefore, to use the error correcting codes it is necessary to spread the energy from the reference message across all the codeword symbols to maintain a fair comparison. Each codeword symbol is therefore transmitted using n k E s energy. It is well known that low rate codes are capable of correcting more errors than high rate codes. This, however, comes at the expense of reduced energy per codeword symbol. Reduced energy per codeword symbol results in more errors that need to be corrected, which counteracts the error correcting improvement. The fact that error correction codes offer a gain over uncoded systems indicates that the error correcting improvement is greater than the energy reduction. This is not true for all code rates though. Consider a trivial (15,1) code. The same symbol is transmitted 15 times with an energy of 1 15 E s. For all practical purposes this is equivalent to transmitting a single symbol with energy E s, which is

5 130 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.107 (3) September 2016 the same as an uncoded system. There is therefore a point at which decreasing the code rate is not going to provide any further improvements in error rate. The error rate curves for RS codes of length 15 with HD decoding in a SISO AWGN channel are shown in figure 5. The (15,9) RS code offers the best performance, but the (15,11) and (15,7) codes offer comparable performance at an SER of. The (15,5) code performs 0.9 db worse than the (15,9) code. This is not the case for all channel models as we demonstrate in subsequent sections. rate curves for a SISO block Rayleigh fading channel. As anticipated, the results are identical to Figure 5, with (15, 9) RS codes exhibiting the best performance. On the other hand, MIMO channels exhibit somewhat different characteristics. Figures 7, 8 and 9 respectively show the results of comparing various RS code rates Figure 5: Comparison of code rates in a SISO AWGN channel Figure 6: Comparison of code rates in a SISO Rayleigh fading channel For practical purposes, the SISO equivalent of the Rayleigh fading channel used in this paper is an AWGN channel. The Rayleigh fading coefficient can be thought of as a scalar H which is multiplied with the received vector to effect a change in magnitude and phase. Since the MIMO channel model requires that E [ H ij 2] = 1, the magnitude of H in the SISO case is unity. The transmitted vector thus only experiences a phase shift, but since perfect channel state information is assumed at the receiver, this phase shift is reversed at the receiver. The only effect that the channel has is thus to rotate the white Gaussian noise, which has no significant effect. To verify this, Figure 6 shows the error Figure 7: Comparison of code rates in an Alamouti MIMO system Using the Alamouti scheme, it is evident from Figure 7 that the (15,9) RS code achieves a SER of at the lowest SNR. It outperforms the (15,11) and (15,7) codes by 0.18 db and 0.27 db respectively. The (15, 5) code performs 1 db worse than the best performing code ((15, 9)). These results are very similar to the results for a SISO AWGN channel because the (15, 9) code outperforms the other codes by the same margins. The Alamouti scheme is thus effectively eliminating the effect of the multipath fading, making the MIMO channel behave in the same way as a SISO AWGN channel Figure 8: Comparison of code rates in a VBLAST MIMO system The results are somewhat different when utilising the VBLAST scheme as depicted in Figure 8. In contrast to the SISO and Alamouti results, the (15,5) code offers the best performance at an SER of, closely followed by

6 Vol.107 (3) September 2016 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 131 the (15,7) code. The (15,9) code performs 1.4 db worse than the (15, 5) code. This phenomenon is explained by considering the propagation of symbol errors within the VBLAST structure. When the first symbol in the VBLAST decoding process produces an error, the wrong value is used in the cancelling process. The second symbol is thus decoded using erroneous information, dramatically increasing the error probability of the second symbol as well. This interdependence of symbols results in additional errors, requiring greater error correcting capability from the channel code. Although the interleaver minimises the impact of error propagation on single codewords, the effect remains noticeable due to cross-propagation of errors from different codewords. containing enough poor fading blocks to overwhelm the error correcting capability decreases for low rate codes, resulting in a lower error floor. In order to establish whether bad channels affect both symbol streams, simulations were run using an interleaver which transmitted two symbols from the same codeword in each time slot. Figure 10 shows the error floors for the (15,13), (15,11) and (15,9) codes at an SER of.93,.75 and.35 respectively. These floors are significantly higher, indicating that the error correction ability of the code is more easily overwhelmed due to both symbols in a time slot becoming corrupted. If only one symbol was being corrupted the error floor would be lower than in Figure 9, since the likelihood of n k poor fading blocks per 8 blocks is lower than the likelihood of the same number of poor blocks per 15 blocks Figure 9: Comparison of code rates in a TBLAST MIMO system TBLAST exhibits similar properties to VBLAST, in that the optimal code rate is lower than that for the SISO AWGN channel. Once again the (15,5) code performs the best, achieving an SER of at 18.8 db. The (15,3) and (15,7) codes offer the next best performance at around 19.5 db. The (15,9) code performs 3.3 db worse than the best performing code. An interesting difference between TBLAST and VBLAST is that an error floor is evident when using TBLAST in conjunction with higher code rates. The (15,11) code exhibits an error floor at an SER of.44 while the error floor for the (15,13) code is as high as.37. Lower rate codes would also experience error floors, although these occur below the simulation SER cut-off of. Even when there is no noise present, there is thus a limit to the achievable error rate for each code rate. Errors under noise free conditions were found to occur when the channel matrix had strong spatial correlation. Due to the channel matrix being generated as a set of four independent and identically distributed Rayleigh random variables, there is some probability that the channel for any given fading block is spatially correlated. These bad channel states often resulted in both symbol streams producing errors. Although the interleaver ensures that no fading block allows more than one symbol per codeword, the errors caused by multiple poor channels overwhelmed the error correcting capability of the RS codes in some cases. The probability of a codeword Figure 10: Error floors using TBLAST while transmitting symbols from the same codeword in each time slot The absence of an error floor in the VBLAST curves is indicative of the manner in which errors are caused. In some cases, an incorrect error in the first decoded symbol will result in the second decoded symbol also being incorrect. If this propagating error was due to spatial correlation, the errors would be present even at very high SNRs, resulting in an error floor. Since no error floor is evident even for high rate RS codes, the propagating errors are therefore induced by noise. A correlated channel could potentially aggravate the effect of noise for the second decoded symbol, but due to the ordering of detection it has a minimal impact on the first symbol. Based on the results in this section, the best error rates are achieved using (15, 5) RS codes with the VBLAST and TBLAST schemes and (15, 9) codes with the Alamouti scheme. The optimal rates are not expected to be changed by using soft decision decoding, despite lower rates experiencing a greater SD gain. For VBLAST and TBLAST, the (15, 3) code performs approximately 0.8 db worse than the (15, 5) code. Considering the soft-decision gains found in section 3.3, the (15,3) code is unlikely to be optimal even with SD decoding. When using the Alamouti scheme, the gap to the next lowest rate code is only 0.3 db. Extrapolating the results in section 3.3, once again suggest

7 132 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.107 (3) September 2016 that the (15,7) code will at best match the performance of the (15, 9) code. A natural selection for two codes where the ratio of rates is 2 is thus the (15,5) and (15,9) codes, with (15, 10) codes being used with soft-decision decoding. 3.2 VBLAST vs TBLAST The two BLAST based schemes use an identical transmission structure, that is, transmitting independent streams of information over each antenna. TBLAST offers better performance on systems with many transmit and receive antennas as shown in [6]. This is due to improved handling of CAI. On systems with few antennas, such as the 2 2 MIMO system used in this paper, CAI does not constitute as large a fraction of the total received power. VBLAST is therefore less likely to erroneously decode the first symbol for systems with few antennas than systems with many antennas (15,5) VBLAST, HD (15,9) VBLAST, HD (15,5) TBLAST, HD (15,9) TBLAST, HD Figure 11: Comparison of VBLAST and TBLAST using a 2 2 MIMO system and the B-M decoding algorithm (15,5) VBLAST, SD (15,9) VBLAST, SD (15,5) TBLAST, SD (15,9) TBLAST, SD Figure 12: Comparison of VBLAST and TBLAST using a 2 2 MIMO system and the KV decoding algorithm Figures 11 and 12 show the performance of the two BLAST based schemes using hard and soft decision decoding respectively. It is evident that VBLAST and TBLAST offer similar performance, but VBLAST offers marginally better performance at an error rate of for all four rate/decoding combinations. With hard-decision decoding VBLAST outperforms TBLAST by 0.3 db and 1.3 db for the (15, 5) and (15, 9) codes respectively. With SD decoding, VBLAST outperforms TBLAST by 0.1 db and 0.4 db for the low and high rate codes. The difference in margins is attributable to the channel induced errors experienced by TBLAST as discussed in the previous section. As a result, the increased error correction capability provides a more significant benefit to TBLAST. Although TBLAST may offer a significant gain for large antenna systems, this performance increase is not evident when there are only two transmitting and receiving antennas. This is explained by considering the energy contribution of the desired symbol and the CAI. With two transmitting antennas, the expected CAI energy is equal to the expected symbol energy, resulting in a signal-to-cai ratio (SCR) of 0 db. With 16 transmitting antennas, the expected CAI energy is 15 times greater than the expected symbol energy, which is an SCR of 11.8 db. It is clear that handling of CAI becomes significantly more important when a large number of antennas are present. The technique of ordering symbol decoding by post-detection SNR as used in VBLAST thus provides more of a benefit than iteratively estimating the CAI. It is also shown in section 3.1 that TBLAST is more likely than VBLAST to cause error propagation when the channel is spatially correlated. 3.3 Soft decision decoding gain Soft-decision (SD) RS decoding offers some improvement in error rate performance at the expense of significantly increased complexity. Analysing the complexity performance trade-off is not the objective of this paper, but the performance increase is quantified in this section. The asymptotic error correcting capability of the GS algorithm used by KV is n 1 (k 1)n, compared to a hard-decision bound of n k 2. The benefit of using soft-decision decoding thus increases as the code rate decreases. The KV algorithm offers performance exceeding the bound of the GS algorithm. The error correcting capability can however not be easily quantified, since some symbol reliability patterns allow more errors to be corrected than others. The soft-decision gain obtained using the KV algorithm is investigated for all three MIMO schemes using (15, 5), (15,9) and (15,10) RS codes. Since a (15,10) code has t = 2 while a (15,9) code has t = 3, the lower hard-decision performance of the former code should result in a larger soft-decision gain as the SD decoder is not affected by the odd number of parity symbols. Additionally, it is expected that the (15,5) code will exhibit a larger SD gain than the higher rate codes due to the increased error correcting capability. Figure 13 shows error rate performance for the three code rates over an Alamouti scheme with hard and soft decoding. This demonstrates the effect of using soft-decision decoding on RS codes when there are an odd number of parity symbols. At a SER of, the (15,9)

8 Vol.107 (3) September 2016 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 133 code exhibits a soft-decision gain of 0.9 db, compared to 1.3 db for the (15,10) code, implying a difference of 0.4 db. It can thus be concluded that the soft-decision decoder does not suffer a significant penalty from an odd number of parity symbols as compared to a hard-decision decoder. To analyse the effect of code rate on soft-decision gain, it is thus not practical to include codes with an odd (n k). This analysis is therefore performed using only the (15, 5) and (15,9) RS codes. Figures 13, 14, and 15 show the performance of the two codes in conjunction with the Alamouti, VBLAST and TBLAST schemes respectively. The soft-decision gain for all combinations of code rates and MIMO schemes is summarised in table 2. (15,5) TBLAST, SD (15,9) TBLAST, SD (15,5) TBLAST, HD (15,9) TBLAST, HD Figure 15: Comparison of soft and hard decision RS decoding with the TBLAST scheme (15,5) Alamouti, SD (15,9) Alamouti, SD (15,10) Alamouti, SD (15,5) Alamouti, HD (15,9) Alamouti, HD (15,10) Alamouti, HD Figure 13: Comparison of soft and hard decision RS decoding with the Alamouti scheme (15,5) TBLAST, SD (15,9) TBLAST, SD (15,5) TBLAST, HD (15,9) TBLAST, HD Figure 14: Comparison of soft and hard decision RS decoding with the VBLAST scheme Table 2: Summary of soft-decision gains in db at SER = (15, 5) (15, 9) Difference Alamouti VBLAST TBLAST On average, the soft-decision gain for the (15,5) code is 0.7 db higher than for the (15,9) code. This is attributable to the improved error correcting capability of soft-decision decoding for low rate codes. The three MIMO schemes exhibit approximately similar soft-decision gain for the low rate code. For both high and low rate code, VBLAST experiences the lowest SD gain compared to the Alamouti STBC and TBLAST schemes. This is due to error propagation in VBLAST increasing the likelihood of the error correcting capability being exceeded. Additionally, this indicates that VBLAST is generating less accurate soft information in comparison to the Alamouti STBC and TBLAST schemes. If an error occurs in the first decoded symbol, the error propagates to the second symbol. Both symbols have errors and are likely to have low reliability, but this does not necessarily hamper the performance of the SD decoder. However, the scenario where the first symbol is decoded correctly but the second symbol is decoded incorrectly can cause problems. The feedback mechanism which generates the soft information for the first symbol will thus back-propagate the error, resulting in the first (correct) symbol having poor reliability. The soft decision decoder is then prone to ignore the correct symbol, potentially favouring some error symbols in its place. These properties result in there being slightly less benefit to using soft-decision decoding in conjunction with VBLAST. 3.4 Transmit diversity vs code rate The primary objective of this paper is to investigate the feasibility of using low rate codes as an alternative to diversity. VBLAST is used as the rate 2 MIMO scheme, as it is shown in section 3.2 to offer better performance than TBLAST. The VBLAST scheme is paired with a (15,5) RS code and is compared to the Alamouti STBC with (15, 10) RS coding, keeping the overall rate of the two systems constant. These code rates were also shown in section 3.1 to be optimal or close to optimal for each MIMO scheme. Soft-decision RS decoding is used with both systems. Figure 16 shows a comparison between these two schemes using the KV decoding algorithm. The high diversity system outperforms the low diversity

9 134 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vol.107 (3) September 2016 (15,5) VBLAST, SD (15,10) Alamouti, SD Figure 16: Comparison of MIMO systems with equal overall rate system by 6.2 db at a symbol error rate of. This indicates a very large SER gain, implying that the high diversity system can achieve the same error rates as the low diversity system while using about four times less power. Thus, low rate channel codes are thus not a feasible alternative to transmit diversity in the MIMO systems which were evaluated. It can be concluded from this result that increasing transmit diversity is significantly more effective than decreasing code rate for the system simulated. Although this conclusion can only be drawn under the conditions of this simulation, the magnitude of the margin suggests that similar results are likely to be achieved using different channel models and MIMO schemes. The mechanism by which the high rate system achieves such good performance is that it reduces the number of symbol errors in the received vector before it is passed to the RS decoder. The number of pre-decoding symbol errors eliminated by high diversity exceeds the increase in error correcting capability of the low rate code by a significant margin. 4. CONCLUSION In this paper, the performance of three 2 2 MIMO systems have been studied for different channel code rates. Given the computer simulation results, three primary conclusions can be drawn from this study. Firstly, low rate channel codes are not a feasible alternative to transmit diversity in a 2 2 MIMO systems. Using error correction as an alternative to diversity is thus not a practical solution. Secondly, while TBLAST may offer significant performance improvements over VBLAST in systems with a large number of antennas, the two schemes perform virtually identically for a 2 2 system. Finally, there is no appreciable difference in soft-decision gain between low diversity and high diversity 2 2 MIMO schemes. ACKNOWLEDGEMENT The financial assistance of the National Research Foundation (NRF) of South Africa towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the NRF. The financial support of the Centre for Telecommunication Access and Services (CeTAS), the University of the Witwatersrand, Johannesburg, South Africa is also acknowledged. REFERENCES [1] ETSI, 3GPP TS Evolved Universal Terrestrial Radio Access (E-UTRA); Physical layer procedures, Tech. Rep., [2] IEEE Standard for Information technology Local and metropolitan area networks Specific requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 5: Enhancements for Higher Throughput, IEEE Std n-2009 (Amendment to IEEE Std as amended by IEEE Std k-2008, IEEE Std r-2008, IEEE Std y-2008, and IEEE Std w-2009), pp , Oct [3] IEEE Standard for Information technology Telecommunications and information exchange between systems. Local and metropolitan area networks Specific requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 4: Enhancements for Very High Throughput for Operation in Bands below 6 GHz. IEEE Std ac-2013 (Amendment to IEEE Std , as amended by IEEE Std ae-2012, IEEE Std aa-2012, and IEEE Std ad-2012), pp , Dec [4] L. A. M. Guzman, A Study on MIMO Mobile-To-Mobile Wireless Fading Channel Models, Master s thesis, School of Engineering and Physical Sciences, Heriot Watt University, June [5] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela, V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel, in Signals, Systems, and Electronics, ISSSE URSI International Symposium on, Sep. 1998, pp [6] M. Sellathurai and S. Haykin, TURBO-BLAST for high-speed wireless communications, in Wireless Communications and Networking Confernce, WCNC IEEE, vol. 1, 2000, pp vol.1. [7] S. Alamouti, A simple transmit diversity technique for wireless communications, Selected Areas in Communications, IEEE Journal on, vol. 16, no. 8, pp , Oct [8] I. G. Reed and G. Solomon, Polynomial codes over certain finite fields, J. Soc.Ind. Appl. Maths., pp. 8: , June [9] E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, Inc., 1968.

10 Vol.107 (3) September 2016 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 135 [10] J. Massey, Shift-register synthesis and bch decoding, Information Theory, IEEE Transactions on, vol. 15, no. 1, pp , Jan [11] Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, A method for solving key equation for decoding goppa codes, Information and Control, vol. 27, no. 1, pp , [12] L. R. Welch and E. R. Berlekamp, Error correction for algebraic block codes, Patent US , Dec 30, [13] R. Koetter and A. Vardy, Algebraic soft-decision decoding of Reed-Solomon codes, Information Theory, IEEE Transactions on, vol. 49, no. 11, pp , Nov [14] J. Jiang and K. Narayanan, Iterative Soft-Input Soft-Output Decoding of Reed amp; #8211;Solomon Codes by Adapting the Parity-Check Matrix, Information Theory, IEEE Transactions on, vol. 52, no. 8, pp , Aug [15] O. Ur-rehman and N. Zivic, Soft decision iterative error and erasure decoder for Reed #8211;Solomon codes, Communications, IET, vol. 8, no. 16, pp , [16] W. C. Jakes and D. C. Cox, Eds., Microwave Mobile Communications. Wiley-IEEE Press, [17] V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes, in Foundations of Computer Science, Proceedings. 39th Annual Symposium on, Nov 1998, pp [18] O. Ogundile and D. Versfeld, Improved reliability information for rectangular 16-QAM over flat rayleigh fading channels, in Computational Science and Engineering (CSE), 2014 IEEE 17th International Conference on, Dec 2014, pp [19] L. Godara, Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations, Proceedings of the IEEE, vol. 85, no. 8, pp , Aug [20] C. Han, X. Zhang, and Y. Chen, A novel full density QSTBC scheme with low complexity, in Image and Signal Processing (CISP), th International Congress on, vol. 03, Dec 2013, pp [21] H. Lee, Robust full-diversity full-rate quasi-orthogonal STBC for four transmit antennas, Electronics Letters, vol. 45, no. 20, pp , September [22] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol. 6, pp , 1998.