Performance analysis of MIMO-SESS with Alamouti scheme over Rayleigh fading channels

Transcription

1 University of Neraska - Lincoln DigitalCommons@University of Neraska - Lincoln Faculty Pulications from the Department of Electrical and Computer Engineering Electrical & Computer Engineering, Department of 211 Performance analysis of MIMO-SESS with Alamouti scheme over Rayleigh fading channels Shichuan Ma University of Neraska-Lincoln, sma@huskers.unl.edu Lim Nguyen University of Neraska-Lincoln, lnguyen1@unl.edu Yaoqing Lamar Yang University of Neraska-Lincoln, yyang3@unl.edu Won Mee Jang University of Neraska-Lincoln, wjang1@unl.edu Follow this and additional works at: Part of the Computer Engineering Commons, and the Electrical and Computer Engineering Commons Ma, Shichuan; Nguyen, Lim; Yang, Yaoqing Lamar; and Jang, Won Mee, "Performance analysis of MIMO-SESS with Alamouti scheme over Rayleigh fading channels" 211). Faculty Pulications from the Department of Electrical and Computer Engineering This Article is rought to you for free and open access y the Electrical & Computer Engineering, Department of at DigitalCommons@University of Neraska - Lincoln. It has een accepted for inclusion in Faculty Pulications from the Department of Electrical and Computer Engineering y an authorized administrator of DigitalCommons@University of Neraska - Lincoln.

2 RESEARCH Open Access Performance analysis of MIMO-SESS with Alamouti scheme over Rayleigh fading channels Shichuan Ma *, Lim Nguyen, Yaoqing Lamar) Yang and Won Mee Jang Astract Self-encoded spread spectrum SESS) is a novel modulation technique that acquires its spreading sequence from the random input data stream rather than through the use of the traditional pseudo-noise code generator. It has een incorporated with multiple-input multiple-output MIMO) systems as a means to comat fading in wireless channels. In this paper, we present the analytical study of the it-error rate BER) performance of MIMO-SESS systems under Rayleigh fading. The BER expressions are derived in closed form, and the veracity of the analysis is confirmed y numerical calculations that demonstrate excellent agreement with simulation results. The performance analysis shows that the effects of fading can e effectively mitigated y taking advantage of spatial and temporal diversities. For example, a 2 2 MIMO-SESS system can achieve aout 7 db performance improvement at 1-4 BER over a MIMO PN-coded spread spectrum system. Keywords: self-encoded spread spectrum, iterative detection, multiple-input multiple-output, Alamouti scheme 1 Introduction Conventional direct sequence spread spectrum system employs pseudo-noise PN) code generators which are typically linear feedack shift register circuits that generate maximal length or related sequences. In contrast, y deriving its spreading sequences from the user data stream and oviating the need for PN code generators, self-encoded spread spectrum SESS) provides a feasile implementation of random-coded spread spectrum and has a numer of unique features such as the potential for enhanced transmission security and anti-jamming capaility, multi-rate applications, modulation gain and inherent time diversity [1-7]. The modulation memory associated with self-encoding was exploited y iterative detection to achieve a significant gain [6]. Recently, SESS was incorporated with multiple-input multiple-output MIMO) Alamouti scheme to further improve the performance [8]. The spatial diversity from multiple antennas at the transmitter TX) and receiver RX) enales MIMO techniques to e effective against channel fading [9-11] and the technology has een standardized [12-14]. In a recent report [15], we have extended our work in [8] to orthogonal space-time * Correspondence: sma@huskers.unl.edu Department of Computer and Electronics Engineering, University of Neraska-Lincoln, Omaha, NE 68182, USA lock codes with multiple iterations and have shown via simulation that MIMO-SESS provides a novel means to comat fading in wireless channels y exploiting diversities in oth space and time domains. However, the theoretical analysis of MIMO-SESS in Rayleigh fading channels has not een conducted. In this paper, we thus extend our previous work in [8] to present the analytical study of the BER performance of SESS with MIMO-Alamouti scheme under Rayleigh fading. We first derived BER of the correlation detector in closed form and verified the performance improvement due to spatial diversity. Next, we derived the closed-form BER expression of the time-diversity SESS detector that further improves the performance from the comined spatial and temporal diversities. We then investigate the performance of the iterative detector that comines the outputs of the correlation and time-diversity detectors. By approximating the proaility density function pdf) associated with the time-diversity detector y a Dirac delta function, we otained a lower-ound expression for the BER of the iterative detector and verified with simulations that the ound is tight even for relatively small spreading lengths. This paper is organized as follows. MIMO-SESS system with iterative detection is descried in the next section. Section 3 presents the theoretical analysis of the 211 Ma et al; licensee Springer. This is an Open Access article distriuted under the terms of the Creative Commons Attriution License which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited.

3 Page 2 of 13 system. The BER performance of the correlation and diversity detectors is analyzed in closed form, and a lower ound for the iterative detector is derived. Section 4 presents the numerical results to validate the analysis, and Section 5 concludes the paper. Notations: The superscripts * and T represent the conjugate and the transpose operations, respectively., E{ }, and diag ) denote the asolute value of a scalar, the expectation operation, and the diagonal vector of a matrix, respectively. Matrix vector) is represented y capital small) old letter. The Q-function, gamma function, and incomplete gamma function are denoted y Q x), Γa), and Γa, x), respectively. Dirac delta function is represented y δ ). 2 System model In this section, the transceiver of the MIMO-SESS system with iterative detection is descried. 2.1 MIMO-SESS transmitter The lock diagram of the proposed transmitter is shown in Figure 1, the rounded corner locks represent N delay registers and T is the it interval. Alamouti scheme-ased space-time lock coding STBC) is utilized to achieve transmit diversity. Thesourceinformation is assumed to e i-polar values of ± E /2,E is the average transmit energy per it. It should e mentioned that the energy per transmit antenna is one half of the total transmit energy in order to make the multi-antenna system comparale to a single antenna system. The its are first spread y the self-encoded spreading sequence of length N at a chip rate of N/T. This sequence is constructed from the user s information stored in the delay registers that are updated every T. Thus, with a random input data stream, the sequence is also random and time-varying from one it to another. For example, the spreading sequence for the kth it, k), is given as sk) =[k 1)k 2),...,k N)] T, 1) and the spreading chips are given as ck = k)sk). 2) To facilitate the description in the sequel, let ck, n) denote the nth chip of the kth it, which can e expressed as ck, n) =k)k n). 3) Figure 1 Block diagram of the transmitter of the proposed MIMO-SESS system.

4 Page 3 of 13 The spreading chips are then divided into two streams y applying Alam-outi scheme, a lock of two consecutive chips, ck, 2i) andck, 2i +1),aretransmitted y sending ck, 2i), -c*k, 2i +1)tothefirst antenna, and ck, 2i + 1), c*k, 2i) to the second antenna. Here, i Î {,1,...,N/2-1} is the lock index for the chips of the kth it. The signals are then transmitted over a MIMO fading channel. The channel etween each transmit and receive antenna pair is assumed to undergo Rayleigh fading that remains constant over T ut is independent from it to it [6,16]. The channels for different transmit/receive antenna pair are independent. 2.2 MIMO-SESS receiver Figure 2 shows the lock diagram of the receiver. For the mth antenna, the received signals within the ith lock are given as x m k,2i) =h 1,m k)ck,2i)+h 2,m k)ck,2i +1)+e m k,2i),4) x m k,2i +1)= h 1,m k)c k,2i +1)+h 2,m k)c k,2i)+e m k,2i +1), 5) h 1,m k) andh 2,m k) respectively denote the normalized complex channel impulse response coefficients for the kth it etween the 1st and 2 nd transmit antennas and the mth receive antenna; e m is a Gaussian noise with zero mean and variance NN /2.Itshouldenoted that the noise is roadand ecause it is sampled at chip level. Its variance is thus the narrow-and noise variance N /2 multiplied y the spreading factor N Diversity comining We assume that the delay registers in the receiver have een synchronized with the transmitter [7], and that perfect channel knowledge is availale at the receiver. Under these assumptions, diversity comining from the M receiver antennas is carried out over two consecutive chip intervals according to [9,17] as yk,2i) = M m=1 h 1,m k)x mk,2i)+h 2,m k)x m k,2i +1))+wk,2i) = αk)ck,2i) + wk,2i), M yk,2i +1)= h 2,m k)xmk,2i) h1,mk)x m k,2i +1))+wk,2i +1) m=1 = αk)ck,2i +1) + wk,2i +1), αk) = and 6) 7) M h 1,m k) 2 + h 2,m k) 2 ) 8) m=1 M h 1,m k)e mk, n)+h 2,m k)e m k, n +1)), ifn is even m=1 wk, n) = M 9) h 2,m k)e mk, n 1) h 1,m k)e m k, n)), if n is odd m=1 Because the channel coefficients and the white noise are independent, the noise term w is a random variale with zero means and variance MNN. The comined signals given in 6) and 7) can e written in one equation as yk, n) =αk)ck, n) +wk, n), 1) and further in vector form as yk) =αk)ck)+wk), 11) Figure 2 Block diagram of the receiver of the proposed MIMO-SESS system with iterative detection.

5 Page 4 of 13 yk) =[yk,1)yk,2),..., yk, N)] T, 12) and wk) =[wk,1)wk,2),..., wk, N)] T. 13) Correlation detection After diversity comining, the chips are despread and detected. The correlation estimate of the kth it is given as ˆk) = 1 N yt k)sk) = αk)k) + v 1 k) v 1 k) = 1 N 14) N wk, n)k n) 15) n=1 is a random variale with zero mean and variance MN. The conditional SNR of the correlation detection given the channel coefficients is γ 1 = αk)k) 2 E{ v 1 k) 2 }. 16) Because the channel fading coefficients, the information its, and the thermal noise are independent of each other, it is easy to see that Equation 16) is reduced to γ 1 = E N αk). 17) A hard decision is then performed on the correlation output, which in turn is fed ack to the receiver delay registers in order to update the dispreading sequence for the next it. { 1, ˆk) > rcl k) =sgn[ˆk)] = 1, ˆk) 18) < This also provides an estimate of the corresponding it if no iterative detection is employed Time-diversity detection SESSprovidesauniqueencodingschemethatcane utilized to achieve temporal diversity. To etter illustrate this mechanism, let us write the following N 2 chips in a square matrix as αk +1)k +1)k) αk +1)k +1)k 1)... αk +1)k +1)k N +1) αk +2)k +2)k +1) αk +2)k +2)k)... αk +2)k +2)k N) Pk) = αk + N)k + N)k + N 1) αk + N)k + N)k + N 2)... αk + N)k + N)k) 19) each element represents a chip as given in 1) and each row includes the chips of one it. For simplicity, the noise term is omitted. We oserve that the kth it, k), is present in the diagonal elements of the matrix Pk). This means that the information of the kth it is effectively transmitted in the next N its, which is a unique characteristic of the SESS modulation scheme. By defining dk) = diag P k)), the diversity estimate of k) can e otained from the hard decisions of the next N its, [ k +1) k +2),..., k + N)], as k) = 1 N dk)[ k +1) k +2),..., k + N)] T + v 2 k) = 1 N αk + n)k)+v 2 k), N v 2 k) = 1 N n=1 2) N wk + n, n) k + n) 21) n=1 is a random variale with zero mean and variance MN. Notice that the effect of the possile it errors from the correlation detection has een ignored - this is justified with sufficient SNR, as shown in Section 4. The conditional SNR, given the channel coefficients for the diversity detection, is 1 N 2 n=1 N αk + n)k) 22) γ 2 = E{ v 2 k) 2. } Because ak +n), n Î {1,2,...,N}, k), and v 2 k) are independent of each other, Equation 22) reduces to γ 2 = E N αk + n). 23) NN n=1 It should e noted that the diversity detection introduces a time delay of NT and that the structure is similar to a Rake receiver employing N fingers to exploit the temporal diversity of SESS signals Iterative detection Theestimateofk) can e improved iteratively with the summation of the correlation estimate, ˆk),and the diversity estimate, k). Thus, we define k) =ˆk)+ k) = αk)+ 1 N ) N αk + n) k)+vk), n=1 24)

6 Page 5 of 13 vk) =v 1 k) +v 2 k). The conditional SNR of the iterative detection, conditioned on the channel coefficients, is the summation of the SNR of the two independent signals, as follows: αk)+ 1 ) N n=1 N αk + n) 2 k) γ = E{ vk) 2 } [ ] = E αk)+ 1 N 25) αk + n) N N = γ 1 + γ 2. n=1 The final hard decision is then otained y 1, k) > rcl k) =sgn[ k)] = 1, k) < 26) 3 Performance analysis In this section, we analyze the distriution of the SNRs given in 17), 23) and 25), and the corresponding BER expressions. We proceed with the distriution of the normalized channel gain, hk) 2. It is well known that the coefficients of the Rayleigh fading channel is generated as hk) =h r + jh i )/ 2,h r and h i are real values and normally distriuted as N, 1). The denominator 2 is for power normalization and the channel gain hk) 2 = h r 2 + h i 2 )/2, is then gamma-distriuted as Γ1, 1) [18]. Because the summation of independent and identically distriuted i.i.d.) gamma random variales still follows a gamma distriution [18], the spatial diversity gain ak) givenin8)hasaγ2m, 1) distriution. Furthermore, the conditional SNR of the correlation detector given in 17) follows Γ2M, E /N ) distriution, i.e., γ γ 2M 1 e E /N p γ1 γ )= 2M 1)!E /N ) 2M, γ. 27) Now the it-error proaility for inary phase shift keying BPSK) in AWGN with an SNR of E /N is given in [16] as P = Q 2E N ) 28) Given the conditional SNR of p γ1 γ ), the BER expression for the correlation detection is given y [[16], Eq ]: P e,corr = = ) Q 2γ p γ1 γ )dγ [ ] 1 2M 2M μ 1) i= 2M 1+i i )[ 1 ] i 29) μ 1) E /N μ 1. 3) 1+E /N A similar analysis shows that the conditional SNR of the time-diversity detector given in 23) follows a Γ2NM, E /N /N) distriution, i.e., γ 2NM 1 e E /N /N p γ2 γ )= / / 2NM 2NM 1)!E, γ. 31) N N) and the BER expression for the diversity detection is then otained as P e,div = = ) Q 2γ p γ2 γ )dγ [ ] 1 2NM 2NM μ 2) i= γ 2NM 1+i i )[ 1 ] i 32) μ 2) / / E N N μ 2 / /. 33) 1+E N N It should e noted that the correlative detector has an M-fold spatial) diversity, as the time-diversity detector has an MN-fold spatial and temporal) diversity. For the iterative detection, since the conditional SNR g is the summation of the independent conditional SNRs g 1 and g 2,thepdfofg is the convolution of the gamma pdfs of g 1 and g 2. This is written as follows: p γ γ ) = p γ1 γ ) p γ2 γ ) γ M 1 e γ = ) γ NM 1 e γ ) E/N/N E/N 34) M 1)!NM 1)!E /N ) M / / NM E, N N) γ. The BER expression can then e otained y averaging Q 2γ ) over the conditional pdf of g. The complexity of such a calculation can e oviated with the oservation that the pdf p γ2 γ ) approaches a Dirac delta function as N ecomes large. This stems from the fact that the gamma distriution approaches a Gaussian distriution if the degree of freedom, here 2NM, is large [18]. Furthermore, the Gaussian distriution tends toward a Dirac delta function as its variance, here 2NME /N /N) 2 = 2ME /N ) 2 /N, ecomes very small [18]. It follows that if the spreading factor N is sufficiently large, the distriution of p γ2 γ ) can e approximated y an impulse located at the mean value of g 2, which is given as 2NME /N /N )=2M E / N ). In other words, p γ2 γ ) is approximatively equal to δg -2ME /N ).

7 Page 6 of 13 So as N ecomes large, the pdf of g 2 approaches an impulse and the pdf of g can e otained as the shifted pdf of g 1 : / p γ γ ) p γ1 γ ) δγ 2ME N ) / γ 2ME N / = γ 2ME/N) 2M 1 e E N, γ 2M E 2M 1)!E/N) 2M N else. 35) The BER of the iterative detector can e calculated as Pe = ) Q 2γ pγ γ )dγ ) / ) 2M 1e γ 2ME/N γ 2ME N E/N Q 2γ / 2M 1)!E N) 2M dγ 2ME/N ) = Q 4M E N ) 2M 1 μ1 N k k 2π e2m k E k k= k! μ1) k l 2M E l N l= ) l Ɣ k l + 12 )),2M 1+ E N 36) Appendix A presents the derivation of this result. We note that the impulse approximation of the pdf yields a lower ound for the BER, ecause it can only e approached when the spreading factor N is sufficiently large. 4 Analytical and simulation results 4.1 Proailistic models In this section, we present the distriution of the conditional SNRs and discuss the delta approximation related to the iterative detection. Figure 3 shows the proaility density functions of the conditional SNRs of the correlation detection g 1, diversity detection g 2, and iterative detection g, for different scenarios. Each su-figure includes an extra curve that shows the distriution of g 1 that has een right-shifted y 2M E N. The coordinates for different su-figures have een adjusted in order to illustrate the distriutions in detail. As analyzed in γ 1 γ 2 γ shifted γ γ 1 γ 2 γ shifted γ 1 Proaility density Proaility density Conditional SNR Conditional SNR a) 2 2, N=8, SNR=1 db ) 2 1, N=8, SNR=1 db.35 γ γ γ 2 γ shifted γ γ 2 γ shifted γ 1 Proaility density.2.15 Proaility density Conditional SNR Conditional SNR c) 2 2, N=64, SNR=1 db Figure 3 Distriution of the conditional SNRs. d) 2 2, N=64, SNR= db

8 Page 7 of 13 Section 3, the mean and variance of the conditional SNR of the diversity detection g 2 are 2M E N and 2M N E N ) 2, respectively. This means that while the numer of receive antennas and the SNR affect oth position and shape of the pulse, the spreading length only influences its shape. Figure 3a and 3 show that the position and the shape of the pulse vary with the numer of antennas. In oth cases, the pulse is not very narrow due to the small value of N = 8, resulting in a discernale difference etween the shifted version of the distriution of g 1 dotted curve) and the distriution of g dashed curve). The pulse looks sharper in Figure 3c since N = 64 is much larger. Because the area under the pdf is fixed, this sharper pulse approximates a delta function such that there is negligile difference etween the shifted distriution of g 1 and the distriution of g. Also, Figure 3c and 3d shows that the delta approximation is not very sensitive to SNR, although it is more accurate with lower SNR. 4.2 Performance comparisons Next,wepresenttheBERperformanceofMIMO-SESS and compared with a conventional MIMO, PN-coded spread spectrum MIMO-PNSS) system. The channel etween each transmit and receive antenna pair is assumed to e Rayleigh fading that remains constant over T ut is independent from it to it [6,16]. In addition, the channels for different transmit/receive antenna pair are independent. The length of the spreading sequence is set to N = 64 unless noted otherwise. For each scenario and it signal-to-noise ratio SNR), 1 runs of 1, its are simulated to evaluate the average it error rate BER). Figure 4 compares the BER of MIMO-SESS with correlation detection to a MIMO-PNSS system. It is BER MIMO PNSS 2x1 MIMO PNSS 2x2 MIMO SESS 2x1 Corr Sim MIMO SESS 2x2 Corr Sim MIMO SESS 2x1 Corr Ana MIMO SESS 2x2 Corr Ana SNR db) Figure 4 Comparison etween MIMO-PNSS and MIMO-SESS with correlation detection, N = 64.

9 Page 8 of 13 oserved that without iterative detection, there is no time diversity gain, and thus, the performance with or without self-encoding is the same as expected under high SNR. The performance degradation of MIMO- SESS at low SNR is due to error propagation [1]. In particular, Figure 4 shows that the effect of error propagation is quite severe for a BER greater than 1%. The effect of error propagation can e efficiently alleviated with more antennas, as shown y the example 2 2 scenario, since larger spatial diversity reduces BER to elow 1% even at low SNR. The plots in Figure 4 show excellent agreement etween the analytical and simulation results at high SNR, error propagation ecomes insignificant. The performance of MIMO-SESS with time-diversity detection is shown in Figure 5. The plots show that the proposed system significantly outperforms conventional MIMO-PNSS systems. As an example, there is aout 4.5 db gain for the 2 2 configuration at 1-4 BER. This performance improvement can e attriuted to the N- fold time diversity introduced y self encoding. Due to the significantly improved BER, error propagation is effectively mitigated as shown y the 2 2 scenario. Moveover, the agreement with the simulation results verifies that the analysis has een justified in ignoring error propagation. Figure 6 compares the BER with the iterative detector. Again, there is excellent agreement etween the analytical and simulation results at sufficiently high SNR. The performance gain over conventional MIMO-PNSS systems is almost 7 db at 1-4 BER for the 2 2 configuration. The excess gain eyond the expected 3 db SNR improvement can e attriuted to the diversity gain from the time-diversity detection. In fact, the plot shows that a 2 2 MIMO-SESS would require an SNR of only 3dBtoachievea1-4 BER. The results verify the BER MIMO PNSS 2x1 MIMO PNSS 2x2 MIMO SESS 2x1 T D Sim MIMO SESS 2x2 T D Sim MIMO SESS 2x1 T D Ana MIMO SESS 2x2 T D Ana SNR db) Figure 5 Comparison etween MIMO-PNSS and MIMO-SESS with diversity detection, N = 64.

10 Page 9 of 13 veracity of the performance analysis and demonstrate that MIMO-SESS can completely mitigate the effect of Rayleigh fading. The difference etween the numerical and simulation results at low SNR is due to error propagation. This can e easily demonstrated in simulation y feeding the original transmitted) its into the delay registers at the receiver. The results of this simulation are compared with the numerical calculations in Figure 7. The excellent agreement etween the simulation and numerical results demonstrates the validity of the statistical models and approximations. Figure8showsthenumericalresultsofMIMO-SESS with the iterative detection for different spreading lengths. For oth 2 1 and 2 2 cases, the performance approaches the lower ound when the spreading length increases. It should e noted that the lower ound is very tight for N 64. Also, the lower ound is tighter for the 2 1 case and at lower SNR. These results are consistent with the discussion in Section 3. It can e seen y comparing the lower ound and the performance of BPSK in AWGN that 2 2 MIMO-SESS can provide aout 5.2 db gain over BPSK at 1-4 BER. This significant performance improvement is clearly due to the comined temporal and spatial diversities associated with MIMO-SESS design. 4.3 Remarks The effect of error propagation is largely confined to high BERs >1-2 )astheiterrorsarefedacktothe de-spreading chips and propagate through the receiver delay registers. Thus, the enefit of MIMO reducing the effect of error propagation occurs in the low SNR regions primarily through the 3 db array gain) as shown y the simulation results. However, the performance gaps etween error propagation and no error propagation true its) remain for low BER. The BER MIMO PNSS 2x1 MIMO PNSS 2x2 MIMO SESS 2x1 Iter Sim MIMO SESS 2x2 Iter Sim MIMO SESS 2x1 Iter Num MIMO SESS 2x2 Iter Num SNR db) Figure 6 Comparison etween MIMO-PNSS and MIMO-SESS with iterative detection, N = 64.

11 Page 1 of BER MIMO SESS 2x1 Iter Sim TrueBits) MIMO SESS 2x2 Iter Sim TrueBits) MIMO SESS 2x1 Iter Num MIMO SESS 2x2 Iter Num SNR db) Figure 7 Comparison etween numerical results and simulation results without error propagation, N = 64. comparison with the analytical results also showed that the discrepancies are primarily in high BER, low SNR regions. This suggests that in practical systems the BER is sufficiently low, the effect of error propagation and hence its mitigation with MIMO scheme are rather minimal and relatively insignificant. For example at 1-3 BER, the performance gap is negligile for correlation and time diversity detectors, and amounts to less than.5 db for the iterative detector. The results showed that MIMO scheme provides spatial diversity under fading that is most eneficial to correlation detection which has no diversity). Time diversity detection on the other hand can achieve N-fold diversity that completely mitigates the effect of fading; so there is very little spatial diversity enefit from MIMO scheme especially under high SNR. Since iterative detection comines time diversity and correlation detection, it also enefit from spatial diversity ut to a lesser extent. The advantage of the comination of MIMO and SESS therefore is to exploit the array gain to mitigate error propagation at low SNR and the spatial diversity to improve the overall BER. 5 Conclusion In this paper, we presented the analytical study of the BER performance of MIMO-SESS over Rayleigh fading channels. We derived the closed-form BER expressions for the correlation detector and time diversity detector, and otained a lower ound for the iterative detector. The veracity of the analysis has een confirmed with the numerical calculations ased on the analytical expressions, which have demonstrated excellent agreement with the simulation results. The performance analysis has shown that MIMO-SESS offers almost 7 db of gain at a 1-4 BER compared with a 2 2 MIMO PN-coded spread spectrum system. Furthermore, the system

12 Page 11 of BER BPSK AWGN) MIMO SESS Iter Num N=16 MIMO SESS Iter Num N=32 MIMO SESS Iter Num N=64 MIMO SESS Iter Num N=128 MIMO SESS Iter Num N=256 MIMO SESS Iter LowerBound SNR db) Figure 8 Performance of the proposed MIMO-SESS system with various N values. 2 2 requires only aout 3 db SNR to achieve a BER of 1-4. This demonstrates that MIMO-SESS can provide a very effective means to exploit oth spatial and temporal diversities in order to achieve roust performance in wireless environments. Although we have examined an Alamouti schemeased MIMO system with two transmitting antennas, the MIMO-SESS analysis that has een developed here can e easily applied to other orthogonal space-time lock coded MIMO systems with any numer of antennas at transmitter side. Additional study is eing undertaken to extend this work toward multi-code MIMO- SESS that can improve the transmission rate, system throughput, and spectral efficiency. P e = = a Q Q 2γ ) γ a) L 1 e γ a L 1)! L dγ 2x + a) ) x L 1 e x L 1)! L dx 1 e u2 2π 2x+a) 2 du xl 1 e x L 1)! L dx By changing the order of the integrals, 37) ecomes P e 1 1 2π L 1)! L 2a e 37) u 2 2 P 1 du 38) Appendix A Derivation of Equation 36 This appendix provides a detailed derivation of Equation 36). Define a =2ME /N, = E /N,andL =2M, then the BER of the iterative detector can e written as P 1 = u 2 2 a x L 1 e x dx 39)

13 Page 12 of 13 Let z = x, then P 1 = L u 2 2 a z L 1 e z dx 4) Using the indefinite integral equation given in [[19], Eq ], P 1 can evaluated as P 1 = L e u2 2 + a L 1 L 1)! u 2 k! 2 a ) k ) L L 1)! 1 e u2 2 + a k=1 Sustituting 41) into 38) to determine P e : P e P 2 = 2a 1 e u2 2 du 1 e a 2π 2π 1 e a L 1 2π k=1 1 k! ) = Q 2a 1 e a 2π 2a e u 2 2 ) 1 k e 2 2a L 1 k= 1 k! 2a e u2 2 u ) 1 k P 2 2 ) 1+ du u 2 2a ) k du 41) 42) ) 1+ u 2 2a ) k du 43) This can e further transformed with the sustitution 1+ t a 2 P 2 = 2 k 1 e t t a1+) 2 = 2 k 1 e t t 1 k k 2 1+ l a1+) l= ) 2 k k = 2 k 1 1+ l 1+ l= ) k dt ) 1+ t 2 P 2 = 2 k 1 e t t a1+) 2 = 2 k 1 e t t 1 k k 2 1+ l a1+) l= ) 2 k k = 2 k 1 1+ l 1+ l= ) k l a) l dt ) k l a) l Ɣ k l ) k 1+ t a dt ) 1+ t : ) a1 + ), ) k l a) l dt ) k l a) l Ɣ k l ) a1 + ), 44) Γa, x) denotes the incomplete gamma function with shape a and scale x [[2], Eq ]. Sustituting 44) into 42), we have: ) P e Q 2a 1 e a 2π L 1 1 k k! k= k k l l= 1+ ) 1+ ) k l a) l Ɣ k l + 1 ) 45) a1 + ), 2 The final expression for P e in 36) can e otained with the defined values for a,, and L: ) Pe Q 4M E N ) 2M 1 μ1 N k k 2π e2m k E k k= k! μ1) k l 2M E l N l= μ 1 is given y 3). ) l Ɣ k l + 12 )) 46),2M 1+ E N Acknowledgements This work was funded in part y contract award FA from the US Air Force Office of Scientific Research. The authors wish to thank Dr. J. Sjogren for his support of this study. This paper was presented in part at IEEE International Conference on Communications ICC), Cape Town, South Africa, May 21, and at ISITA/ISSSTA21, Taichung, Taiwan, Oct. 21. Competing interests The authors declare that they have no competing interests. Received: 28 April 211 Accepted: 27 Octoer 211 Pulished: 27 Octoer 211 References 1. L Nguyen, Self-encoded spread spectrum communications, in Proceedings of Military Communications Conference Proceedings MILCOM 1999). 1, ) 2. L Nguyen, Self-encoded spread spectrum and multiple access communications, in Proceedings of IEEE Sixth International Symposium on Spread Spectrum Techniques and Applications, vol. 2. Parsippany, NJ, 2), pp W Jang, L Nguyen, Capacity analysis of m-user self-encoded multiple access system in AWGN channels, in Proceedings of IEEE Sixth International Symposium on Spread Spectrum Techniques and Applications. 1, ) 4. Y Kong, L Nguyen, W Jang, Self-encoded spread spectrum modulation with differential encoding, in Proceedings of IEEE Seventh International Symposium on Spread Spectrum Techniques and Applications. 2, ) 5. WM Jang, L Nguyen, M Hempel, Self-encoded spread spectrum and turo coding. J Commun Netw. 61), ) 6. YS Kim, W Jang, Y Kong, L Nguyen, Chip-interleaved self-encoded multiple access with iterative detection in fading channels. J Commun Netw. 91), ) 7. K Hua, L Nguyen, W Jang, Synchronisation of self-encoded spread spectrum system. Electron Lett. 4412), ). doi:1.149/el: S Ma, L Nguyen, WM Jang, Y Yang, Multiple-input multiple-output selfencoded spread spectrum system with iterative detection, in Proceedings of IEEE International Conference on Communications ICC 1), Cape Town, South Africa, 21) 9. A Paulraj, R Naar, D Gore, Introduction to Space-Time Wireless Communications, Camridge University Press, Camridge, UK, 23) 1. D Gesert, M Shafi, D Shan Shiu, PJ Smith, A Nagui, From theory to practice: an overview of MIMO space-time coded wireless systems. IEEE J Sel Areas Commun. 213), ). doi:1.119/jsac AJ Paulraj, DA Gore, RU Naar, H Bölcskei, An overview of MIMO communications a key to gigait wireless. Proc IEEE. 922), ). doi:1.119/jproc IEEE, IEEE Standard for Local and Metropolitan Area Networks, Part 16: Air Interface for Fixed Broadand Wireless Access Systems. IEEE Std ) 13. IEEE, IEEE Candidate Standard 82.11n: Wireless LAN Medium Access Control MAC) and Physical Layer PHY) Specifications. 24) 14. 3GPP Technical Specification Group Radio Access Network, Overall description of E-UTRA/E-UTRAN; stage 2. Tech Rep TR ) 15. S Ma, L Nguyen, WM Jang, Y Yang, MIMO self-encoded spread spectrum with iterative detection over rayleigh fading channels. Hindawi J Electr Comput Eng. 21, ). Article ID JG Proakis, Digital Communications, 4th edn. McGraw-Hill, New York, 1989) 17. S Alamouti, A simple transmit diversity technique for wireless communicaiton. IEEE J Sel Areas Commun. 168), ). doi:1.119/

14 Page 13 of PG Hoel, SC Port, CJ Stone, Introduction to Proaility Theory Houghton Mifflin, Boston, 1971) 19. L Rade, B Westergren, Mathematics Handook for Science and Engineering, Birkhauser Inc., Boston, Camridge, MA, USA, 1995) 2. M Aramowitz, IA Stegun, Handook of Mathematical Functions With Formulas, Graphs, and Mathematical Tales Dover, New York, 1965) doi:1.1186/ Cite this article as: Ma et al.: Performance analysis of MIMO-SESS with Alamouti scheme over Rayleigh fading channels. EURASIP Journal on Wireless Communications and Networking :145. Sumit your manuscript to a journal and enefit from: 7 Convenient online sumission 7 Rigorous peer review 7 Immediate pulication on acceptance 7 Open access: articles freely availale online 7 High visiility within the field 7 Retaining the copyright to your article Sumit your next manuscript at 7 springeropen.com