Outage Probability and Symbol Error Rate Analysis of MIMO-MRC with Estimation Error, Feedback Delay and Co-Channel Interference

— This correspondence propose the outage probability and symbol error rate (SER) performance of multiple-input multiple-output (MIMO) maximum ratio combining (MRC) in the existence of co-channel interferences (CCIs) with channel estimation error (CEE) and feedback delay (FD). In general, we derive the exact closed-form expressions for the statistical properties of the signal-to-noise-plus-interference ratio (SINR), the outage probability of the SINR and SER of M-ary constellation. Our outcome indicates that the system performance enhances with the increment of number of transceiver antenna, however deteriorates with the harshness of CEE, FD and CCIs. Key words— Channel estimation error (CEE), co-channel interference (CCI), maximum ratio combining (MRC), multiple-input multiple-output (MIMO) systems, outage probability, symbol error rate (SER). 1 Introduction WIRELESS multiple-input multiple-output (MIMO) system have drawn lot of research interest due to their potential to alleviate the performance reduction of channel impairments such as multipath fading and cochannel interference (CCI) via diversity and provide high capacity along with reliability of the system [1]-[8]. Based on the channel state information (CSI), MIMO maximal ratio combining (MRC) scheme maximize the receiver signal-to-noise (SNR) by driving the transmit power along the optimal eigen mode of the MIMO channel, was introduced in [2], [3]. By considering the impact of channel estimation error (CEE) in the absence of CCIs, a MIMO-MRC system was first introduced in [4]. Performance analyses of MIMO-MRC systems in the existence of CCIs with perfect channel estimation were explored in [5]–[6]. In [7], authors evaluated the outage probability performance of MIMO-MRC systems in the existence of CEE and CCIs, however the result derived in [7] was in the form of infinite sums of the Whittaker’s hypergeometric function. All the research work mention above was studied in the absence of feedback delay (FD). Recently, authors derived the performance of MIMO systems with CEE and FD under Ricean fading channels in the absence of CCIs in [8]. In this paper, we examine the outage probability and SER for MIMO-MRC systems in the existence of CCIs with CEE and FD. Our outcomes provide a framework of finite sums, permitting for effortless investigation of MIMO-MRC systems with respect to infinite sums Whittaker’s hypergeometric function [7]. Notation: Matrices and vectors are expressed by bold symbols. . , . . stand for expectation, transpose and complex conjugate transpose. 2 System and Channel Model Considering a MIMO MRC system employing transmit and receive antennas in the existence of independent CCIs. Then, the × 1 signal vector received at the antennas can be expressed as = + + 1 where is the transmitted signal with average received power of the desired user, ∈ !×" is a receiver noise vector and are modeled as ~ $%0, '()* !+, ∈ ,×" is the transmitted signal vector of the CCIs and = . /% ,", ,), ... , ,,+ is the average received power of the . 1 CCIs where . = 1,2, ... , . Let × channel matrix × channel matrix of the interferers are independent and identically-distributed (i.i.d.) complex Gaussian random variables (CGRVs) with zero mean and unit variance and also assume that the channel matrices and noise vector are uncorrelated of each other. In (1) represents × 1 unit energy weight vector at the transmitter, i.e., 3 3) = 1. In practice, transmit weighting factor is calculated at the receiver and forwarded to the transmitter via a feedback channel. The CEE at the WSEAS TRANSACTIONS on COMMUNICATIONS Sudakar Singh Chauhan, Sanjay Kumar E-ISSN: 2224-2864 506 Volume 14, 2015 4 = 5)6 789: ∑ <= >?@A ,BC" DEF G HI ,BD) + ∑ JKLMK?@A O PC" DEF G HI Q ,PD) + 'R)6 + 1 4 _______________________________________________________________________________________________________ receiver and FD between receiver and transmitter is consistently present. Hence, with the existence of CEE, the true channel matrix changes to be G by an estimation error b = [dBe] !× O whose elements are i.i.d. CGRVs with zero-mean and variance 'R). In this correspondence, we consider feedback channel of delay g, thus we model as [8] h = 5 G HI + b + Q 2 where 5 = ijk G Ol=mk G Ono p l=mq r"HLMK , G HI is the estimated channel matrix delayed by g time occurrence whose elements are i.i.d. $%0, 1 − 'R) +. Finally Q = G − 5 G HI is the CEE matrix incurred by the delay, consist of i.i.d $% 1 − 5) 1 − 'R) +. At the transmitter if the weighting vector = EF is picked in order to maximize the receiver SNR, the weighting vector at the receiver is selected to = G HIEF where eigenvector EF corresponding to the highest eigen value 789: of the Wishart matrix G HI G HI . Then the output signal at the receiver combiner can be written as t = = 5 EF G HI G HIEF + 'R EF G HI b EF +5'R EF G HI Q EF + EF G HI +EF G HI 3 The output signal-to-interference-plus-noise ratio (SINR) 4 of the MIMO-MRC system in the existence of CCIs with CEE and FD can be written as in (4), at the top of the page. In (4), ,B means . 1 column of the matrix , Q ,P means v 1 column of the matrix Q , 6 = '() ⁄ and 6B = B '() ⁄ . Therefore, the SINR in (4) can be expressed as the ratio of two independent random variables and given by 4 = 7 x + 1 5 where 7 = z789:, x = { + |, { = ∑ }= >?@A ,BC" DEF G HI ,BD), | = ∑ ~N >?@A O PC" DEF G HI Q ,PD)in which z = JK 7 = ˆ ˆ žPŸ Γ + 1 (8H)P P ŸC(H8 8 PC" ¡vz¢ Ÿ" 7ŸdH£¤> 10 where ¥ = ¥. , ¦ and = ¥ { , ¦ and coefficient žPŸ have been defined in [3]. Using (8) and (10), the distribution of (5) can be computed as ‚§ 4 = ̈ x + 1 © ‚>%4 x + 1 +‚‡ x x 11 = ˆ ˆ ˆ ˆ ˆ ‰BežPŸ Γ Œ Γ + 1 Ÿ" C (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" × Γ š + Œ B e ¡vz¢ a + 1 š « 4Ÿd H£¤§ a1 + P§ ¬‹=« e 12 The pdf in (12) can be evaluated by using binomial expansion as a finite sum + { ( = ∑ a v« {P (HP (PC and the integral ­ (H"dHŠ˜ © = “H(Γ . 4 Performance Analysis 4.1) Outage Probability The nonergodic capacity of MIMO-MRC systems can be written as [3] = ›/) 1 + 4 13 The outage probability may be defined as the probability that the random variable drops under particular SINR threshold 4 1, hence the outage probability is given by WSEAS TRANSACTIONS on COMMUNICATIONS Sudakar Singh Chauhan, Sanjay Kumar E-ISSN: 2224-2864 507 Volume 14, 2015 ® ̄ 4 1 = š < 4 1 = ­ ‚§ 4 ± 4, where 2 = 2§O3 − 1. ® ̄ 4 1 = ̈ ̈ x + 1 © ‚>%4 x + 1 +‚‡ x x ± 4 14 = ˆ ˆ ˆ ˆ ‰BežPŸ Γ Œ Γ + 1 (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" ̈ ̈ 1 + x Ÿ" © xeH"dH‹=‡4ŸdH £¤ ‡" § x ± 4 15 The integral ­ ́(H"dH ́ ˜ = μ , can be used to simplify (15), where μ , is incomplete gamma function and is defined as[9, (8.350.1)] μ , = Γ ¶1 − dH˜ ∑ a v« ˜£ P! (PC ·. Hence interchanging the order of integration and utilizing the binomial expansion 1 + ( as a finite sum as well as integral ­ ‹H"dHŠ˜ © = “H‹Γ  [9, 3.351.3 ], the closedform expression of the outage probability is given as ® ̄ 4 1 = ˆ ˆ ˆ ˆ ‰BežPŸ B e (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" × ̧1 − dH£¤± ˆ ˆ ¡v2 z ¢ ̄ ̄ C Ÿ ̄C aš 1« Γ š + Œ Γ Œ 1! 1 aP± ¬ + B« eo (16) 4.2) Symbol Error Rate (SER) The SER of MIMO MRC with CEE and FD in the existence of CCI over Rayleigh fading is computed by averaging the instantaneous SER » 4 over the pdf of 4. R = ̈ » 4 ∞ ‚§ 4 4 17 4.2.1 SER for M-PSK: The » 4 of M-PSK is given as [10]-[11] » 4 = 21⁄4 1⁄2r24 . ) 3⁄4 ¿À − 1 3⁄4 ̈ dHÁÂ=Ã Ä Å ÆÇÂKÈ ÄK ÄKH Ä Å É 18 For high SNR and for high value of M the » 4 of MPSK is approximated as » 4 ≅ Ë% 2Ì4+ 19 where = 2, Ì = . ) Í ». Substituting (11) and (19) into (17), we get R»HÎÏÐ = ˆ ˆ ˆ ˆ ‰BežPŸ Γ Œ B e (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" ̈ xeH"dH‹=‡ © × Ñ Ò2Ì, vz 1 + x , + 1 Ó x 20 whereÑ {, |, = |( Γ ⁄ ­ Ë © % {‚+dH…Ô‚(H" ‚, in which Ë is defined as Ë ‚ = " Í ­ dHÔK )˜B(KÕ ⁄ É ÄK [9]. Applying the identity [9] ­ {‹H"dHŠ: © { = “H‹Γ  , Ñ {, |, can be written as Ñ {, |, = Ö aÍ) , a : )…« , « where Ö ×, Ø, = 1 3⁄4 ⁄ ­ . )É . )É + Ø ⁄ ( Ù É ‚›š − 3⁄4 ≤ × ≤ 3⁄4. Utilizing the result in [11,(18)], for integer the function Ö ×, Ø, has closed form solution [12, (5A.24)]. Let a function Û , ž, ×, ¥, is defined as Û , ž, ×, ¥, = ̈ x8H" © dHÜ‡Ö a×, ž 1 + x , « x (21) For × = Í), the function Ö aÍ) , Ý "‡ , « can be decomposed as Ö a3⁄42 , ž 1 + x , « = 12 − ž 2 ˆ ˆ ÞPŸ P ŸC (H" PC xŸ ¡ 1 1 + ž + x¢ PßK 22 where ÞPŸ = a2v v « av « a"à«P. Inserting (22) into (21), the function Û , ž, ×, ¥, has a closed form expression as given below Û , ž, ×, ¥, = Û" − Û) 23 where Û" = םH8Γ ¥ 3⁄4 24 Û) = ž 2 ˆ ˆ ÞPŸ P ŸC (H" PC 1 + ž 8ŸHPH ßKΓ ¥ + × μ 1⁄2¥ + ; ¥ + − v + 12 ;  1 + ž À 25 where μ ; Ì; Ø is the confluent hypergeometric function defined by [9,(9.211.4)]. Utilizing (21) to (25), (20) can be evaluated as R»HÎÏÐ = ˆ ˆ ˆ ˆ ‰BežPŸ Γ Œ B e (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" × Û ¡ 1 B , Ìz v , 3⁄42 , Œ, + 1¢ = ˆ ˆ ˆ ˆ ‰BežPŸ 2 (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" â1 − ¡Ìz v ¢ ßK Ž WSEAS TRANSACTIONS on COMMUNICATIONS Sudakar Singh Chauhan, Sanjay Kumar E-ISSN: 2224-2864 508 Volume 14, 2015 × 1 B e ˆ ˆ a21 1 « a1š« ¡14¢ ̄ C Ÿ ̄C × Γ š + Œ Γ Œ ¡1 + Ìz v ¢e H ̄H ßK Ž× μ Ҍ + š; Œ + š − 1 − 12 ; 1 B ¡1 + Ìz v ¢Óã 26 4.2.2 SER for M-QAM: The rectangular QAM is equal to two PAM signal on quadrature carrier. Therefore SEP of M-QAM can be computed as [10] R»Häå» = 1 − a1 − R√»HÎ廫) 27 The R√»HÎå» can be obtained as R√»HÎå» = ̈ √» 4 ∞ ‚§ 4 4 28 where √» 4 = 2 a1 − " √»« Ë% 2Ì4+ in which = 2 a1 − " √»« Ì = ç ) »H" . Substituting √» 4 and (11) into (17) and utilizing (21)-(25) we get R√»HÎå» = ˆ ˆ ˆ ˆ ‰BežPŸ 2 (8H)P P ŸC(H8 8 PC" Š= eC" ) BC" â1 − ¡Ìz v ¢ ßK Ž × 1 B e ˆ ˆ a21 1 « a1š« ¡14¢ ̄ C Ÿ ̄C × Γ š + Œ Γ Œ ¡1 + Ìz v ¢e H ̄H ßK Ž× μ Ҍ + š; Œ + š − 1 − 12 ; 1 B ¡1 + Ìz v ¢Óã 29 Hence inserting (29) into (27), we can achieve SER of M-QAM. 5 Numerical Results This correspondence examines the analytical outage probability and SER performance of MIMO-MRC over uncorrelated Rayleigh fading channels. Fig. 1 illustrates the outage probability against average received SINR per branch with = 6, = '() = 1, = 5, 4 1 = 10 è, and , = 2, 2 Ì , = 4, 4 configuration in case of perfect CEE and perfect FD 5 = 1, 'R) = 0 and various imperfect cases 5 = 0.99, 'R) = 0.01 , 5 = 0.95, 'R) = 0.15 and 5 = 0.90, 'R) = 0.25 . It can be observed that the outage performance improves in case of perfect CEE and FD and with increase of antenna configuration. Fig. 2 depicts the SER against average received SINR per branch with = 4, = '() = 1, = 5, 5 = 0.99, 'R) = 0.01 for different antenna configuration using 4PSK constellation. It can be noticed that SER deteriorates on increasing the antenna configuration. 6. Conclusion In this correspondence we have analyzed the performance of MIMO-MRC system in the existence of CCI with CEE and FD. We have evaluated outage probability and SER for M-ary constellation. It can be observed that the system performance enhances with the increment of number of transceiver antenna, however deteriorates with the harshness of CEE, FD and CCIs. The performance metrics were derived analytically and avoids the necessity of integration methods. Our outcome provide a framework of finite sums, permitting for effortless investigation of MIMO-MRC systems with respect to infinite sums Whittaker’s hypergeometric function [7]. WSEAS TRANSACTIONS on COMMUNICATIONS Sudakar Singh Chauhan, Sanjay Kumar E-ISSN: 2224-2864 509 Volume 14, 2015 -5 0 5 10 15 20 25 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Average received SINR(dB)per branch O u ta g e P ro b a b ili ty