Joint complex regularised zero‐forcing equalisation and CFO compensation for MIMO SC‐FDMA systems
2016; Institution of Engineering and Technology; Volume: 10; Issue: 16 Linguagem: Inglês
10.1049/iet-com.2016.0059
ISSN1751-8636
AutoresM. Ibrahim, Fatma Newagy, I.M. Hafez,
Tópico(s)Advanced Power Amplifier Design
ResumoIET CommunicationsVolume 10, Issue 16 p. 2245-2251 Research ArticlesFree Access Joint complex regularised zero-forcing equalisation and CFO compensation for MIMO SC-FDMA systems Mohamed Mostafa, Corresponding Author Mohamed Mostafa m.mostafa@sha.edu.eg Department of Electronics and Communications, The Higher Institute of Engineering, El Shorouk Academy, Cairo, EgyptSearch for more papers by this authorFatma Newagy, Fatma Newagy Department of Electronics and Communications, Faculty of Engineering, Ain Shams University, Cairo, EgyptSearch for more papers by this authorIsmail Hafez, Ismail Hafez Department of Electronics and Communications, Faculty of Engineering, Ain Shams University, Cairo, EgyptSearch for more papers by this author Mohamed Mostafa, Corresponding Author Mohamed Mostafa m.mostafa@sha.edu.eg Department of Electronics and Communications, The Higher Institute of Engineering, El Shorouk Academy, Cairo, EgyptSearch for more papers by this authorFatma Newagy, Fatma Newagy Department of Electronics and Communications, Faculty of Engineering, Ain Shams University, Cairo, EgyptSearch for more papers by this authorIsmail Hafez, Ismail Hafez Department of Electronics and Communications, Faculty of Engineering, Ain Shams University, Cairo, EgyptSearch for more papers by this author First published: 01 November 2016 https://doi.org/10.1049/iet-com.2016.0059Citations: 16AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In single carrier frequency division multiple access (SC-FDMA) systems, frequency domain equalisers suffer performance degradation due to several channel impairments such as inter-symbol interference, inter-carrier interference (ICI), especially in multiple access/multiple input multiple output (MIMO) systems. This study presents a joint complex regularised zero forcing (JCRZF) equaliser that is able to compensate the channel impairments and to mitigate the ICI induced by the carrier frequency offset (CFO) in mobile channel environments. JCRZF equaliser is also able to eliminate the inter antenna interference resulting from the MIMO channel coupling by transforming the MIMO channel into equivalent SISO channels. Simulation results show that the JCRZF equaliser outperforms ZF based equalisers in terms of bit error rate while preserving their potential complexity and appropriate robustness to CFO estimation errors. 1 Introduction Recently, many reports have focused on the analysis and evaluation of the transmission performance of SC-FDMA systems [1-3]. Two main issues have been considered as the source of performance degradation when investigating the transmission performance of SC-FDMA systems, namely, the peak to average power ratio (PAPR) at the transmitter side and the carrier frequency offset (CFO) at the receiver side. SC-FDMA systems can be implemented in two possible formats, the localised and the interleaved FDMA. The impact of the CFO on both types of SC-FDMA systems have been studied in [4, 5]. In general, the subcarriers orthogonally relationship can be destroyed by CFO which lead to inter-carrier interference (ICI) [6, 7]. However, interleaved FDMA systems are more sensitive to CFO [4]. Moreover, in the case of interleaved SCDMA, another two common sources of performance degradation in both types of SC-FDMA systems are the single user interference and the multiuser interference. For the localised SC-FDMA, the two interference sources are almost comparable, while in interleaved SC-FDMA, the BER is worse by almost two orders of magnitude [5]. In recent years, many techniques have been reported to investigate the problem of CFO induced in SC-FDMA systems [8-10]. However, few of these techniques have been concerned with the problem of joint channel equalisation and CFO compensation, particularly in multi user systems employing MIMO diversity. The impact of CFO in SC-FDMA systems can be mitigated using several SC-FDE techniques, which are preferred as they preserve low PAPR [11]. Zero forcing is one of the most frequently used detectors in SC-FDMA systems [12]. Despite its low complexity, it suffers noise enhancement problems. The single user detectors suffer from increasing the system complexity sense it require a DFT block for each user [13]. Circular convolution detectors are also used to compensate for the CFO, however, they suffer from the error floor problem as a result of multi access interference (MAI) [14]. In [15], a multi user SISO equaliser has been proposed for joint CFO compensation and channel equalisation. This equaliser suffers from its high complexity as the equaliser is implemented in the frequency domain before the de-mapping process where the number of subcarriers is larger than that after the de-mapping. In [16], a JLRZF equaliser is proposed in order to overcome the noise enhancement problem in ZF equaliser. Multiple access interference and inter antenna interference (IAI) terms are combined to form a single effective noise term. The BER is optimised with respect to the signal to effective noise ratio. The authors in [16] could overcome the noise enhancement problem by defining a fixed real valued regularisation parameter which mitigates the CFO induced ICI by equalising the impact of the CFO jointly with the channel induced inter-symbol interference (ISI). This paper presents a novel FDE equalisation scheme in which the ICI can be mitigated more effectively by separating the CFO induced ICI from the desired user signal and associating it to the effective noise. In contrast to [16], the ICI becomes a part of the effective noise rather than the MIMO channel matrix and hence; could be mitigated more effectively using a complex regularisation parameter instead of the real valued parameter introduced in [16]. The ICI can be separated from the desired signal by properly truncation of a Taylor series expansion for the CFO matrix. Moreover, the bit error rate (BER) is optimised with respect to a complex valued regularisation parameter, which in turn depends not only on the signal to noise ratio (SNR) as in [16], rather, on the signal to interference ratio and on the signal to effective noise ratio. Theoretical analysis, validated by simulation results, show that the proposed equaliser outperforms the conventional ZF [12] and the JLRZF [16] equalisers in terms of the BER. The rest of the paper is organised as follows. The MIMO SC-FDMA systems model is reviewed in Section 2. Section 3 presents the mathematical formulation of the proposed equaliser joint complex regularised zero forcing (JCRZF). In Section 4, complexity of the proposed JCRZF scheme is derived. In Section 5, simulation results are presented to validate the theoretically obtained expressions in Section 3. The whole paper is concluded in Section 6. Throughout the rest of the paper, boldface symbols and letters are used to designate vectors and matrices. The symbols represent the matrix Hermition and matrix inverse, respectively. The Kronicker product is denoted by ⊗. 2 MIMO SC-FDMA system model The structure of a spatial multiplexing SC-FDMA 2 × 2 system is shown in Fig. 1. This transceiver configuration is the same for all U users in the system. Time synchronisation is assumed among all users. The data of each user is spatially de-multiplexed via two independent transmit antennas, emitted over the channel and received by two receive antennas of the corresponding receiver. Throughout the rest of the paper, the user which has the index k is the desired user of interest while the index u refers to a general user, u ε {1, 2, …, U}. Fig. 1Open in figure viewerPowerPoint 2 × 2 SM SC-FDMA transceiver a Transmitter for user u b Receiver In the transmitter of a user u, binary data emitted from the information source is encoded by a convolutional encoder, mapped to complex base band QPSK symbols according to its constellation diagram in order to form a modulated data vector , demultiplexed into two sequences on a symbol-by-symbol basis. Each output sequence of the de-multiplexing block where j ε {1, 2} is transformed into frequency-domain via an N-points discrete Fourier transform (DFT) matrix as and is the unitary DFT matrix whose entries are defined as where n, m =, 0, 1, …, N − 1 and therefore, . The N samples representing the orthogonal subcarriers in the resulting frequency domain vector are mapped into M subcarriers as . . Q Is the bandwidth expansion factor and is the mapping matrix defined as follows (1)where denotes Q′ × N all-zero matrix and denotes a unit column vector of length N. Non-allocated subcarriers in the mapping matrix are forced to zero. The mapping process is followed by an M-points IDFT operation which transforms frequency domain symbol back to the time domain . Again, the M-points matrix is the inverse of the corresponding DFT matrixFM and is a unitary matrix whose entries are given by therefore, . Each component vector is extended by a cyclic prefix (CP) vector of NC samples, inserted at the head of each symbol. In matrix form, the cyclic prefix addition is performed by a multiplying by a CP matrix where and IM denotes an M × M identity matrix. The resulting vectors are transmitted via the transmit antenna pair of the uth user. (2)Both vectors emitted by each of the transmit antennas are received by each of the receive antennas due to the coupling nature of the MIMO channel. At each receive antenna, the CP vector is suppressed and an M-points DFT matrix FM converts the time domain vector to the frequency domain as (3)where is a 2M × 1 vector representing the transmitted frequency-domain samples from the uth user after the mapping process. is the frequency domain MIMO channel matrix which can be written as (4)where is a diagonal matrix whose entries are identical and independent distributed (i.i.d) Rayleigh fading random variable representing the frequency response of the channel between the jth transmit antenna and the ith receive antenna. Moreover, the channel is assumed to show a quasi-static behaviour over single symbol duration. is the frequency domain noise vector whose entries are i.i.d complex Gaussian and is the noise power. is the MIMO interference matrix, and is a circulant matrix representing the CFO experienced by the subcarriers of the uth user channel. is a diagonal matrix whose entries are defined as , m = 0,…, M−1, which describes the CFOs matrix of the mth subcarrier of the uth user. ɛu Is the CFO of the uth user normalised by the subcarriers spacing. A demapping matrix is applied to the frequency domain vector, is determined by the transposition of such that . The vector resulting from the demapping process is given by (5)where is the received vector resulting from the demapping process, is the frequency domain MIMO channel matrix after the demapping process, defined as , Denotes the complex noise after the demapping process, and . For small values of ɛu CFO matrix can be approximated by its truncated Taylor series expansion where [D]m,m = 2πm/M, and is an M × M identity matrix. , Thus let . Thus . The effective MIMO channel matrix of the uth user can be defined as or equivalently However, for the desired user, the channel effective matrix can be decomposed as follows, where = . For a MIMO receiver, the matrix product can be decomposed as The received frequency domain vector in (5) is given by (6)where is the noise plus MAI and ICI vector. The effective MIMO channel matrix can be transformed into the CFO free channel matrix by properly employing a compensation process for the impressed CFO. The constituting matrices of are given by . The compensation matrix is easy to construct at the receiver and introduced in the effective MIMO channel matrices . Consequently, the CFO free channel matrix can be perfectly estimated once the CFO is known. The matrix can be used to compensate for the CFO as follows However, the estimation of CFO is subject to error and ɛ may be estimated as . The accuracy of the estimated matrices depends on the estimated CFO. Thus, the matrix will be estimated as where is the estimated CFO. The term should equal I iff . If there is an error in the CFO, the estimated CFO free channel matrix becomes where Δɛ is the error in the estimated CFO. The variation of the BER performance with the error in the estimated CFO is studied in Section 5 of the simulation results and analysis. Throughout the paper, is referred to as the effective noise vector. The index M refers to the demapping results. In Section 3, the demapped frequency domain vector is to be equalised by the proposed FDE equaliser in order to mitigate the ISI, MAI, ICI and eliminate the IAI. N-points IDFT operation is applied to the equalised frequency domain vector for purpose of symbol detection in the time domain. The resulting time domain vector is subsequently demultiplexed, demodulated and finally decoded for the purpose of BER calculation. 3 Proposed JCRZF equaliser As observed from (6), the two components of the transmitted vectorX are linearly combined by the MIMO channel matrices and . Due to the mutual IAI. The proposed FDE equalisation scheme is performed in a two-step process. First a matrix W1 is applied to the received vector in order to eliminate the channel coupling caused by the IAI in the MIMO channel such that the MIMO channel is transformed into two separate SISO channels. In the second step, the role of equaliser is then to compensate for the effect of equivalent SISO channels as seen by equaliser. 3.1 Transformation of the MIMO channel into two equivalent SISO channels Ideally, the components of the transmitted vector X should be received by their respective antennas 1 and 2 of the transmitter, However, due to the IAI caused by the MIMO channel matrices each antenna receives a hybrid signal from the transmitter. The noise free received vector at each receive antenna after the CP suppression and applying the CFO compensation matrix processes is given by (7) (8)The MIMO channel can be transformed into two equivalent and independent SISO channels by applying a matrix , which is given by (9)where , and . Assuming perfect knowledge of the channel state information, the matrix can be constructed and the received vector is thus given by (10)where and are the two equivalent SISO channels of the kth user. 3.2 Equalisation and ICI mitigation The equalisation and ICI mitigation matrix W2 arises as a natural solution for the matrix that leads to the minimisation of the Euclidian distance mean squared error cost function given by (11)The MMSE solution is given by (12)where , and are the covariance matrices of the effective noise at the first and the second antennas, respectively. and are the variances of transmitted signals from the first and the second antennas, respectively. The effective noise terms and at the input of the first and second antennas, respectively, is given by (13a) (13b)where and . Now let us define (14a) (14b)Note that the ratio is defined as the regularisation parameter α as in [16]. In order to derive a mathematical expression for the regularisation parameter, the covariance matrices of the effective noise terms in (13a), (13b) should be evaluated. The covariance matrix of the effective noise at any of the receive antennas is given by (15)where i = {1, 2} corresponding to the first and second receive antennas, respectively. From the properties of the expectation operator, the covariance matrix of the effective noise is given by (16)where is ICI covariance matrix and is the cross correlation between effective noise and the ICI. Note that, in contrast to [16], the autocorrelation matrix has real and imaginary components due to considering the additive ICI term in (6). The real and imaginary components are given by and respectively. Let and . Similarly, for the second receive antenna and where is calculated as the resulting regularisation parameter in this case will be referred to as the complex regularisation parameter and is given by . A major difference between the equalisers proposed in [12, 16] is that, α = 0 in the conventional ZF equaliser [12], leading to the noise enhancement problem associated with ZF equalisation, while in [16] α is a pure real valued parameter that accounts for the noise plus MAI. In our proposed scheme, an imaginary part of the regularisation parameter arises due to considering the additive ICI term in (6). Note that, without CFO, the ICI matrix and the regularisation parameter is again real valued and the proposed equalisation scheme is reduced to JLRZF [16]. Note that all equalisation schemes are equivalent to conventional ZF equaliser and can account for the ISI of the channel by setting b = 0. Effective noise and ICI can be controlled only through tuning the regularisation parameter. In terms of the complex valued regularisation parameter, the proposed FDE-MMSE equaliser is given by (17) 4 Complexity analysis Obviously, the MIMO equalisation matrix consists of two independent diagonally concatenated N × N SISO equalisation matrices. The first matrix is responsible for equalising the vector , while the second matrix is responsible for equalising the vector . The direct implementation of the proposed FDE-MMSE equaliser using the full channel matrices and requires the inverse of these two matrices. Thus, the total number of complex arithmetic operations involved in the whole equalisation process is in the order of [17], which is too large for large N and not suitable for practical high data rate SC-FDMA systems. However, the full channel matrices and can be approximated by banded matrices with bandwidth τ [18], if the resulting banded matrices do not have a significant influence on the BER performance. The banded matrix approximation allows the use of lower number of complex arithmetic operations. According to Cao et al. [17], the complexity of equalisation is expended in calculating the banded matrix–matrix multiplication , which costs approximately flops, lower- upper (LU) factorisation followed by forward and backward substitution. Since has a bandwidth of 2τ, from Table 1 [17], this step requires flops and finally the banded matrix–vector product, which requires about flops. Thus, the total complexity required for the proposed equalisation scheme becomes approximately flops. Table 1. Arithmetic operation counts for Banded matrix Banded matrix Multiplication Division Addition Subtraction matrix-matrix production matrix-vector production LU factorisation 2Nτ2 2Nτ 2Nτ2 forward substitution 2Nτ 2Nτ back substitution 2Nτ 2N 2Nτ 5 Simulation results and analysis This section is devoted to the performance evaluation of the proposed equaliser. First, simulation parameters setup are introduced. Then simulation results are obtained and analysed with the BER as the adopted performance metric of interest. 5.1 Simulation setup We consider a convolutional encoder with a coding rate rC of 1/2, a memory length of 7, and an octal generator polynomial Coefficient set (133,171) is considered. Encoded bits are forwarded to a QPSK modulator and mapped to a number of N = 128 frequency domain subcarriers. The frequency domain samples are transformed back to the time domain after a mapping process to allocate the frequency domain samples to M = 512 subcarriers. A 2 × 2 MIMO TX-RX antenna pair is assumed , and U = 4. The performance of the proposed equalisation scheme is evaluated over Rayleigh fading channels and assuming vehicular-A model [19]. The Rayleigh fading statistics has been proved to be adequate model for the channel frequency response [20] when dealing with SC-FDMA systems. The power delay profile of the considered channel model is shown in Table 2, the maximum CFO considered here is ɛmax = 0.1. Table 2. Power delay profile of vehicular A channel model Symbol Multipath component power, dB 0 −1 −9 −10 −15 −20 delay, µs 0 0.31 0.71 1.09 1.73 2.51 To evaluate the BER performance, Monte Carlo simulation technique is employed to generate a number of 104 channel realisations, the received vector is then detected and the errors in the detection process are averaged over this large number of realisations. 6 Results and discussion This section is devoted to the performance evaluation for the proposed equalisation scheme as compared with the state of the art equalisation scheme in [16]. Different key simulation parameters are identified and varied in order to assess the performance limits of the proposed equaliser. First, we start our evaluation by obtaining optimum value of the complex regularisation parameter . The optimum value of the complex regularisation parameter is defined as the value which leads to a global minimum BER value at a given SNR value. Then, we study the system performance at the obtained optimum value(s). The BER variation with different values of real regularisation parameter αr for SNR = 9,12,15 dB are shown in Fig. 2. there is a minimum BER at αr = 0.1 for the proposed JCRZF and JLRZF [16] which considers only the real part of regularisation parameter. Fig. 2Open in figure viewerPowerPoint BER Performance against the real part of the complex regularisation parameter at different SNR values Fig. 3 depicts simultaneous optimisation for the real and the imaginary parts of the complex regularisation parameter . We calculate the optimum value of the real and imaginary parts of simultaneously by evaluating the BER over a range of values, 1 × 10−4 to 1, for different SNR values. It is clear that, the minimum BER performance is achieved at for SNR = 9 dB and at for SNR = 12,15 dB. These are the two values which will be used for the rest of the evaluation process. Fig. 3Open in figure viewerPowerPoint BER Performance against the real and imaginary part of the complex regularisation parameter at different SNR values a BER Performance against at SNR = 9 dB b BER Performance against at SNR = 12 dB c BER Performance against at SNR = 15 dB d BER Performance against at SNR = 9, 12, 15 dB The performance of the proposed scheme with errors in the estimated CFO is studied in Fig. 4. The error in the estimated CFO is obtained by adding a uniformly distributed random variable to the true values of the CFO. The BER corresponding to each estimated CFO value is evaluated at different SNR values. The variation of the BER with the estimated CFO starts to appear at SNR ≥ 10 dB. Fig. 4Open in figure viewerPowerPoint BER Performance against the relative error of the estimated CFO at different SNR values The performance of JCRZF starts to degrade as the relative CFO estimation error becomes greater than 0.1 which shows appropriate robustness to CFO estimation errors. Finally, we compare the performance of the proposed JCRZF equalisation scheme with JLRZF [16] in the presence of MAI, ICI and CFO at different values of SNR. Fig. 5 demonstrates that the proposed JCRZF outperforms the JLRZF equaliser in both the coded and uncoded SC-FDMA MIMO systems. The imaginary part of the regularisation parameter accounts for the ICI, which is not considered in JLRZF equalisation scheme [16]. The required SNR = 18, 18.9 and 21 dB for coded No CFO, JCRZF and JLRZF, respectively. Accordingly, the proposed equalisation scheme outperforms [16] by gain of about 3 dB and approaches the performance of the No CFO performance. Fig. 5Open in figure viewerPowerPoint BER Performance against the SNR for different equalisation cases. Solid lines correspond to the BER with coding while dashed lines correspond to the BER without coding 7 Conclusions This paper proposes a novel FDE equalisation scheme for SC-FDMA MIMO systems. The proposed equaliser accounts for both the MAI and ISI as the conventional ZF based equalisers. However, our proposed scheme accounts for the ICI induced by the CFO in mobile channel environments by considering the structure of the ICI matrices in the received signal model. Theoretical analysis, verified by simulations, show that the proposed equalisation scheme outperforms the conventional ZF based equalisers with the BER adopted as the performance metric of interest. References 1Rana M.M.Saiful Islam Md.Kouzani A.Z.: ' Peak to average power ratio analysis for LTE systems'. 2010 Second Int. Conf. Communication Software and Networks, ICCSN '10. pp. 516– 520 2Md. 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