Spectrum Efficient MIMO-FBMC System using Filter Output Truncation

This article has been published in IEEE journal. The final version is a vailable at http://dx.doi.org/10.1109/TVT .2017.2771531 Spectrum Ef ficient MIMO-FBMC System using Filter Output T runcation Adnan Zafar , Lei Zhang, Pei Xiao and Muhammad Ali Imran Abstract —Due to the use of an appropriately designed pulse shaping prototype filter , filter bank multicarrier (FBMC) system can achieve low out of band (OoB) emissions and is also robust to the channel and synchronization errors. However , it comes at a cost of long filter tails which may reduce the spectral efficiency significantly when the block size is small. Filter output truncation (FO T) can reduce the overhead by discarding the filter tails but may also significantly destroy the orthogonality of FBMC system, by introducing inter carrier interference (ICI) and inter symbol interference (ISI) terms in the received signal. As a result, the signal to interference ratio (SIR) is degraded. In addition, the presence of intrinsic interference terms in FBMC also pro ves to be an obstacle in combining multiple input multiple output (MIMO) with FBMC. In this paper , we present a theoretical analysis on the effect of FOT in an MIMO-FBMC system. First, we derive the matrix model of MIMO-FBMC system which is subsequently used to analyze the impact of finite filter length and FO T on the system performance. The analysis reveals that FOT can a void the ov erhead in time domain but also intr oduces extra interference in the receiv ed symbols. T o combat the interference terms, we then propose a compensation algorithm that considers odd and even overlapping factors as two separate cases, where the signals are interfered by the truncation in different ways. The general form of the compensation algorithm can compensate all the symbols in a MIMO-FBMC block and can improve the SIR values of each symbol for better detection at the recei ver . It is also shown that the proposed algorithm requires no overhead and can still achieve a comparable BER perf ormance to the case with no filter truncation. Index terms – filter bank multicarrier , wavef orm, per- formance analysis, filter output truncation, intrinsic inter - ference I . I N T RO D U C T I O N F IL TER bank multicarrier (FBMC) has illustrated profound advantages ov er con ventional multicarrier modulation (MCM) schemes such as orthogonal frequency di vision multi- plexing (OFDM) in time and frequency dispersive channels [1]–[4]. Such advantages come from the fact that OFDM suffers from large out of band (OoB) emissions and thus require large guard bands to protect neighboring channels, hence reducing the efficiency of the system. This presents a major source of problem that limits the applicability of OFDM in some present and future communication systems [5]. FBMC, on the other hand, is a promising technique that over - comes this problem by utilizing a specially designed prototype A. Zafar and P . Xiao are with Institute for Communication Systems (ICS), Univ ersity of Surrey , Guildford, UK. Emails: { a.zafar , p.xiao } @surrey .ac.uk. A. Zafar is also affiliated with Institute of Space T echnology , Islamabad, Pakistan. Email: adnan.zafar@ist.edu.pk L. Zhang and M. A. Imran are with School of Engineering, Univ ersity of Glasgo w , Glasgow , UK. Email: { lei.zhang, muham- mad.imran } @glasgow .ac.uk filter such as isotropic orthogonal transform algorithm (IOT A) which is well localized both in time and frequency [6]. This prototype filter enables FBMC to provide best OoB emission among the new wav eforms proposed for future networks, such as generalized frequency di vision multiplexing (GFDM) [7], univ ersal filtered multi-carrier (UFMC) [8], [9], filtered orthogonal frequency division multiplexing (FOFDM) [10] and their variants [11]. This adv antage enables FBMC systems to utilize the fragmented spectrum more efficiently [12]. Other main advantage of FBMC include higher spectral efficienc y compared to con ventional OFDM systems. It is due to the good time and frequency localization properties of the prototype filter in FBMC that ensures inter carrier interference (ICI) and inter symbol interference (ISI) are negligible without the use of cyclic prefix (CP) [1]. The strict synchronization requirements in con ventional OFDM based systems are also much relaxed for FBMC system. This facilitates low com- plexity implementation of multi-user (MU) access in uplink transmissions for FBMC systems [13], [14]. Due to these advantages, FBMC is considered as a key area of research for the past sev eral years and one of the most promising wav eform candidate for future wireless networks [15], [16]. Unlike con ventional OFDM, the FBMC system utilizes orthogonal QAM symbols as the system is non-orthogonal in complex plane. Ho wev er , FBMC requires more complex receiv er structure, particularly when combined with MIMO as compared to the MIMO-OFDM based systems. Moreov er , FBMC system may encounter residual interference terms in the form of ICI and ISI if a low complexity channel equal- ization is used for highly dispersiv e channels. The impact of doubly dispersi ve channel on a SISO-FBMC system with both zero forcing (ZF) and minimum mean squared error (MMSE) based one tap equalization schemes is analyzed in [17]. It is proposed that a complex multi-tap equalization may be required as the performance of the FBMC system is severely limited by strong doubly dispersive channel impact. The authors in [18] hav e in vestigated the performance degeneration of OFDM and FBMC systems in doubly-selective channels using a closed-form bit error probability (BEP) expression. It is shown that FBMC performs better than CP-OFDM in highly time-varying channels due to the use of well localized prototype filter . Unlike OFDM, the use of FBMC in multi-antenna con- figurations is not as straightforward and the applications are very limited. T ensubam et al. in [19] hav e presented a study on recent adv ancements in MIMO-FBMC and suggest that filtered multitone (FMT) based FBMC systems offer the same This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 flexibility as OFDM in adopting MIMO technology . Howe ver , it is spectrally inefficient compared to other variants of FBMC like cosine modulated multitone (CMT) and staggered mod- ulated multitone (SMT) as it requires guard bands between the subcarriers. But unlike conv entional OFDM system, the receiv ed symbols in CMT and SMT based FBMC systems are contaminated with pure imaginary intrinsic interference. This interference proves to be a huge obstacle in combining MIMO techniques with FBMC. A two step recei ver for MIMO-FBMC is proposed in [20], where the first stage estimates and cancels this intrinsic interference, while the second stage improves the estimation using widely linear processing. The authors in [21] ha ve sho wn strong correlation between the real and imaginary components of FBMC signal and hav e proposed a new equalizer structure by exploiting the imaginary intrinsic interference components. A scheme referred as FFT -FBMC, is proposed by Rostom Zakaria et al. in [22] and is applied to multiple antenna systems. Although, FFT -FBMC technique can address the issue of intrinsic interference by using a CP , howe ver , it has a poor bit error rate (BER) performance as compared to the con ventional OFDM systems. It was shown in [23] that the FFT precoded signals in FFT -FBMC can reduce the frequency band occupied by each subcarrier by reducing the interference power in the immediate adjacent sub band as compared to conv entional FBMC. Jayasinghe et al. in [24] hav e analyzed the ef fect of intrinsic interference in FBMC system and proposed a precoder based on signal to leakage plus noise ratio (SLNR) at the transmitter side to overcome its effects on the FBMC system. It is shown that the proposed precoder design at the transmitter outperforms the equalization based FBMC and OFDM systems. Recent de velopments in combining FBMC with massiv e MIMO are discussed in [25]. There has been in vestigations on the performance of MIMO- FBMC system in frequency selective channels. V arious pre- coding and equalization techniques are proposed to achie ve robustness against channel frequenc y selectivity and to im- prov e the spectral efficienc y in a MIMO-FBMC system [26]. The authors in [27] have presented a single-tap precoder and decoder design for multiuser MIMO-FBMC system for frequency selective channels by optimizing the MSE formula under ZF and MMSE design criteria. Mestre et al. in [28] hav e proposed a novel architecture for MIMO-FBMC system by exploiting the structure of the analysis and synthesis filter banks using approximation of an ideal frequency-selecti ve precoder and linear receiv er . Another precoding and decoding technique for MIMO-FBMC system is proposed in [29] to enable multi-user transmission in frequency selectiv e chan- nels. Soysa et al. in [30] have e valuated the performance of precoding and receiv er processing techniques for mul- tiple access MIMO-FBMC system for an extended ICI/ISI scenario in uplink and downlink. A. Ikhlef et al. in [31] proposed successive interference cancellation (SIC) to extract the transmitted information in a MIMO-FBMC system. The proposed solution outperforms the classical one tap equalizers in case of moderate and high frequency selective channels. Chang et al. in [32] hav e presented a precoded SISO-FBMC system without CSI at the transmitter . The proposed system is limited by the assumption of perfect equalization at the receiv er whereas, imperfect equalization can lead to residual ISI and ICI terms. The authors then analyzed the effect of interference from imperfect equalization in [33]. The results suggest that multi-tap equalization is required to reduce the effect of interference in FBMC system for highly frequency selectiv e channels. Inaki Estella et al. in [34] provided a comparison between multi-stream MIMO based OFDM and FBMC systems and suggested that OFDM achie ves a lower energy-ef ficiency than the FBMC. Howe ver , unlike OFDM, the use of multiple streams increases interference in FBMC which require new equalization techniques. Ana I. Perez-Neira et al. ha ve presented a detailed and comprehensi ve o vervie w of various challenges in MIMO-FBMC systems and their known solutions in [35]. The aforementioned studies are mainly focused on the performance of MIMO-FBMC systems in frequency selectiv e channels and its spectral efficienc y compared to OFDM based systems. Ho wev er , despite the fact that FBMC does not require a CP , it is not completely free from overhead as the filter bank itself introduces extra tails in the FBMC block that affects the spectral ef ficiency of the system. A recent study has considered improving the spectral ef ficiency in FBMC system by tackling the over head (tails) caused by the filter operation. The authors in [36] have introduced non data symbols (virtual symbols) before and after each FBMC data packet for shortening the ramp-up and ramp-down periods. In [12], it is pointed out that filter output truncation (FO T) or tail cutting can improv e the spectral efficienc y of FBMC system but require one extra tail to be transmitted as o verhead along with the FBMC block. In this paper , we pro vided an in-depth analysis of FO T in a MIMO-FBMC system. W e in vestigated the possibili- ties of completely discarding all the tails (overhead) to im- prov e the spectral ef ficiency of the MIMO-FBMC system. T o achiev e this, we represented the complete MIMO-FBMC system in a matrix form including the filter operation, tail cutting/truncating, channel con volution, equalization, detection etc. The interference terms like ICI and ISI introduced by the FOT along with the intrinsic interference terms are then deriv ed using the MIMO-FBMC matrix model. In light of the analytical results, we proposed a compensation algorithm to overcome the interferences caused by FO T . The proposed algorithm enables the complete elimination of the overhead in a MIMO-FBMC system by compensating the truncation affect at the receiv er . As a result, the spectral efficienc y of MIMO- FBMC systems is improv ed. The contributions of this paper are summarized as follows. • W e first deri ve a compact matrix model of MIMO-FBMC system which lays the ground for the subsequent in- depth analysis of the effect of FO T on the detection performance in terms of the SIR and BER. • Based on the matrix model, we then analyze the impact of finite filter length and different types of FO T on the system performance. Through simulation results, it is sho wn numerically that FO T can overcome the high ov erhead but significantly degrade the SIR of the symbols This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 at the edges. • Thirdly , based on the observ ations made in the aforemen- tioned numerical analysis, a compensation algorithm is designed to compensate the symbols in a MIMO-FBMC block to improv e the SIR of each symbol. The advantage of the algorithm is that it requires no ov erhead but can still achieve a similar performance compared to the case with no FOT . Notations: V ectors and matrices are denoted by lowercase and uppercase bold letters. { . } H and { . } T represent conjugate transpose (hermitian conjugate) and transpose operations. F and F H denote the normalized N point DFT and IDFT matrices. A ⊗ B represents kronecker product of A and B . < ( A ) and = ( A ) are the real and imaginary part of scalar/vector/matrix A . I N represents an identity matrix for dimension N × N . A ∗ B represents the linear con volution of A and B . A ↓ l represents l sample delayed version of the vector A with zero padding at the front. W e use { ¯ . } or { ˜ . } ov er any variable to represent the real and imaginary part of that scalar/vector/matrix respectiv ely . I I . P RO B L E M F O R M U L AT I O N A. MIMO-FBMC System Model In our analysis of MIMO-FBMC system, we assumed N t transmit antennas are used to transmit multiple FBMC signals simultaneously which are receiv ed by N r receiv ed antennas, where N t ≤ N r . The block diagram for both transmitter and recei ver of a MIMO-FBMC is shown in Fig. 1, where real and imaginary branches i.e. I and Q branches are processed simultaneously and independently . The MIMO-FBMC system model follows a block based processing approach where each block contains M FBMC symbols with each symbol containing N subcarriers in fre- quency domain. Therefore, we can represent each MIMO- FBMC block as S = [ s 0 , s 1 , · · · , s M − 1 ] ∈ C N N t × M where s m = [ s m, 0 , s m, 1 , · · · , s m,N − 1 ] T ∈ C N N t × 1 . The transmitted signal on the n th subcarrier in a MIMO system is an N t × 1 vector , i.e., s m,n = [ s m,n, 1 , s m,n, 2 , · · · , s m,n,N t ] T ∈ C N t × 1 . Each s m,n,j represents a complex signal on n th subcarrier of m th FBMC symbol transmitted by j th transmitting antenna. Hence, M N N t QAM symbols are transmitted in one FBMC block. Note that a precoding scheme such as ZF can be applied at the transmitter side when the channel state information (CSI) is av ailable. In such cases the performance of a system can be further enhanced. Howe ver , the focus of this paper is to analyze the performance of MIMO-FBMC system with finite filter length and FOT . Therefore, the analysis presented in Section IV is based on unitary precoding matrix but is easily extend-able to the precoding case as well. Moreover , the power of modulated symbols s m,n,j is represented as δ 2 i.e. E {k s m,n,j k 2 } = δ 2 . The real and imaginary parts of s m are represented as ¯ s m and ˜ s m respectiv ely . B. MIMO Channel Impulse Response W e assume the system operates ov er a slo wly-varying f ading channel i.e. quasi-static fading channel. In such a scenario, it is plausible to assume that the duration of each transmitted data block is smaller than the coherence time of the channel, therefore, the random fading coefficients stay constant over the duration of each block [37]. In this case, we define the multipath channel as a l -tap channel impulse response (CIR) matrix with the l th -tap power being ρ 2 l . It is also assumed that the average power remains constant during transmission of whole block. The CIR matrix H is defined as H = [ H 0 , H 1 , · · · , H L − 1 ] T = [ ρ 0 Z 0 , ρ 1 Z 1 , · · · , ρ L − 1 Z L − 1 ] T (1) where H l defines the l th matrix valued CIR coefficient of the channel between all the antennas and is represented as H l = ρ l Z l = ρ l    z 11 ( l ) · · · z 1 N t ( l ) . . . . . . . . . z N r 1 ( l ) · · · z N r N t ( l )    ∈ C N r × N t (2) The random variable z ij ( l ) with complex Gaussian distri- bution as C N (0 , 1) represents the multipath fading factor for l th tap of the quasi-static rayleigh fading channel between j th transmit antenna and i th receiv e antenna. Note that we consider co-located transmit and receive antennas to simplify our analysis. Ho we ver , if we consider either transmit or receiv e antennas to be geographically separated, the analysis can easily be extended by considering the common coefficient ρ l to be different among the antennas. C. Prototype F ilters / F ilter Matrices Ideally , an infinite filter length ( K = ∞ ) is required to provide the best performance. Ho wev er , a finite filter length (e.g. overlapping f actor K = 4 ∼ 6 ) is used in practice in a FBMC system to achieve comparable system perfor- mance. T o generalize the deriv ation, the filter overlapping factor is taken as K , therefore, K N is the total length of the prototype filter i.e. ¯ w = [ ¯ w 0 , ¯ w 1 , · · · , ¯ w K − 1 ] = [ ¯ w 0 , ¯ w 0 , · · · , ¯ w K N − 1 ] ∈ R 1 × K N . The I branch filter matrix ¯ P orig ∈ R ( K + M − 1) N N t × M N N t can be expressed as ¯ P orig =             ¯ P i F ¯ P ¯ P i R             =                             ¯ W 0 0 0 · · · 0 0 ¯ W 1 ¯ W 0 0 · · · 0 0 . . . . . . ¯ W 0 · · · 0 0 ¯ W t − 1 ¯ W t − 2 . . . · · · . . . . . . ¯ W t ¯ W t − 1 ¯ W t − 2 · · · 0 0 ¯ W t +1 ¯ W t ¯ W t − 1 · · · ¯ W 0 0 . . . ¯ W t +1 ¯ W t · · · ¯ W 1 ¯ W 0 ¯ W K − 1 . . . ¯ W t +1 · · · . . . ¯ W 1 0 ¯ W K − 1 . . . · · · ¯ W K − t . . . 0 0 ¯ W K − 1 · · · . . . ¯ W K − t . . . . . . . . . · · · ¯ W K − 1 . . . 0 0 0 · · · 0 ¯ W K − 1                             (3) where, ¯ W k = diag ( ¯ w k ) ∈ R N × N for k = 0 , 1 , 2 , · · · , K − 1 and ¯ w k = [ ¯ w kN , ¯ w kN +1 , · · · , ¯ w kN + N − 1 ] ∈ R 1 × N while t = b K 2 c . The value of t defines the truncated matrix ¯ P as shown in (3). The prototype filter matrix for the Q branch This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 Fig. 1: Blocks diagrams for MIMO-FBMC transmitter and receiver in matrix operation form is defined in the same manner . The only dif ference is that the Q branch filter is a shifted version of the I branch filter i.e. ˜ w = [ ˜ w 0 , ˜ w 1 , · · · , ˜ w K − 1 ] = [ ˜ w 0 , ˜ w 0 , · · · , ˜ w K N − 1 ] = [ ¯ w N 2 , ¯ w N 2 +1 , · · · , ¯ w K N − 1 , ¯ w 0 , ¯ w 1 , · · · , ˜ w N 2 − 1 ] ∈ R 1 × K N . Shifting prototype filter in the Q branch instead of offset- ting the QAM symbols makes the overall design simpler . Similarly , the Q branch truncated filter matrix ˜ P is defined in the same manner as described for the I branch with ˜ W k = diag ( ˜ w k ) ∈ R N × N for k = 0 , 1 , 2 , · · · , K − 1 and ˜ w k = [ ˜ w kN , ¯ w kN +1 , · · · , ˜ w kN + N − 1 ] ∈ R 1 × N . I I I . M I M O - F B M C M A T R I X M O D E L The MIMO-FBMC matrix model is deriv ed by extending our previous work on a SISO-FBMC system [17]. It is worth mentioning that the derivation of MIMO-FBMC matrix model is not a simple SISO to MIMO mapping. Signal definition, transmit processing, channel modeling, as well as receive processing including channel equalization has to be redefined. The derived MIMO-FBMC model also incorporates FO T as well as the proposed compensation algorithm at the receiver . A. T ransmit Processing W e will only focus on the real branch in detail since the imaginary branch will follow the same procedure. 1) Real Branch Pr ocessing: According to Fig. 1, the signal ¯ s m is first multiplied by a phase shifter matrix ¯ Φ m symbol by symbol i.e., ¯ a m = ( ¯ Φ m ⊗ I N t ) ¯ s m = ¯ Φ k,m ¯ s m ∈ C N N t × 1 (4) where ¯ Φ m is a diagonal matrix i.e. ¯ Φ m = diag [ e − j π (0+2 m ) 2 , e − j π (1+2 m ) 2 , · · · , e − j π ( N − 1+2 m ) 2 ] ∈ C N × N . Note that ¯ Φ k,m represents the kronecker product ¯ Φ m ⊗ I N t that yields a matrix of size N N t × N N t . 2) Real Branc h IDFT Pr ocessing: Signal after the phase shifter matrix will pass through an N point IDFT (in verse discrete Fourier transform) block F H i.e. ¯ b m = ( F H ⊗ I N t ) ¯ a m = F H k ¯ a m ∈ C N N t × 1 (5) where F H k = F H ⊗ I N t . Signal after the IDFT block can be represented as ¯ b = [ ¯ b 0 ; ¯ b 1 ; · · · ; ¯ b M − 1 ] = [ F H k ¯ a 0 ; F H k ¯ a 1 ; · · · ; F H k ¯ a M − 1 ] ∈ C M N N t × 1 . Here IDFT is a block wise operation since each modulated subcarrier is a column vector of size N t × 1 and F H k is a generalized N N t point IDFT matrix. 3) Real Branch Pr ototype F ilter: The signal is then passed through a prototype filter in I and Q branches independently . In general, prototype filters are linearly conv olved with the input signal. In order to represent a complete system in matrix form we have defined a prototype filter matrix ¯ P in a manner that when this filter matrix is multiplied by v ector ¯ b ; the multiplication of matrices is equiv alent to the required linear con volution process. The output of the I branch filter can be written as ¯ o = ¯ P k,or ig ¯ b (6) where ¯ P k,or ig = ¯ P orig ⊗ I N t . Note that the output of the real branch filter ¯ o has ( K − 1) N N t more samples than the input due to the linear con volution process. Hence, to keep the orthogonality (minimum interference from other subcarriers and symbols), all of these samples have to be transmitted to the receiv er side. Howe ver , the transmission efficienc y η will drop by the proportion of η = M K + M − 1 (7) It can be seen from (7), that transmission efficienc y η is high only if M is large. Another way to achie ve higher η is to truncate ¯ P orig to improve the spectral efficienc y . Howe ver , truncation may lead to interferences in the system that can significantly degrade the system performance. W ithout any compensation, the maximum allow able cut off symbols would be K − 2 so as to keep the signals detectable [12]. Howe ver , with compensation we can completely discard all the K − 1 symbols while still keeping the signals detectable. The truncation should take place at the first i F and the last i R rows of ¯ P orig such that i F + i R ≤ K − 1 as shown in (3), where ¯ P i F is first i F N ro ws and ¯ P i R is the last i R N ro ws of ¯ P orig . The middle part of ¯ P orig i.e. ¯ P which is the truncated This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 filter matrix will be used at transmitter side to improv e the spectral efficiency of the system. The performance loss due to the truncation of ¯ P orig will be compensated at the receiver side and is discussed later in Section IV. The output of real branch truncated filter can be written as ¯ o = ( ¯ P ⊗ I N t ) ¯ b = ¯ P k ¯ b ∈ C M N N t × 1 (8) 4) Imaginary Branch Pr ocessing Including Pr ototype F il- tering: Similar process is follo wed for the Q branch i.e. the signal ˜ s m is first multiplied by a phase shifter matrix ˜ Φ m = j ¯ Φ m symbol by symbol i.e., ˜ a m = ( ˜ Φ m ⊗ I N t ) ˜ s m = ˜ Φ k,m ˜ s m ∈ C N N t × 1 (9) After the phase shifter matrix, the signal will pass through an N point IDFT block F H as ˜ b m = ( F H ⊗ I N t ) ˜ a m = F H k ˜ a m ∈ C N N t × 1 (10) The signal after IDFT block can be represented as ˜ b = [ ˜ b 0 ; ˜ b 1 ; · · · ; ˜ b M − 1 ] = [ F H k ˜ a 0 ; F H k ˜ a 1 ; · · · ; F H k ˜ a M − 1 ] ∈ C M N N t × 1 . Likewise, the following matrix multiplication of truncated filter matrix ˜ P and the signal vector ˜ b represents the linear conv olution of the imaginary branch prototype filter and the imaginary branch input signal. ˜ o = ( ˜ P ⊗ I N t ) ˜ b = ˜ P k ˜ b ∈ C M N N t × 1 (11) B. P assing thr ough the Channel The real and imaginary branch signals ¯ o and ˜ o after the prototype filtering are added together and is then passed through the channel H . The receiv ed signal is no w represented as r = H ∗ ( ¯ o + ˜ o ) + n (12) where n = [ n 1 , n 2 , · · · , n N r ] T ∈ C M N N r × 1 is a Gaussian noise vector with each element ha ving zero mean and v ariance σ 2 . T o represent the con volution process giv en in (12) as matrix multiplication, we define the l th tap multipath fading factor Z l of the MIMO channel as a block diagonal matrix by taking the kronecker product of an identity matrix I ( K + M − 1) N with Z l as Z l,blk = I ( K + M − 1) ⊗ Z k,l (13) where Z k,l = I N ⊗ Z l ∈ C N N r × N N t . The block diagonal matrix Z l,blk ∈ C ( K + M − 1) N N r × ( K + M − 1) N N t has Z l as its diagonal sub matrices. The definition of Z l,blk implies that each FBMC symbol in a block experiences the same channel i.e. Z l . Hence, we can write (12) as r = L − 1 X l =0 ρ l Z l,blk ( ¯ o ↓ N t l + ˜ o ↓ N t l ) + n (14) where ¯ o ↓ N t l , ˜ o ↓ N t l represents N t l samples delayed version of ¯ o and ˜ o with zero padding in the front i.e. ¯ o ↓ N t l = [ 0 N t l × 1 ; ¯ o q ,N t l ] and ˜ o ↓ N t l = [ 0 N t l × 1 ; ˜ o q ,N t l ] respectively . Note that ¯ o q ,N t l and ˜ o q ,N t l represents the first ( K + M − 1) N N t − N t l elements of ¯ o and ˜ o respectively . From (8) and (11) we can write ¯ o ↓ N t l = ¯ P ↓ N t l k ¯ b and ˜ o ↓ N t l = ˜ P ↓ N t l k ˜ b , where ¯ P ↓ N t l k = [ 0 N t l × M N N t ; ¯ P k,q ] and ˜ P ↓ N t l k = [ 0 N t l × M N N t ; ˜ P k,q ] . Here ¯ P k,q and ˜ P k,q are the first ( K + M − 1) N N t − N t l rows of ¯ P k and ˜ P k respectiv ely . Eq (14) can thus be reformed as r = L − 1 X l =0 ρ l Z l,blk ( ¯ P ↓ N t l k ¯ b + ˜ P ↓ N t l k ˜ b ) + n (15) The abov e equation indicates that the truncated filter matrix ¯ P k and ˜ P k are distorted because of the channel multipath effect and are represented as ¯ P ↓ N t l k and ˜ P ↓ N t l k respectiv ely . T o represent (15) in a point-wise multiplication form in frequency domain, we apply the circular conv olution prop- erty by first introducing a block diagonal e xchange matrix X N t l ∈ R M N N t × M N N t as X N t l =      X sub,N t l 0 · · · 0 0 X sub,N t l · · · 0 . . . . . . . . . . . . 0 0 · · · X sub,N t l      (16) with X sub,N t l =  0 N t l × ( N N t − N t l ) I N t l × N t l I ( N N t − N t l ) × ( N N t − N t l ) 0 ( N N t − N t l ) × N t l  (17) As X T N t l X N t l = I , we have ¯ o ↓ N t l = ¯ P ↓ N t l k ¯ b = ¯ P ↓ N t l k X T N t l X N t l ¯ b = ¯ P ↓ N t l k,e ¯ b ↓ N t l e (18) The matrix X T N t l and X N t l are used to exchange the locations of elements of ¯ P ↓ N t l k and ¯ b respectiv ely , such that ¯ P ↓ N t l k,e = ¯ P ↓ N t l k X T N t l and ¯ b ↓ N t l e = X N t l ¯ b . By multiplying the matrix X N t l with ¯ b , the last N t l symbols of its each sub- vector ¯ b m will be moved to the front, i.e., ¯ b ↓ N t l e,m = [ b m,N N t − N t l · · · , b m,N N t − 1 , b m, 0 , · · · , b m,N N t − N t l − 1 ] T ∈ C N N t × 1 (19) Like wise, ¯ b ↓ N t l e = [ ¯ b ↓ N t l e, 0 ; ¯ b ↓ N t l e, 1 ; · · · ; ¯ b ↓ N t l e,M − 1 ] ∈ C M N N t × 1 (20) The effect is similar when multiplying X T N t l with ¯ P ↓ N t l k . X T N t l only changes the elements location in ¯ P ↓ N t l k . Similarly , we can write ˜ o ↓ N t l as ˜ o ↓ N t l = ˜ P ↓ N t l k ˜ b = ˜ P ↓ N t l k X T N t l X N t l ˜ b = ˜ P ↓ N t l k,e ˜ b ↓ N t l e (21) Substituting (18) and (21) into (15) yields r = L − 1 X l =0 ρ l Z l,blk ( ¯ P ↓ N t l k,e ¯ b ↓ N t l e + ˜ P ↓ N t l k,e ˜ b ↓ N t l e ) + n (22) It can be observed that the non zero elements of ¯ P ↓ N t l k,e and ¯ P k are very close i.e. the nonzero elements of ¯ P ↓ N t l k,e are only delayed by N t l elements as compared to the elements in ¯ P k . If the non-zero i th row and k th column element of ¯ P k is w n , then the element of ¯ P ↓ N t l k,e at the same location will be w n + N t l . This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 Since N  L , the difference between w n and w n + N t l is very small as the adjacent elements of the prototype filter are close to each other . Eq (22) can thus be written as r ≈ L − 1 X l =0 ρ l Z l,blk ( ¯ P k ¯ b ↓ N t l e + ˜ P k ˜ b ↓ N t l e ) + n (23) C. Receive Pr ocessing On the receiv er side, the signal r is receiv ed by N r receiv ed antennas and is fed to the real and imaginary branches of the receiv er as shown in Fig. 1 for independent processing. 1) Real Branch Pr ocessing: Following the similar ap- proach, we will focus on the real branch processing and the imaginary branch processing follows the same procedure. In the real branch, signals from N r receiv ed antennas are passed through the real branch recei ved filters leading to the follo wing output ¯ x = ¯ P H k r = ¯ P H k ¯ P k L − 1 X l =0 ρ l Z l,blk ¯ b ↓ N t l e + ¯ P H k ˜ P k L − 1 X l =0 ρ l Z l,blk ˜ b ↓ N t l e + ¯ P H k n (24) Autocorrelation and cross-correlation matrices of ¯ P k and ˜ P k are defined as ¯ ¯ G k = ¯ P H k ¯ P k , ¯ ˜ G k = ¯ P H k ˜ P k , ˜ ˜ G k = ˜ P H k ˜ P k and ˜ ¯ G k = ˜ P H k ¯ P k . Here ¯ ¯ G k , ¯ ˜ G k , ˜ ˜ G k and ˜ ¯ G k ∈ R M N N r × M N N t The abov e equation (24) can now be written as ¯ x = ¯ ¯ G k L − 1 X l =0 ρ l Z l,blk ¯ b ↓ N t l e + ¯ ˜ G k L − 1 X l =0 ρ l Z l,blk ˜ b ↓ N t l e + ¯ P H k n (25) 2) Real Branc h DFT Pr ocessing and Phase Shifting: The signal v ector at the output of the real branch filter matrix i.e. ¯ P H k is represented as ¯ x = [ ¯ x 0 , ¯ x 1 , · · · , ¯ x M N N r − 1 ] T ∈ C M N N r × 1 and is then passed through a serial to parallel con version to split the vector into M segments, each of which has N N r elements to perform N -point DFT and phase shifting process. The m th segment of the vector ¯ x is represents as ¯ x m = [ ¯ x mN N r , ¯ x mN N r +1 , · · · , ¯ x mN N r + N N r − 1 ] T ∈ C N N r × 1 for m ∈ 0 , 1 , · · · , M − 1 . The signal is now represented as ¯ x = [ ¯ x 0 , ¯ x 1 , · · · , ¯ x M − 1 ] ∈ C N N r × M where ¯ x m = [ ¯ x m, 0 , ¯ x m, 1 , · · · , ¯ x m,N − 1 ] T ∈ C N N r × 1 in which ¯ x m,n = [ ¯ x m,n, 1 , · · · , ¯ x m,n,N r ] T ∈ C N r × 1 . Each ¯ x m,n,i represents the real signal on n th modulated subcarrier for m th FBMC symbol receiv ed by i th receiving antenna. Using equation (25), we can write signal vector ¯ x m as ¯ x m = M − 1 X i =0 ¯ ¯ G k,m,i L − 1 X l =0 ρ l Z k,l ¯ b ↓ N t l e,i + M − 1 X i =0 ¯ ˜ G k,m,i L − 1 X l =0 ρ l Z k,l ˜ b ↓ N t l e,i + ¯ P H k,m n (26) where ¯ ¯ G k,m,i and ¯ ˜ G k,m,i are the m th row and i th column sub-matrices of ¯ ¯ G k and ¯ ˜ G k respectiv ely . The signal after DFT and phase shifting is represented as ¯ y m = ¯ Φ H k,m F k ¯ x m (27) where ¯ Φ H k,m = ¯ Φ H m ⊗ I N r and F k = F ⊗ I N r ∈ C N N r × N N r . Hence, ¯ y m ∈ C N N r × 1 can now be simplified by substituting (26) into (27) as follows before the channel equalization. ¯ y m = ¯ Φ H k,m F k M − 1 X i =0 ¯ ¯ G k,m,i L − 1 X l =0 ρ l Z k,l ¯ b ↓ N t l e,i | {z } ¯ u R,m + ¯ Φ H k,m F k M − 1 X i =0 ¯ ˜ G k,m,i L − 1 X l =0 ρ l Z k,l ˜ b ↓ N t l e,i | {z } ¯ u I ,m + ¯ Φ H k,m F k ¯ P H k,m n (28) In (28), the third term is the noise processed by the proto- type filter, DFT and the phase shifter . The term ¯ u I ,m is the interference caused by the imaginary part of the signal ( ˜ s m ). Whereas, the first term ¯ u R,m contains the actual desired sym- bol ( ¯ s m ). In ¯ u R,m , we can write P L − 1 l =0 ρ l Z k,l ¯ b ↓ N t l e,i = H cir ¯ b i . The matrix H cir is an N N r × N N t block circulant matrix. In general, an N N r × N N t block circulant matrix is fully defined by its first N N r × N t block matrices. In our case, H cir is deter - mined by [ H 0 , H 1 , · · · , H L − 1 , 0 ( N − L ) N r × N t ] T ∈ C N N r × N t ¯ u R,m = ¯ Φ H k,m F k  M − 1 X i =0 ¯ ¯ G k,m,i H cir ¯ b i  = ¯ Φ H k,m F k M − 1 X i =0 ¯ ¯ G k,m,i F H k F k H cir F H k F k ¯ b i (29) where F H k F k = I . Then we can use the circular conv olution property as follows (pp.129-130) [38]. ¯ u R,m = ¯ Φ H k,m F k M − 1 X i =0 ¯ ¯ G k,m,i F H k F k H cir F H k | {z } C F k ¯ b i (30) where C is the frequenc y domain channel coef ficients in block diagonal matrix form and is gi ven as C = blkdiag [ C 0 , C 1 , · · · , C N − 1 ] ∈ C N N r × N N t . The n th block diagonal element in the frequency response of the MIMO channel can be represented as C n = P L − 1 l =0 H l e − j 2 π N nl ∈ C N r × N t . F k ( ¯ b i ) denotes the DFT processing of ¯ b i and according to (5) and (4), we hav e F k ( ¯ b i ) = ¯ a i = ¯ Φ k,i ¯ s i , substituting it into (30) leads to ¯ u R,m = ¯ Φ H k,m F k M − 1 X i =0 ¯ ¯ G k,m,i F H k C ¯ Φ k,i ¯ s i = M − 1 X i =0 ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i C ¯ s i (31) The order of C and ¯ Φ k,i are exchangeable since both are diagonal, we can thus obtain the following expression ¯ u R,m = M − 1 X i =0 ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i C ¯ s i (32) Similarly using the same method we can deri ve the expression for ¯ u I ,m as ¯ u I ,m = M − 1 X i =0 ¯ Φ H k,m F k ¯ ˜ G k,m,i F H k ˜ Φ k,i C ˜ s i (33) This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 Substituting (32) and (33) into (28) yields ¯ y m = ¯ Φ H k,m F k M − 1 X i =0 ¯ ¯ G k,m,i F H k ¯ Φ k,i C ¯ s i + ¯ Φ H k,m F k M − 1 X i =0 ¯ ˜ G k,m,i F H k ˜ Φ k,i C ˜ s i + ¯ Φ H k,m F k ¯ P H k,m n (34) W e can further reduce (34) as ¯ y m = ¯ Φ H k,m F k ¯ ¯ G k,m,m F H k ¯ Φ k,m C ¯ s m + M − 1 X i =0 ,i 6 = m ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i C ¯ s i + M − 1 X i =0 ¯ Φ H k,m F k ¯ ˜ G k,m,i F H k ˜ Φ k,i C ˜ s i + ¯ Φ H k,m F k ¯ P H k,m n (35) 3) Channel Equalization: W e represent one tap channel equalizer as a block diagonal matrix E and is applied to the real branch signal ¯ y m as ¯ u m = E¯ y m (36) Substituting (35) into (36) we get the equalized signal ¯ u m as ¯ u m = E  ¯ Φ H k,m F k ¯ ¯ G k,m,m F H k ¯ Φ k,m C ¯ s m + M − 1 X i =0 ,i 6 = m ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i C ¯ s i + M − 1 X i =0 ¯ Φ H k,m F k ¯ ˜ G k,m,i F H k ˜ Φ k,i C ˜ s i  + E ¯ Φ H k,m F k ¯ P H k,m n (37) Eq (37) can be written as ¯ u m = EC  ¯ Φ H k,m F k ¯ ¯ G k,m,m F H k ¯ Φ k,m ¯ s m + M − 1 X i =0 ,i 6 = m ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i ¯ s i + M − 1 X i =0 ¯ Φ H k,m F k ¯ ˜ G k,m,i F H k ˜ Φ k,i ˜ s i  + E ¯ Φ H k,m F k ¯ P H k,m n (38) where E can be either ZF or MMSE based linear channel equalizer [39] E = C H ( CC H + ν σ 2 /δ 2 I ) − 1 (39) where ν = 0 for ZF while ν = 1 is for MMSE case. W ith a simple ZF equalization i.e. E = ( C ) − 1 , we can write (38) as ¯ u m = ¯ Φ H k,m F k ¯ ¯ G k,m,m F H k ¯ Φ k,m ¯ s m + M − 1 X i =0 ,i 6 = m ¯ Φ H k,m F k ¯ ¯ G k,m,i F H k ¯ Φ k,i | {z } ¯ ¯ Q k,m,i ¯ s i + M − 1 X i =0 ¯ Φ H k,m F k ¯ ˜ G k,m,i F H k ˜ Φ k,i | {z } ¯ ˜ Q k,m,i ˜ s i + E ¯ Φ H k,m F k ¯ P H k,m n | {z } ¯ u noise,m (40) I V . F I N I T E F I LT E R L E N G T H A N D F I L T E R O U T P U T T RU N C AT IO N A NA LY S I S This section presents the impact of finite filter length and FO T on the system performance. W e will first consider the case with infinite filter length with no FO T and then we will extend our findings to deri ve the interferences caused by truncating the infinite filter length. A. Infinite F ilter Length ( K = ∞ ) with no FO T In this case the autocorrelation and cross correlation matri- ces used in (40) can no w be written as ¯ ¯ G k = ¯ P H k,or ig ¯ P k,or ig and ¯ ˜ G k = ¯ P H k,or ig ˜ P k,or ig respectiv ely . According to the orthogonality of FBMC with infinite filter length [17], ¯ ¯ Q k,m,i and ¯ ˜ Q k,m,i defined in (40) have the following property: ¯ ¯ Q k,m,i =  I + j ={ ¯ ¯ Q k,m,i } for i = m j ={ ¯ ¯ Q k,m,i } for i 6 = m ¯ ˜ Q k,m,i = j ={ ¯ ˜ Q k,m,i } for i = 0 , · · · , M − 1 (41) Using the property of infinite filter length giv en in (41), we can write (40) as ¯ u m = ¯ s m + j h M − 1 X i =0 ={ ¯ ¯ Q k,m,i } ¯ s i + M − 1 X i =0 ={ ¯ ˜ Q k,m,i } ˜ s i i | {z } ¯ u intri,m + ¯ u noise,m (42) The ¯ u intri,m term is the pure imaginary intrinsic interfer- ence that is inherent in the FBMC system. This interference can be av oided by taking the real part of (42). Hence, we can write (42) as <{ ¯ u m } = ¯ s m + <{ ¯ u noise,m } (43) Eq (43) sho ws that with infinite filter length and no truncation, the actual transmitted symbol i.e. ¯ s m can be recov ered without any ISI or ICI. The term < ( ¯ u noise,m ) is the real part of the processed noise. If we take the real part of (41), the property is then simplified as <{ ¯ ¯ Q k,m,i } =  I for i = m 0 for i 6 = m <{ ¯ ˜ Q k,m,i } = 0 for i = 0 , · · · , M − 1 (44) The simplified property satisfies the result obtain in (43). This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 1 2 3 4 5 6 7 8 -6 -5 -4 -3 -2 -1 0 (a) Signal power in I branch 1 2 3 4 5 6 7 8 -70 -60 -50 -40 -30 -20 -10 0 (b) Interference power in I branch 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 70 (c) Output SIR in I branch 1 2 3 4 5 6 7 8 -0.5 -0.4 -0.3 -0.2 -0.1 0 (d) Signal power in Q branch 1 2 3 4 5 6 7 8 -70 -60 -50 -40 -30 -20 (e) Interference power in Q branch 1 2 3 4 5 6 7 8 20 30 40 50 60 70 (f) Output SIR in Q branch Fig. 2: Signal and interference power with output SIR in real and imaginary branches ( K = 6 ) B. Finite F ilter Length ( K 6 = ∞ ) with FO T As it is impractical to use infinite filter length from imple- mentation point of view , we now consider the practical case where we consider a finite filter length ( K 6 = ∞ ) with FOT . In this case the autocorrelation and cross correlation matrices giv en in (40) are no w defined using the truncated matrices define in (3) i.e. ¯ ¯ G k = ¯ P H k ¯ P k and ¯ ˜ G k = ¯ P H k ˜ P k respectiv ely . In this case, (44) will now be modified as <{ ¯ ¯ Q k,m,i } =  I + <{ ∆ ¯ ¯ Q k,m,m } for i = m <{ ∆ ¯ ¯ Q k,m,i } for i 6 = m <{ ¯ ˜ Q k,m,i } = <{ ∆ ¯ ˜ Q k,m,i } for i = 0 , · · · , M − 1 (45) where ∆ ¯ ¯ Q k,m,i = ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,i F H k ¯ Φ k,i and ∆ ¯ ˜ Q k,m,i = ¯ Φ H k,m F k ∆ ¯ ˜ G k,m,i F H k ˜ Φ k,i in which ∆ ¯ ¯ G k,m,i and ∆ ¯ ˜ G k,m,i are the error matrices due to the finite filter length and truncating ef fect. Hence, Eq (43) will now be modified as <{ ¯ u m } = ¯ s m + M − 1 X i =0 <{ ∆ ¯ ¯ Q k,m,i } ¯ s i + M − 1 X i =0 <{ ∆ ¯ ˜ Q k,m,i } ˜ s i + <{ ¯ u noise,m } (46) The v ariance of elements in the error matrices not only depends on the filter length K and the truncation number i F and i R , b ut more importantly on the odd or e ven value of K . The truncation causes the filter correlation matrices to be unsaturated at both the edges i.e. the symbols at the start and at the end of the block will e xperience truncation ef fect while the truncation causes the filter correlation matrix to be saturated in the middle part. Hence, the symbols in the middle of the filtered MIMO-FBMC block are least ef fected. This can be confirmed from [12], where we have demonstrated that with finite filter length ( K = 6) , the filter output contains K − 1 symbols and that these extra tails at the edges of the FBMC block hav e small av erage energy compared to the middle part of the block. C. Filter Output T runcation (FO T) Analysis T o analyze the impact of these factors on the filter output truncation, we consider the following cases. W e first consider the e ven v alue of filter length ( K = 6) , which will introduce K − 1 tails i.e. 5 extra symbols at the output of the transmit filter . Also we have assumed M = 8 i.e. symbols per block at the input of the filter . Note that this value of M is considered just as an example and does not affect the outcomes of the analysis . a) Use it all: No cut at all ( i F = 0 , i R = 0 ), i.e., input 8 symbols and output 13 symbols. b) One symbol (front and end): Cut 2 at the front and 1 at the end ( i F = 2 , i R = 1 ), i.e., input 8 symbols and output 10 symbols. c) One symbol (front): Cut 2 at the front and 2 at the end ( i F = 2 , i R = 2 ), i.e., input 8 symbols and output 9 symbols. d) One symbol (end): Cut 3 at the front and 1 at the end ( i F = 3 , i R = 1 ), i.e., input 8 symbols and output 9 symbols. e) The same length: Cut the front 3 and last 2 symbols ( i F = 3 , i R = 2 ), to keep the number of symbols the same i.e. input 8 symbols and output 8 symbols. Fig. 2 sho ws the desired signal and interference po wers for real and imaginary branches in case of finite filter length ( K = 6) with different FO T scenarios. The observations drawn from This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 Fig. 2 regarding the aforementioned cases are discussed as follows • Use it all case i.e. no truncation can achiev e very good performance for both real or imaginary branches. As in this case, the second and third terms in (46) will not exist and therefore the desired symbols are free from interference terms. • One symbol (fr ont and end) case can achieve similar performance as in use it all case, only marginal dif ference is at the edge symbols. This is because the one symbol at the front has significant ener gy as compared to the other tails [12]. Leaving this symbol at the front will significantly reduce the interference lev el and the effect of cutting other two symbols at the front and one at the end has much less affect on the neighboring symbols as can be seen from Fig. 2b. • One symbol (fr ont) case introduces interference at the last symbols i.e. m = 7 , 8 compared to the one symbol (fr ont and end) case. This loss is tolerable as the signal power loss and the increase in the interference level for m = 7 and m = 8 are insignificant as can be seen from Fig. 2b. These losses are acceptable as we are av oiding an extra symbol ov erhead compared to the one symbol (front and end) case. This performance loss at the last symbols is due to the truncation at the end of the filter that introduces interference in the last two symbols. • Howe ver , One symbol (end) case does not work as the signal po wer for m = 1 is reduced and the interference lev el has increased significantly which are both unaccept- able. It is because in this case we are truncating the front part of the filter that discards all the symbols at the front of the block and introduces significant interference in the neighboring symbols. Hence, leaving one symbol at the end is not a good strategy . • In the same length case, the desired signal power and interference po wer for the symbols at the edges ( m = 1 , 2 and 7 , 8) are af fected significantly . This is because the extra symbols at the start and the end of the block are truncated that affects their neighboring symbols. In this case, the second and third terms in (46) will exist and as a result, the detected symbols will be effected by these interference terms. The output signal to interference ratio (SIR) for real and imaginary branches is illustrated in Fig. 2c and Fig. 2f respectiv ely , where we can see that with a finite filter length ( K = 6) , the best SIR can be obtained with use it all case; howe ver , the ov erhead is quite high in this case. While the same length case can completely remov e the overhead but significantly reduces the SIR of the symbols at the edges. A good balance is to adopt the one symbol (fr ont) case for e ven K which offers an acceptable trade-off between the overhead and the performance. Howe ver , the observations are totally re versed when we consider the odd number of filter length e.g. ( K = 5) . In this case the last symbol in the imaginary branch is significantly affected by the FO T as can be seen from Fig. 3. The one symbol (end) case is now more effecti ve in case of odd filter length as it provides better SIR compared to the other cases as can be seen from Fig. 3c and Fig. 3f. Since the target branch and symbol are totally different for odd and ev en K , in the next section, we will focus on the ev en K only for proposing the compensation algorithm. The compensation algorithm for the odd K can be deriv ed using the same approach. V . P R O P O S E D C O M P E N S AT I O N A L G O R I T H M Although adding one symbol (fr ont) case can provide ac- ceptable performance (SIR > 20dB). Howe ver , this approach is valid only when the block size M is large. For instance when M = 20 , the total overhead is only 5 % and this percentage further drops when M goes to larger value [12]. Howe ver , considering different traf fic models and also the latency of the data, the solution that one symbol (fr ont) case may cause significant ov erhead e.g. with moderate M = 5 , the total ov erhead is 20 % , which is very inef ficient. In order to overcome this inefficienc y for moderate M , we propose a compensation approach which allows complete remov al of the overhead caused by the filtering operation. Note that when K is ev en, if we consider the same length case only the first symbol on the I branch has unacceptable lev el of SIR whereas the corresponding symbol on the Q branch has sufficient SIR lev el (20dB) as can be see from Fig. 2c and Fig. 2f respectively . While in the odd K case, the situation is opposite (only the last symbol on the Q branch has unacceptable le vel of SIR) as can be seen from Fig. 3f. W ith this observation, we can state that all of the other symbols (both real and imaginary , except the first real symbol for ev en K or last imaginary symbol for odd K ) can be easily detected. Considering the e ven K case with the assumption that the channel is known and that we only need to compensate the first symbol in the real branch to have sufficient SIR v alue to detect all the symbols. According to (46), the first I branch symbol can be written as <{ ¯ u 0 } = ¯ s 0 + M − 1 X i =0 <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 ,i F H k ¯ Φ k,i } ¯ s i + M − 1 X i =0 <{ ¯ Φ H k, 0 F k ∆ ¯ ˜ G k, 0 ,i F H k ˜ Φ k,i } ˜ s i = ¯ s 0 + <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 } ¯ s 0 + M − 1 X i =1 <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 ,i F H k ¯ Φ k,i } ¯ s i + M − 1 X i =0 <{ ¯ Φ H k, 0 F k ∆ ¯ ˜ G k, 0 ,i F H k ˜ Φ k,i } ˜ s i (47) The first term in (47) is the desired signal, the second term is the ICI and the third and fourth terms are the ISI caused by the I and Q branches respectively . For simplicity , we omit the noise term. In order to improve the SIR of the first symbol in the I branch, we need to compensate the ICI and the ISI terms at the receiver . For this, we need to find the compensation matrices i.e. ∆ ¯ ¯ G k, 0 ,i and ∆ ¯ ˜ G k, 0 ,i in (47). Note This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 1 2 3 4 5 6 7 8 -0.5 -0.4 -0.3 -0.2 -0.1 0 (a) Signal power in I branch 1 2 3 4 5 6 7 8 -50 -45 -40 -35 -30 -25 -20 -15 -10 (b) Interference power in I branch 1 2 3 4 5 6 7 8 10 15 20 25 30 35 40 45 50 (c) Output SIR in I branch 1 2 3 4 5 6 7 8 -10 -8 -6 -4 -2 0 (d) Signal power in Q branch 1 2 3 4 5 6 7 8 -40 -35 -30 -25 -20 -15 -10 -5 (e) Interference power in Q branch 1 2 3 4 5 6 7 8 -10 0 10 20 30 40 (f) Output SIR in Q branch Fig. 3: Signal and interference power with output SIR in real and imaginary branches ( K = 5 ) that the ∆ ¯ ¯ G k, 0 ,i and ∆ ¯ ˜ G k, 0 ,i are caused by the FO T which brings significant SIR reduction for some symbols. T o derive the matrices, we define the perfect autocorrelation matrices ¯ ¯ G k,or ig and ¯ ˜ G k,or ig as follo ws ¯ ¯ G k,or ig = ¯ P H k,or ig ¯ P k,or ig =  ¯ P H k,i F ¯ P H k ¯ P H k,i R    ¯ P k,i F ¯ P k ¯ P k,i R   = ¯ P H k,i F ¯ P k,i F + ¯ P H k ¯ P k + ¯ P H k,i R ¯ P k,i R (48) Similarly , ¯ ˜ G k,or ig = ¯ P H k,i F ˜ P k,i F + ¯ P H k ˜ P k + ¯ P H k,i R ˜ P k,i R (49) W e can write the compensation matrices ∆ ¯ ¯ G k and ∆ ¯ ˜ G k us- ing the perfect autocorrelation matrices ( ¯ ¯ G k,or ig and ¯ ˜ G k,or ig ) and the truncated autocorrelation matrices ( ¯ ¯ G k = ¯ P H k ¯ P k and ¯ ˜ G k = ¯ P H k ˜ P k ) as: ∆ ¯ ¯ G k = ¯ ¯ G k,or ig − ¯ ¯ G k = ¯ P H k,i F ¯ P k,i F + ¯ P H k,i R ¯ P k,i R (50) ∆ ¯ ˜ G k = ¯ ˜ G k,or ig − ¯ ˜ G k = ¯ P H k,i F ˜ P k,i F + ¯ P H k,i R ˜ P k,i R (51) Now for ev en K case, we propose the following compen- sation algorithm to determine ∆ ¯ ¯ G k, 0 ,i and ∆ ¯ ˜ G k, 0 ,i in (47) for compensating ISI in the first real symbol. Using (50) and (51), we can determine ∆ ¯ ¯ G k, 0 ,i = ∆ ¯ ¯ G 0 ,i ⊗ I N r and ∆ ¯ ˜ G k, 0 ,i = ∆ ¯ ˜ G 0 ,i ⊗ I N r for i = 0 · · · M − 1 using (3) as ∆ ¯ ¯ G 0 , 0 = ¯ W H 0 ¯ W 0 + ¯ W H 1 ¯ W 1 + ¯ W H 2 ¯ W 2 ∆ ¯ ¯ G 0 , 1 = ¯ W H 1 ¯ W 0 + ¯ W H 2 ¯ W 1 ∆ ¯ ¯ G 0 , 2 = ¯ W H 2 ¯ W 0 ∆ ¯ ¯ G 0 ,j = 0 for 3 ≤ j ≤ M − 1 (52) and ∆ ¯ ˜ G 0 , 0 = ¯ W H 0 ˜ W 0 + ¯ W H 1 ˜ W 1 + ¯ W H 2 ˜ W 2 ∆ ¯ ˜ G 0 , 1 = ¯ W H 1 ˜ W 0 + ¯ W H 2 ˜ W 1 ∆ ¯ ˜ G 0 , 2 = ¯ W H 2 ˜ W 0 ∆ ¯ ˜ G 0 ,j = 0 for 3 ≤ j ≤ M − 1 (53) A. Compensating the Real Branch Signal: The real branch signal is affected by ISI and ICI terms as shown in (47). The proposed algorithm can compensate these two interferences as follows 1) Compensating the ISI: The third and fourth terms in (47) are the ISI terms caused by the I and Q branch symbols. Using (52) and (53), we can compensate these ISI terms at the recei ver side as ¯ u 0 ,comp = <{ ¯ u 0 } − M − 1 X i =1 <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 ,i F H k ¯ Φ k,i } ¯ s i − M − 1 X i =0 <{ ¯ Φ H k, 0 F k ∆ ¯ ˜ G k, 0 ,i F H k ˜ Φ k,i } ˜ s i = ¯ s 0 + <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 } ¯ s 0 = [ I + <{ ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 } ] ¯ s 0 = < [ I + ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 ] ¯ s 0 (54) 2) Compensating the ICI: Apparently , the term ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 in (54) is also known. Hence, we can compensate this term by using a ZF (or if we consider the noise term in (47) we can use MMSE) equalization at the receiv er to estimate ¯ s 0 with relati vely higher SIR as ˆ ¯ s 0 = ( < [ I + ¯ Φ H k, 0 F k ∆ ¯ ¯ G k, 0 , 0 F H k ¯ Φ k, 0 ]) − 1 ¯ u 0 ,comp (55) This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 1 2 3 4 5 6 7 8 -6 -5 -4 -3 -2 -1 0 1 (a) Signal power in I branch 1 2 3 4 5 6 7 8 -80 -70 -60 -50 -40 -30 -20 -10 0 (b) Interference power in I branch 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 70 80 (c) Output SIR in I branch 1 2 3 4 5 6 7 8 -0.5 -0.4 -0.3 -0.2 -0.1 0 (d) Signal power in Q branch 1 2 3 4 5 6 7 8 -70 -60 -50 -40 -30 -20 (e) Interference power in Q branch 1 2 3 4 5 6 7 8 20 30 40 50 60 70 (f) Output SIR in Q branch Fig. 4: Signal and interference power with output SIR using compensation algorithm ( K = 6 ) It can be seen from (55) that both ICI and ISI can be compensated at the receiv er side. The proposed compensation algorithm can provide the same SIR for the first real symbol as in the use it all case by compensating the effect of FO T as can be seen from Fig. 4c. Further , we can deri ve a generalized expression of (55) which can be used to further improv e the SIR of other symbols as well by compensating their ICI and ISI terms. The generalized expression of (54) can be derived as ¯ u m,comp = <{ ¯ u m } − M − 1 X i =0 ,i 6 = m <{ ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,i F H k ¯ Φ k,i } ¯ s i − M − 1 X i =0 <{ ¯ Φ H k,m F k ∆ ¯ ˜ G k,m,i F H k ˜ Φ k,i } ˜ s i = ¯ s m + <{ ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,m F H k ¯ Φ k,m } ¯ s m = [ I + <{ ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,m F H k ¯ Φ k,m } ] ¯ s m = < [ I + ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,m F H k ¯ Φ k,m ] ¯ s m (56) Similarly , (55) can be generalized as ˆ ¯ s m = ( < [ I + ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,m F H k ¯ Φ k,m ]) − 1 ¯ u m,comp (57) In (56), it is worth mentioning that the term P M − 1 i =0 ,i 6 = m <{ ¯ Φ H k,m F k ∆ ¯ ¯ G k,m,i F H k ¯ Φ k,i } ¯ s i should be treated carefully for i = 0 , since only accurate ˆ ¯ s 0 will bring accurate compensation to other symbols, otherwise errors will be introduced, which implies that ˆ ¯ s 0 should be always compensated first. B. Compensating the Imaginary Branc h Signal: The equalized imaginary branch symbol, ˜ u m can be written as follo ws using the same approach as adopted for the real branch. ˜ u m = ˜ Φ H k,m F k ˜ ˜ G k,m,m F H k ˜ Φ k,m ˜ s m + M − 1 X i =0 ,i 6 = m ˜ Φ H k,m F k ˜ ˜ G k,m,i F H k ˜ Φ k,i | {z } ˜ ˜ Q k,m,i ˜ s i + M − 1 X i =0 ˜ Φ H k,m F k ˜ ¯ G k,m,i F H k ¯ Φ k,i | {z } ˜ ¯ Q k,m,i ¯ s i + E m ˜ Φ H k,m F k ˜ P H k,m n | {z } ˜ u noise,m (58) According to the orthogonality of FBMC with infinite filter length [17], ˜ ˜ Q k,m,i and ˜ ¯ Q k,m,i hav e the follo wing property ˜ ˜ Q k,m,i = ( j I + <{ ˜ ˜ Q k,m,i } for i = m <{ ˜ ˜ Q k,m,i } for i 6 = m ˜ ¯ Q k,m,i = <{ ˜ ¯ Q k,m,i } for i = 0 , · · · , M − 1 (59) Using the property of infinite filter length giv en in (59), we can write (58) as ˜ u m = j ˜ s m + h M − 1 X i =0 <{ ˜ ˜ Q k,m,i } ˜ s i + M − 1 X i =0 <{ ˜ ¯ Q k,m,i } ¯ s i i | {z } ˜ u intri,m + ˜ u noise,m (60) T aking the imaginary part of (60), we hav e ={ ˜ u m } = ˜ s m + ={ ˜ u noise,m } (61) W e can now compensate the Q branch as well using the same approach used for the I branch; howe ver , since all the This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 symbols already hav e good initial SIR, it will be easier to compensate them in this branch compared to the I branch. The compensation approach for the Q branch is as follows ˆ ˜ s m = ( = [ I + ˜ Φ H k,m F k ∆ ˜ ˜ G k,m,m F H k ˜ Φ k,m ]) − 1 ˜ u m,comp (62) where, ˜ u m,comp = ={ ˜ u m } − M − 1 X i =0 ,i 6 = m ={ ˜ Φ H k,m F k ∆ ˜ ˜ G k,m,i F H k ˜ Φ k,i } ˜ s i − M − 1 X i =0 ={ ˜ Φ H k,m F k ∆ ˜ ¯ G k,m,i F H k ¯ Φ k,i } ¯ s i (63) C. Combining Real and Imaginary Branc hes: W ith (57) and (62), we can write the compensated estima- tion of s m as follo ws ˆ s m = ˆ ¯ s m + j ˆ ˜ s m (64) V I . S I M U L A T I O N R E S U L T S In this section, we present a set of simulation results to demonstrate the effecti veness of the proposed compensation algorithm in the the same length case. For simulations, the selected parameters for the MIMO-FBMC system includes the IO T A prototype filter with overlapping factor K = 6 . The number of transmit and receive antennas are N t = N r = 2 . The desired signal is modulated by QPSK with normalized power and input signal to noise ratio (SNR) is controlled by the noise power . The L TE channel model considered in our simulation is the extended pedestrian A model (EP A) [40]. For the equalization, the MMSE based equalizer is selected as it is more generic. As we have concluded in Section V that our main concern is the first real branch symbol which has a very low SIR value of around 2dB. The proposed compensation algorithm giv en in (55) significantly improves the SIR of the first real symbol i.e. the signal power increase from -5.1dB to 0dB while the interference lev el drops from -5dB to -48dB. Hence, increasing the SIR of the first real symbol from 2dB to 48dB as can be seen from Fig. 4. Note that we do not need to compensate the imaginary branch as it already has sufficient SIR values for detecting all the symbols at the receiver as discussed in Section V. Howe ver , the term acceptable SIR value is strongly de- pendent on the modulation order as higher modulation order require high SIR v alues for achieving a specific required performance. The proposed general form of the compensation algorithm given in (57) and (62) is incorporated at the receiv er side of the MIMO-FBMC system. The algorithm significantly improv es the SIR of all the symbols in the real and imaginary branches respectiv ely as can be see from Fig. 4. Compensating all the symbols can help in improving the probability of detection at the receiver . The coded results (con volutional code with code rate 1/2) for the BER performance of various FO T schemes in MIMO-FBMC system with and without compensation algorithm are presented in Fig. 5. It can be observed that the system performance in case of use it all and one symbol (fr ont) has similar BER performance but the latter required only one extra tail compared to the former, which required K − 1 extra tails. The same length case requires no extra tail but has a relativ ely poor BER performance. 0 5 10 15 10 -6 10 -4 10 -2 10 0 12 13 14 15 10 -4 Fig. 5: BER performance of OFDM and FBMC system with and without compensation W e ha ve used con ventional MIMO-OFDM as a baseline scheme to show the advantage of the proposed algorithm over such con ventional multicarrier schemes. For a fair comparison between MIMO-FBMC and MIMO-OFDM systems, the SNR loss, due to the cyclic prefix (overhead) in OFDM, must be considered. For this reason, we have calculated the noise power for both systems as discussed in [12]. The comparison shows the significance of the proposed algorithm especially for higher modulation schemes. It can be seen from Fig. 5 that for low order modulation schemes like QPSK, MIMO- FBMC system without compensation can still perform better than con ventional MIMO-OFDM but if we increase the mod- ulation schemes to higher order like 16QAM or 64QAM, the performance of MIMO-FBMC system without compensation becomes poorer than MIMO-OFDM due to self-interference caused by FOT . In such cases, use of the proposed compensa- tion algorithm is very significant as it not only provides better performance but also improves the spectral efficienc y (SE) of the system. 0 5 10 15 20 5 10 15 20 (a) SE with respect to M 0 5 10 15 20 0 50 100 150 200 250 (b) SE gain with respect to M Fig. 6: Spectral efficienc y of MIMO-FBMC with respect to block size ( M ) This article has been published in IEEE journal. The final version is av ailable at http://dx.doi.org/10.1109/TVT .2017.2771531 0 5 10 15 6 8 10 12 14 16 18 20 (a) SE with respect to E b / N o 0 5 10 15 0 10 20 30 40 50 60 70 80 (b) SE gain with respect to E b / N o Fig. 7: Spectral efficienc y of MIMO-FBMC with respect to SNR ( E b / N o ) The SE of the system has been simulated using Shannon equation [37] which giv es an upper bound of the capacity that the system can achieve i.e. maximum error free transmission rate. Note that the capacity is measured using only the simula- tion and is not the exact r epresentation of achievable capacity . The objectiv e is to provide an idea regarding the spectral efficienc y gain that can be achie ved using the filter output truncation and the compensation algorithm at the receiv er . The spectral efficiency expression with respect to the block size M used in the simulation is giv en as follo ws SE = min { N T , N R } × M M + α ( 1 M M X i =1 log 2 (1 + S I N R m ) ) (65) where α represents the overhead in each case i.e. K − 1 for use it all , 1 for one symbol (fr ont) and 0 for compensate all . The SE in each case is giv en in Fig. 6. It can be observed from Fig. 6a that the SE is independent of M for the compensate all case whereas it is dependent for the use it all and one symbol (fr ont) cases as they have one and K − 1 tails respecti vely with each block. It can also be seen from Fig. 6b that the SE gain obtained using the compensate all case reduces with the increase in M for both use it all and one symbol (fr ont) cases. Hence, the compensation algorithm is best suited for applications that has a frame structure based on moderate M . The SE results for a range of SNR ( E b / N o ) values are also shown in Fig. 7. W ith a fixed block size, the SE of the system increases as input SNR ( E b / N o ) increases as shown in Fig. 7a. It can be observed that SE performance of the compensate all case is better than one symbol (fr ont) and use it all cases, as these FOT schemes require certain ov erhead to achie ve improved BER performance. Howe ver , the proposed compensation algorithm provides similar BER performance by compensating the effects of FO T in the same length case without introducing any overhead. This enables compensate all case to have a certain SE gain compared to other FO T schemes as can be seen from Fig. 7b. V I I . C O N C L U S I O N The impact of finite filter length and different types of FO T has been theoretically analyzed in a MIMO-FBMC system. The analysis is based on a compact matrix model of a MIMO- FBMC system, which was then used for in vestigating the effects of FO T on the detection performance in terms of the SIR and BER. The analysis showed that although FO T can av oid ov erhead but it also destroys the orthogonality in the FBMC system thus introducing interferences. Howe ver , due to the isolation property between the (FBMC) symbols, only real part of the first symbol or the imaginary part of the last symbol are affected by the aforementioned interferences. A general form of compensation algorithm based on the observations made in the theoretical analysis has been de- signed to compensate the symbols in a MIMO-FBMC block to improve the SIR of each symbol. The adv antage of this algorithm is that it improves the spectral ef ficiency of the system as it requires no overhead and at the same time can still achieve similar performance compared to the case without FO T . Howe ver , the spectral ef ficiency gain tends to reduce with the increasing M as the overhead tends to decrease with increase in frame size. 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