Interference Analysis of QAM based Filter Bank Multicarrier System with Index Modulation

1 Interference Analysis of QAM based Filter Bank Multicarrier System with Inde x Modulation Adnan Zafar , Aijun Cao, Mahmoud Abdullahi, Lei Zhang, Pei Xiao and Muhammad Ali Imran Abstract —Index modulation (IM) has r ecently emerged as a promising concept for spectrum and energy-efficient next generation wireless communications systems since it strikes a good balance among error perf ormance, complexity , and spectral efficiency . IM technique when applied to a multicarrier wav e- forms yields the ability to con vey the inf ormation not only by M - ary signal constellations as in con ventional multicarrier systems, but also by the indexes of the subcarriers, which are activated according to the incoming bit stream. Although IM is well studied for OFDM based systems, FBMC with index modulation has not been thoroughly in vestigated. In this paper , we shed light on the potential and implementation of IM technique for QAM based FBMC system. W e start with a mathematical model of the IM based QAM-FBMC system (FBMC/QAM-IM) along with the derivation of interference terms at the recei ver due to channel distortions and noise. The interference terms including the ones intr oduced by the multipath channel are analyzed in terms of MSE and output SINR. It is shown with analytical and simulation results that the interference power in FBMC/QAM- IM is smaller comparing to that of the con ventional FBMC/QAM system as some of the subcarriers are inactive. The perf ormance of FBMC/QAM with IM is inv estigated by comparing the SIR and output SINR with that of the con ventional FBMC/QAM system along with the BER performance which shows that the FBMC/QAM-IM is a promising transmission technique f or future wireless networks. Index terms – FBMC, intrinsic interference, index mod- ulation, maximum likelihood detection, log likelihood ratio detector , interference analysis I . I N T RO D U C T I O N I N D E X modulation technique, which utilizes an innov ative way of con veying information compared to conv entional communication systems, has emerged as a promising candidate scheme for 5G wireless networks [1]. In this scheme, information is not only con veyed by the M -ary signal constellation, but also by the index es of the building blocks of the corresponding communication system. The two main promising applications of IM is in multiple antenna systems like MIMO and multicarrier schemes like OFDM. In MIMO, IM can be applied on the transmit antennas of the MIMO system to con vey additional information. This application of IM to MIMO systems is termed as SM [2]. On the other hand application of IM with a multicarrier A.Zafar , M.Abdullahi and P .Xiao are with Institute for Communication Systems (ICS), Uni versity of Surre y , Guildford, UK. A.Zafar is also with Insti- tute of Space T echnology , Islamabad, Pakistan. Email: { a.zafar , m.abdullahi, p.xiao } @surrey .ac.uk A.Cao is with ZTE R&D Center , 164 51 Stockholm, Sweden. Email: cao.aijun@zte.com.cn L.Zhang and M.A.Imran are with School of Engineering, Univ ersity of Glasgow , Glasgow , UK. Email: { lei.zhang, muham- mad.imran } @glasgow .ac.uk scheme like OFDM considers subcarrier indexes as a source of con ve ying additional information [3]. IM based OFDM (OFDM-IM) system can also be considered as a frequency domain extension of the SM concept. Recent studies have shown that OFDM-IM can offer appealing advantages over classical OFDM and is also being proposed as a strong wa veform candidate for future wireless networks [4]. The performance improvement in OFDM-IM system comes from the fact that the system utilizes the index es of the active subcarriers to con ve y additional information bits. At the transmitter side of the OFDM-IM system, the subcarriers in each symbol are divided in to sub-groups comprising of activ e and inactive subcarriers. The indexes of these activ e subcarriers are used as a source of additional information which may leads to improvement in the transmission efficienc y of the system. At the recei ver side, the activ e subcarrier indexes can either be detected by a ML detector or a lo w-complexity LLR detector [5]. Furthermore, in IM based multicarrier systems, the po wer of inactive subcarriers can either be reallocated to acti ve subcarriers in each sub-group to improv e the BER performance of the system and can also be sa ved to improve the energy efficiency of the system [6]. This has enabled IM to emerge as a promising spectral and energy ef ficient modulation schemes for future wireless communications. A generalization of OFDM-IM ha ve also been proposed in recent years to achie ve higher spectral efficiencies [7] and impro ved system performance [8]. A direct combination of OFDM-IM with MIMO transmission techniques, called as MIMO-OFDM-IM, is proposed in [9], [10]. It is sho wn that the proposed MIMO-OFDM-IM system can achiev e a linear increase in spectral ef ficiency (SE) of the system. Despite several advantages, conv entional OFDM based systems have various short comings including poor OoBR and strict synchronization requirements. In recent years, FBMC/OQAM has emerged as a promising candidate wav eform for future wireless networks due to its profound advantages over con ventional OFDM [11]–[13]. Howe ver , con ventional FBMC/OQAM suffers from pure imaginary intrinsic interference caused by neighboring symbols. Some initial work on the application of IM with FBMC/OQAM (FBMC/OQAM-IM) has been in vestigated and it is shown that FBMC/OQAM-IM has improved performance compared to con ventional FBMC/OQAM system. Howe ver , when the IM scheme is applied to FBMC/OQAM system, the intrinsic interferences are partly eliminated and the remaining interferences still affect the BER performance [14]–[16]. Although IM is well studied for OFDM based systems and some literature on FBMC with OQAM modulation is 2 av ailable, FBMC/QAM with index modulation (FBMC/QAM- IM) has not been thoroughly in vestigated. The main work in this contrib ution is summarized as follows • A mathematical matrix model of the index modulation based FBMC/QAM system is presented along with the deriv ation of interference terms at the receiver due to channel distortions and the intrinsic beha vior of the transceiv er model. • The interference terms including the ones introduced by the multipath channel are analyzed in terms of MSE with and without IM. It is analytically shown that the inter - ference power in FBMC/QAM-IM is smaller comparing to that of the conv entional FBMC/QAM system as some subcarriers are inactiv e. • The performance of FBMC/QAM-IM is ev aluated by comparing the SIR and output SINR with that of the con ventional FBMC/QAM system. • An improved LLR detector is then proposed based on the proposed interference model. It is shown with simu- lations that the proposed detector provide improved BER performance in FBMC/QAM-IM system compared to the con ventional FBMC/QAM system. The rest of this paper is organized as follows: A mathematical model of the IM based QAM-FBMC system is presented in Section II. The interference terms including the ones introduced by the multipath channel and the intrinsic behavior of the FBMC/QAM transceiv er model are analyzed in terms of MSE with and without IM and are presented in Section III. W e proposed an improved LLR detector based on the interference analysis in Section IV . The simulation results including MSE, SIR, output SINR and BER are provided in Section V . Lastly , conclusion is provided in Section VI. Notations: V ectors and matrices are denoted by lower - case and uppercase bold letters. b x c represent the greatest integer smaller than x. {·} H , {·} T , {·} ∗ stand for the Hermitian conjugate, transpose and conjugate operation, respectively . E { A } denotes the expectation operation of A . F and F H represents the po wer normalized N point discrete Fourier transform (DFT) and inv erse DFT matrices. I m × m refers to m dimension identity matrix and for some cases the subscript will be dropped for simplification. A ⊗ B represents kronecker product of A and B. k A k 2 n means taking the n th diagonal element of matrix k A k 2 = AA H . W e use ∗ as a linear con volution operator . Tr { A } denotes the trace of matrix A . I I . F B M C / Q A M S Y S T E M W I T H I N D E X M O D U L A T I O N In this section we define the index modulation based FBMC/QAM system in matrix form which will be subse- quently used to analyze the interference reduction due to the introduction of IM in conv entional FBMC/QAM system. A. System Model The system model of FBMC/QAM-IM is developed as an extension to the FBMC/QAM model presented in [17]. W e will focus on the index modulation block in detail while the FBMC/QAM model will be briefly discussed. 1) Index Modulation: The block diagram of the FBMC/QAM-IM system is illustrated in Fig. 1. Since, index modulation utilizes index es of certain subcarriers according to the incoming bit stream to con vey information, we define the subcarriers bearing M -ary signal constellation symbols as active subcarriers and the rest as inactiv e. The proposed model of FBMC/QAM-IM follows a block based processing approach where each block contains M symbols and each symbol has N subcarriers in the frequency domain. Let us consider that transmitted information of each block has T bits. These information bits are then divided into g groups each containing p bits which are then mapped to an FBMC/QAM subblock of length n , where n = N /g . Each group of p bits can be split into two parts. The first p 1 bits are used for index selection in which k out of n av ailable subcarrier are acti vated according to a predefined look up table [5]. The remaining p 2 bits are mapped on to the M -ary signal constellation to define the data symbols that will modulate the activ e subcarriers. The number of bits p 1 carried by the indexes of the activ e subcarriers are defined as p 1 = b log 2 C n k c , (1) where C n k denotes the binomial coefficient. In other words, selected indexes can have c = 2 p 1 possible realizations. The number of bits carried by the M -ary constellation symbols are defined as p 2 = k log 2 M , (2) The total number of bits that are transmitted by one FBMC/QAM-IM block can be defined as T = ( p 1 + p 2 ) g = ( b log 2 C n k c + k log 2 M ) g (3) In other w ords, in FBMC/QAM-IM system, the information is con veyed by both the indexes of the activ e subcarriers and also the M -ary constellation symbols modulating these activ e subcarriers. Also we can infer that, as we are not using all of the av ailable subcarriers for the data transmission, the loss in transmission ef ficiency is compensated by transmitting additional bits in the spatial domain of the FBMC/QAM block. From Fig. 1, it can be seen that for each subblock β , the first p 1 bits are used for index selection i.e., k out of n av ailable index es are selected as i β = [ i β , 1 , ..., i β ,k ] (4) where i β ,γ ∈ [1 , ..., n ] , β = 1 , ..., g and γ = 1 , ..., k . T o modulate these activ e subcarriers, the remaining p 2 bits are mapped on the M -ary signal constellation to define the transmitted data symbols as a β = [ a β , 1 , ..., a β ,k ] (5) where a β is the symbol vector transmitted on the acti ve sub-carriers in each sub-block. Such that a β ,γ ∈ S for β = 1 , ..., g and γ = 1 , ..., k , where S is the set of all possible complex symbols from the M -ary constella- tion. Using i β and a β from (4) and (5), the FBMC/QAM block creator generates all of the subblocks A m,β ∈ C n × 1 3 Fig. 1: Block diagram of FBMC/QAM-IM T ransmitter for β = 1 , ..., g and form FBMC/QAM-IM symbol as s m = [ s m, 0 , s m, 1 , ..., s m,N − 1 ] T = [ A m, 1 , A m, 2 , ..., A m,g ] T ∈ { 0 , a β ,γ } . The generated FBMC/QAM-IM block can be ex- pressed as S = [ s 0 , s 1 , ..., s M − 1 ] ∈ C N × M . Furthermore, to hav e unit a verage po wer , we reallocate the power from inacti ve subcarriers to the activ e subcarriers in a subgroup. The av erage power of the modulated symbol s m,n can be represented as E {k s m,n k 2 } = δ 2 . It should be noted that when the number of activ e subcarriers k is equal to the number of carriers in a subgroup i.e., k = n , the FBMC/QAM-IM system will become a con ventional FBMC/QAM system. Hence, the FBMC/QAM system can be vie wed as a special case of the FBMC/QAM- IM system. It should be noted that by properly choosing n and k , the spec- tral efficienc y of FBMC/QAM-IM system can be improv ed. The spectral efficienc y η F B M C/QAM − I M can be calculated as follows η F B M C/QAM − I M = M N n ( b log 2 C n k c + k log 2 M ) N M bits/sec/Hz (6) As we already mentioned that FBMC/QAM-IM system re- duces to FBMC/QAM system in case of k = n , the spectral efficienc y of FBMC/QAM can now be deriv ed using (6) as η F B M C/QAM = log 2 M bits/sec/Hz (7) The spectral efficienc y gain of FBMC/QAM-IM over con ven- tional FBMC/QAM system can be expressed as η g ain = η F B M C/QAM − I M η F B M C/QAM = M N n ( b log 2 C n k c + k log 2 M ) N M log 2 M = b log 2 C n k c n log 2 M + k n (8) W e can see from (8), that by properly selecting n , k and M -ary modulation, we can achieve spectral efficienc y gain η g ain > 1 . For example, when n = 4 , k = 3 and M = 2 i.e., BPSK modulation, we can calculate η g ain = 1 . 25 , which indicates that the spectral efficienc y of FBMC/QAM-IM has exceeded that of con ventional FBMC/QAM. 2) T ransmit Processing: It can be seen from Fig. 1 that after index modulation, the signal s m is passed though the con ventional FBMC/QAM transmitter i.e., N point IDFT processor , parallel to serial con version and the transmit filter matrix P . The structure of P is already defined in [17]. The output of the transmit filter matrix can be expressed as o = Pb ∈ C ( K + M − 1) × 1 , (9) where b is the signal v ector processed symbol by sym- bol through the IDFT block i.e., F H . The signal vec- tor can be represented as b = [ b 0 ; b 1 ; · · · ; b M − 1 ] = [ F H s 0 ; F H s 1 ; · · · ; F H s M − 1 ] ∈ C M N × 1 . It should be noted that the prototype filter matrix P is designed in a manner that when it is multiplied by vector b ; the multiplication of matrices is equiv alent to the required linear conv olution process. As a result the output of the transmit filter will have ( K − 1) N more samples than the signal v ector b as can be seen from (9). 3) P assing thr ough the Channel: W e assume the system operates over a slo wly-varying fading channel i.e., quasi-static fading channel. In such a scenario, we can assume that the duration of each of the transmitted data block is smaller than the coherence time of the channel, therefore the random fading coefficients stay constant ov er the duration of each block [18]. In this case, we define the multipath channel as a L -tap channel impulse response (CIR) with the l th -tap power being ρ 2 l . It is also assumed that the av erage power remains constant during the transmission of the whole block. Let us define the CIR 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 , (10) where h l denotes the channel coefficient of the l th tap in the time domain and the comple x random variable z l with comple x Gaussian distribution as C N (0 , 1) represents the multipath fading factor of the l th tap of the quasi-static rayleigh fading channel. The signal vector o is then passed through the multipath channel h as discussed in [17]. The received signal after passing through the channel can be represented as r = L − 1 X l =0 ρ l Z l Pb ↓ l e + o f d + o I B I + n , (11) The first term in (11) represents the linear conv olution process between the channel and the transmitted signal where Z l implies that each FBMC symbol in a block experiences the same channel i.e., Z l = z l × I ( K + M − 1) N × ( K + M − 1) N 4 and b ↓ l e = X l b . Note that the matrix X l = I M ⊗ X sub,l is a block diagonal exchange matrix where X sub,l = [ 0 l × ( N − l ) , I l ; I N − l , 0 ( N − l ) × l ] and is used to exchange the locations of elements of b . Furthermore, o f d = P L − 1 l =0 ρ l Z l ∆ P ↓ l b ↓ l e is the interference caused by the fil- ter distortion due to channel multipath effect, o I B I = P L − 1 l =0 ρ l Z l r B ,l is the inter-block interference (IBI) caused by multipath channel with r B ,l = [ r p,l ; 0 [( M + K − l ) N − l ] × 1 ] and r p,l ∈ C l × 1 is the interfering signal from the previous FBMC/QAM block and n is a Gaussian noise vector with each element having zero mean and variance σ 2 . 4) Receive Pr ocessing: The recei ver process of FBMC/QAM-IM is shown in Fig. 2. At the receiv er , the received signal r is first passed through the receiv e filter P H , which serve as a matched filter in FBMC/QAM recei ver . The signal after matched filtering is then passed though the in verse filter R to cancel the intrinsic interferences in the receiv ed signal block 1 . The signal after the in verse filtering is then processed symbol by symbol through the DFT processor i.e., F . The resulting frequency domain signal y m is then equalized using ZF or MMSE equalizer E to counter the effects of multipath channel. The complete receiv e processing is already deriv ed in detail in [17]. Following the same receiv er processing steps, the equalized symbol ˆ s m can now be represented as follows ˆ s m = s m |{z} Desired Signal + ( I − β ) s m | {z } MMSE Estimation Bias + E F R m P H m o f d | {z } Filter Distortion by Multipath + E F R m P H m o I B I | {z } IBI by Multipath + E F R m P H m n | {z } Noise (12) where β = EC is a diagonal matrix with its n th diagonal element being defined as β n = E n C n = | E n | 2 | E n | 2 + ν σ 2 /δ 2 , (13) and C is the frequenc y domain channel coef ficients in diagonal matrix form and is giv en as C = diag [ C 0 , C 1 , · · · , C N − 1 ] ∈ C N × N . The n th block diagonal element in the frequenc y response of the channel can be represented as C n = P L − 1 l =0 h l e − j 2 π N nl , 0 ≤ n ≤ N . As we can see from (12) that the transmitted signal s m is free from ICI and ISI terms due to the use of inv erse filter matrix R . Ho wev er , the recei ved symbols are affected by interferences caused by MMSE esti- mation bias, IBI and filter distortion due to multipath channel. Since, we kno w that some of the sub-carriers in FBMC/QAM- IM system are in-acti ve and therefore they will not contribute to the overall interference in the system. In what follows, we will ev aluate the interference and noise power in FBMC/QAM and FBMC/QAM-IM systems to estimate the performance improv ement in term of MSE and output SINR due to the introduction of index modulation. I I I . I N T E R F E R E N C E A N D N O I S E P OW E R E S T I M AT I ON It can be seen from (12), that the transmitted symbol vector s m is accompanied with interference terms caused by the 1 The concept of using in verse filter in FBMC/QAM system to cancel the self-interference (so called intrinsic interference) has been proposed in [17] multipath channel and noise. In this section we estimate the interference and noise power in the FBMC/QAM-IM system. First, let us introduce a diagonal matrix D m ∈ R N × N such that the n th diagonal element of D m is defined as D m,n =  1 if s m,n ∈ a β ,γ 0 if s m,n = 0 (14) The diagonal matrix D m represent the active and inactive index es of the transmitted symbol vector s m . It should be noted that introducing D m will hav e no effect on the system model since D m s m = s m . The purpose of defining D m is to ev aluate the impact of inactiv e subcarriers on the system performance. Since some of the subcarriers in FBMC/QAM- IM are inacti ve, therefore the y will not contribute to the ov erall interference in the system. Furthermore, it should be noted that D m will be an identity matrix in case of con ventional FBMC/QAM system since all of the subcarriers are activ e i.e., n = k and contribute to the overall interference in the system. It can be seen from (12) that the estimated symbol is accompanied with MMSE estimation bias, interference terms like filter distortion and IBI due to multipath channel and noise i.e., ˆ s m = s m |{z} Desired Signal + ψ resd,m | {z } MMSE Estimation Bias + ψ f d,m | {z } Filter Distortion by Multipath + ψ I B I ,m | {z } IBI by Multipath + ψ noise,m | {z } Noise , (15) The MSE of the n -th modulation symbol estimation in the m -th FBMC/QAM-IM symbol can be deriv ed as γ tot,m,n = E || ˆ s m,n − s m,n || 2 = E  k ψ resd,m k 2 n + k ψ f d,m k 2 n + k ψ I B I ,m k 2 n + k ψ noise,m k 2 n  (16) A. V ariance of signal estimation bias The desired signal estimation bias is caused by the MMSE receiv er since it minimizes the MSE between the transmitted and recei ved signal. This leads to residual interference in the estimated signal. From (16) and (12), we can write the variance of the signal estimation bias as γ resd,m,n = E k ψ resd,m k 2 n = E {k ( I − β ) s m k 2 n } = E {k ( I − β ) D m s m k 2 n } = δ 2 k ( I − β ) D m k 2 n = δ 2 D m,n ( I − β n ) 2 (17) where E { s m s H m } = δ 2 I and according to (13), β n = | C n | 2 | C n | 2 + ν σ 2 /δ 2 . Substituting β n into (17) yields γ resd,m,n = δ 2 D m,n ( I − β n ) 2 = δ 2 D m,n ( I − 2 β n + β 2 n ] , = δ 2 D m,n h I − 2 | C n | 2 | C n | 2 + ν σ 2 /δ 2 + | C n | 4 ( | C n | 2 + ν σ 2 /δ 2 ) 2 i , = δ 2 D m,n  ν 2 σ 4 ( δ 2 | C n | 2 + ν σ 2 ) 2  . (18) Apparently , when the ZF receiver is adopted, γ resd,m,n = 0 since ν = 0 . Howe ver , the ZF equalization leads to noise enhancement unlike MMSE receivers. 5 Fig. 2: Block diagram of FBMC/QAM-IM Receiv er B. V ariance of filter distortion due to multipath channel W e can write the variance of the interference caused by filter distortion due to multipath channel from (16) and (12) as γ f d,m = E k ψ f d,m k 2 = E k E F R m P H m o f d k 2 , = E [ E F R m P H m o f d o H f d P m R H m F H E H ] , = E F R m P H m E [ o f d o H f d ] P m R H m F H E H , = E F R m P H m α f d P m R H m F H E H , (19) Using (11), we can determine α f d = E [ o f d o H f d ] as follows α f d = E hn L − 1 X l =0 ρ l Z l ∆ P ↓ l b ↓ l e o { L − 1 X l =0 ρ l Z l ∆ P ↓ l b ↓ l e o H i , = L − 1 X l =0 ρ 2 l E [ Z l ∆ P ↓ l b ↓ l e b ↓ lH e ∆ P ↓ lH Z H l ] = L − 1 X l =0 ρ 2 l E [ Z l ∆ P ↓ l X l F H D m s m s H m D H m F X H l ∆ P ↓ lH Z H l ] (20) From (10), we know that E { Z l Z H l } = 1 since z l ∈ C N (0 , 1) also E { s m s H m } = δ 2 I and k X l F H D m D H m F X H l k n = k n k I N × N k n , consequently α f d = k n δ 2 L − 1 X l =0 ρ 2 l T r { ∆ P ↓ l ∆ P ↓ lH } , = k n δ 2 L − 1 X l =0 ρ 2 l T ↓ l , (21) where T ↓ l = T r [∆ P ↓ l ∆ P ↓ lH ] . Since T ↓ l is a scalar , α f d is also a scalar . Now substituting (21) into (19), yields γ f d,m = α f d E F R m P H m P m R H m F H E H , (22) By taking the n th diagonal element of γ f d,m , we obtain γ f d,m,n = α f d k E F R m P H m P m R H m F H E H k n , = α f d | E n | 2 ζ m,n . (23) where kF R m P H m P m R H m F H k n = ζ m,n k I N × N k n . From (23), we can see that the variance of filter distortion in FBMC/QAM-IM system (where k < n ) is k n times the variance of filter distortion in FBMC/QAM system (where k = n ). Hence, we can say that the inactive subcarriers do not contribute to the total interference in the FBMC/QAM-IM system since k n < 1 in FBMC/QAM-IM system. C. V ariance of IBI W e can write the variance of the interference caused by IBI from (16) and (12) as γ I B I ,m = E k ψ I B I ,m k 2 = E k E F R m P H m o I B I k 2 , = E [ E F R m P H m o I B I o H I B I P m R H m F H E H ] , = E F R m P H m E [ o I B I o H I B I ] P m R H m F H E H , = E F R m P H m α I B I P m R H m F H E H , (24) where α I B I = E [ o I B I o H I B I ] , now using (11), we can determine α I B I as α I B I = E hn L − 1 X l =0 ρ l Z l y B ,l on L − 1 X l =0 ρ l Z l y B ,l o H i , = E h L − 1 X l =0 ρ 2 l Z l E { y B ,l y H B ,l } Z H l i , (25) Since Z l has a comple x Gaussian distribution i.e. C N (0 , 1) and also Z l and y B ,l are uncorrelated, we can write the above equation as follows α I B I = L − 1 X l =0 ρ 2 l E { y B ,l y H B ,l } , (26) E { y B ,l y H B ,l } is dependent on the signal type of the last bock, where we assume it is also occupied by an FBMC symbol with the same power , then we hav e E { y B ,l y H B ,l } = E k P ( l ) b last k 2 = T r  P ( l ) E { b last b H last } P H ( l )  , = T r  P ( l ) E {F H D last s last s H last D H last F } P H ( l )  , = k n δ 2 T r  P ( l ) P H ( l )  = k n δ 2 T r  P corr ( l ) ] , = k n δ 2 P corr ( l ) , (27) where P ( l ) = [ P ( last − l ) ; 0 ( M + K − 1) N − l × M N ] in which P ( last − l ) contains the last l -th rows of P also b last is the symbol (after IDFT) in the last block and E { s last s H last } = δ 2 I 6 and kF H D last D H last F k n = k n k I N × N k n . Substituting (27) into (26), we obtain α I B I = k n δ 2 L − 1 X l =0 ρ 2 l P corr ( l ) , (28) Since P corr ( l ) is a scalar , α I B I is also a scalar . Substituting (28) into (24), yields γ I B I ,m = α I B I E F R m P H m P m R H m F H E H , (29) By taking the n th diagonal element of γ I B I ,m , we deriv e the MSE of IBI as γ I B I ,m,n = α I B I k E F R m P H m P m R H m F H E H k n , = α I B I | E n | 2 ζ m,n . (30) where kF R m P H m P m R H m F H k n = ζ m,n k I N × N k n . It should be noted that if we consider a sufficient guard interval between the data blocks then we can safely assume the inter-block interference to be negligible i.e., γ I B I ,m,n = 0 . Also from (30), we can see that the variance of IBI in FBMC/QAM-IM system is also k n times the variance of IBI in FBMC/QAM system. D. V ariance of Noise W e can write the variance of the noise from (16) and (12) as γ noise,m = E k ψ noise,m k 2 = E k E F R m P H m n k 2 , = E [ E F R m P H m nn H P m R H m F H E H ] , = σ 2 E F R m P H m P m R H m F H E H (31) where E { nn H } = E k n k 2 = σ 2 since n is Gaussian noise with each element having zero mean and variance σ 2 . T aking the n th diagonal element of (31), we have γ noise,m,n = σ 2 k E F R m P H m P m R H m F H E H k n , = σ 2 | E n | 2 ζ m,n . (32) where kF R m P H m P m R H m F H k n = ζ m,n k I N × N k n . Note that the ζ m,n is the noise / interference enhancement factor which is introduced when we use an in verse filter matrix at the receiv er . W e can also see that the v ariance of noise in FBMC/QAM-IM system is the same as the v ariance of noise in FBMC/QAM system as it is independent of the acti ve an inac- tiv e subcarrier selection. Hence, the performance improvement in FBMC/QAM-IM system comes from the reduction in the variance of interferences due to the use of index modulation. I V . I M P RO V E D R E C E I V E R F O R F B M C / Q A M - I M The con ventional receivers in FBMC/QAM system are used for detecting the M -ary symbols to extract the transmitted information. Howe ver , the FBMC/QAM-IM receiv er needs to detect the index es of the activ e sub-carriers and also the cor- responding information ( M -ary) symbol transmitted on those activ e subcarriers. T o detect the activ e sub-carrier index es and the M -ary symbols on the active subcarriers, the receiv ed symbol vector ˆ s m is di vided into g groups by the detection group creator block i.e., ˆ s m = [ B m, 1 , B m, 2 , ..., B m,g ] T ∈ C N × 1 as shown in Fig. 2. Each sub-block B m,g ∈ C n × 1 can now be detected by an optimum ML detector . The output of the detector is then used to extract the information embedded in the index es of the acti ve subcarriers as well as the constellation symbols transmitted on those activ e subcarriers. Howe ver , the optimal ML detector suffers from high complexity . In the following section, using the interference and noise po wer analysis presented in Section III, we propose a low complexity detector based on the LLR approach. A. Maximum likelihood (ML) Detector The ML detector is an optimum detector that considers all possible sub-block realizations by searching for all possible sub-carrier index combinations and signal constellation points to make a joint decision on the activ e indexes and the constellation symbols for each sub-block by minimizing the following metric ˆ A m,β = arg min A m,g ∈ Γ   B m,β − A m,β   2 (33) Thus, the ML detector chooses the sub-block A m,g ∈ Γ , where Γ is the set of all the possible sub-blocks, that yields the smallest distance with the receiv ed sub-block B m,g to estimate the transmitted sub-block. From the estimated sub- block ˆ A m,β , the index bits and M -ary symbol bits can then be decoded using the index decoder and M -ary demodulator as shown in Fig. 2. Although the ML detector can provide optimal performance but its complexity is unaffordable i.e., ∼ O ( 2 p 1 M k ). Therefore, to reduce the complexity of the ML detector , we propose an improved LLR detector based on the interference and noise power analysis provided in Section III. The complexity of a LLR detector is ∼ O ( M ) which makes it less complex than ML detector [15]. B. Log-likelihood Ratio (LLR) Detector In this section we proposed an improved detector for FBMC/QAM-IM based on the interference analysis giv en in Section III. A general LLR detector provides the logarithm of the ratio of a posteriori probabilities of the frequency domain symbols by considering the fact that their values can either be zero or non-zero depending upon the sub-carrier being active or inactiv e. T o determine the status of any subcarrier being activ e or inactiv e, we can use the following ratio λ m,n = log e P M χ =1 P ( s m,n = a χ   ˆ s m,n ) P ( s m,n = 0   ˆ s m,n ) (34) where a χ ∈ S and λ m,n is the LLR value of the n th subcarrier of m th symbol in a FBMC/QAM-IM block. It should be noted that a larger value of λ m,n means it is more probable that the n th subcarrier under consideration was selected by the index selection block at the transmitter or in other words the subcarrier was activ e. The LLR expression given in (34) can be simplified by applying Bayes’ formula as follows λ m,n = log e P M χ =1 P ( ˆ s m,n   s m,n = a χ ) P ( s m,n = a χ ) /P ( ˆ s m,n ) P ( ˆ s m,n   s m,n = 0) P ( s m,n = 0) /P ( ˆ s m,n ) = log e P M χ =1 P ( ˆ s m,n   s m,n = a χ ) P ( s m,n = a χ ) P ( ˆ s m,n   s m,n = 0) P ( s m,n = 0) (35) 7 As we already know that M X χ =1 P ( s m,n = a χ ) = k n (36) and, P ( s m,n = 0) = n − k n (37) Using (36) and (37), we can update (35) as λ m,n = log e k P M χ =1 P ( ˆ s m,n   s m,n = a χ ) ( n − k ) P ( ˆ s m,n   s m,n = 0) (38) Eq. (38) can be further simplified as λ m,n = log e ( k ) − log e ( n − k ) − log e P M χ =1 P ( ˆ s m,n   s m,n = a χ ) P ( ˆ s m,n   s m,n = 0) | {z } θ m,n (39) According to (12), the equalized symbol vector can be mod- eled as ˆ s m = s m + ψ tot,m (40) where ψ tot,m = ψ resd,m + ψ f d,m + ψ I B I ,m + ψ noise,m is the sum of interference terms and the processed noise in the FBMC/QAM-IM system. It should be noted that the noise term ψ noise,m is independent of all other terms and interference; the IBI contribution ψ I B I ,m is also independent of all other terms since the interference comes from the previous FBMC/QAM- IM block. Howe ver , the MMSE estimation bias error ψ resd,m and filter distortion due to multipath channel ψ f d,m are correlated since they both depend on the desired signal s m . A ZF equalizer can be used to avoid the MMSE estimation bias error and to have all the remaining interference terms and noise independent with each other . W e can no w write the third term i.e., θ m,n in (39) as follows θ m,n = log e P M χ =1 P ( ˆ s m,n   s m,n = a χ ) P ( ˆ s m,n   s m,n = 0) = log e P M χ =1 1 π γ tot,m,n exp  −| ˆ s m,n − a χ | 2 γ tot,m,n  1 π γ tot,m,n exp  −| ˆ s m,n | 2 γ tot,m,n  = log e P M χ =1 exp  −| ˆ s m,n − a χ | 2 γ tot,m,n  exp  −| ˆ s m,n | 2 γ tot,m,n  = −| ˆ s m,n | 2 γ tot,m,n + log e ( M X χ =1 exp  −| ˆ s m,n − a χ | 2 γ tot,m,n  ) (41) Substituting (41) into (39) yields, λ m,n = log e ( k ) − log e ( n − k ) + | ˆ s m,n | 2 γ tot,m,n + log e ( M X χ =1 exp  −| ˆ s m,n − a χ | 2 γ tot,m,n  ) (42) where γ tot,m,n is the total noise plus interference power of the n th subcarrier of the m th symbol in a FBMC/QAM-IM block and can be calculated using (16). After calculating the n LLR values of each sub-block of the m th FBMC/QAM-IM symbol, the k subcarriers in each sub-group which hav e maximum LLR value are assumed to be active. After detection of activ e sub- carrier index es in each sub-group, the information is passed to the index decoder which provides the estimate of the index selecting p 1 bits based on the indexes of the acti ve sub- carriers in each sub-group. The M -ary constellation symbols transmitted on each activ e sub-carriers is demodulated by the M -ary demodulator in a conv entional manner to estimate the remaining p 2 bits. The bit combiner block then combines p 1 and p 2 bits from all the sub-blocks to generate the transmitted bit vector ˆ T as sho wn in Fig. 2. V . S I M U L A T I O N R E S U LT S In this section we present the simulation results for MSE, output SINR and SIR in FBMC/QAM system with and without index modulation along with the BER performance comparison of index modulation based FBMC/QAM system and conv entional FBMC/QAM system. A. MSE and output SINR Since we know that not all of the subcarriers in FBMC/QAM-IM system are activ e i.e., k < n unlike con- ventional FBMC/QAM system where k = n . In this case the interference po wer will be smaller than the con ventional FBMC/QAM system. For our analysis, we consider n = 4 and k = 3 FBMC/QAM-IM system with QPSK as the M - ary signal constellation. The individual interference terms like noise, residue from the MMSE equalization, IBI and filter distortion due to multipath channel in the proposed FBMC/QAM and FBMC/QAM-IM systems are deriv ed in Section III and the results are presented in Fig. 3a and Fig. 3b respectiv ely . The results show the power of each interference component that is affecting the multicarrier system. It can be seen that the contribution of ICI and ISI (intrinsic interference) is quite insignificant with the use of in verse filter at the recei ver i.e., ISI is around -320dB and ICI cannot be ev en displayed on the same scale. Howe ver , the system is still affected by residue from the MMSE equalization, IBI and filter distortion due to multipath channel. Since some of the subcarriers in IM based FBMC/QAM system are in-active, they will not contribute to these residual interferences. As a result, the interference le vel would be smaller compared to con ventional FBMC/QAM system. The performance in terms of total MSE and output SINR in a FBMC/QAM system with and without index modulation is presented in Fig. 4. It can be seen from Fig. 4a that the MSE in FBMC/QAM system is improved with the use of IM. The improvement gain depends on the selection of n and k v alues. In this case we have considered a n = 4 and k = 3 which result in a gain around 10 log 10 ( n k ) i.e., ∼ 1.25dB. The improvement in MSE performance can be enhanced by a higher n k ratio. The output SINR of the system also improv es with the use of IM as can be seen from the Fig. 4b. It can also be confirmed 8 10 20 30 40 50 60 70 80 -350 -300 -250 -200 -150 -100 -50 0 (a) Individual MSE Components (FBMC/QAM) 10 20 30 40 50 60 70 80 -350 -300 -250 -200 -150 -100 -50 0 (b) Individual MSE Components (FBMC/QAM-IM) Fig. 3: Interference components in FBMC/QAM and FBMC/QAM-IM (4,3,QPSK) that the interference terms in the system model giv e in (12) completely matches with the simulation results, which verifies the accuracy of the deriv ed analytical model. B. SIR P erformance The output SIR of FBMC/QAM system with and without the IM is presented in Fig. 5. As we have discussed earlier that since some of the subcarriers are inacti ve in FBMC/QAM- IM subblock. Their power can either be saved to improve the energy efficienc y of the system or it can be reallocated to the activ e subcarriers in a subgroup to impro ve the system BER performance. In our case we have considered the later option and distribute the po wer of inactiv e subcarriers to the acti ve subcarriers. W e hav e already established in Section III that the interference power in FBMC/QAM-IM system has been reduced by 10 log 10 ( n k ) ∼ 1 . 25 dB compared to con ventional FBMC/QAM system. W e can also see that the interference power is affecting the middle symbols more than the symbols 10 20 30 40 50 60 70 80 -35 -30 -25 -20 -15 -10 -5 (a) MSE performance comparison 10 20 30 40 50 60 70 80 5 10 15 20 25 30 35 (b) Output SINR performance comparison Fig. 4: Performance comparison of FBMC/QAM and FBMC/QAM-IM (4,3,QPSK) at the edges of the FBMC/QAM block. The main reasons for this behavior is the intrinsic interference in FBMC/QAM system. As we know that symbols in FBMC/QAM system ov erlap each other both in time and frequency domain due to per subcarrier filtering. So it is obvious that the symbols at the edges of the block will experience less interference from the neighboring symbols compared to the symbols in the middle. Secondly , it has been already established that the use of in verse filter enhances the residue interferences in the FBMC/QAM system and that the enhancement factor affects the middle symbols more than the symbols at the edges as discussed in [17]. It can be seen in Fig. 5 that the use of IM with FBMC/QAM improv es the SIR of the system by reducing the variance of the interferences existing in the conv entional FBMC/QAM system. 9 2 4 6 8 10 12 14 31 32 33 34 35 36 37 38 39 40 Fig. 5: Output SIR performance comparison of FBMC/QAM and FBMC/QAM-IM (4,3,QPSK) with inv erse filter C. BER P erformance The results for the BER performance of FBMC/QAM system with and without IM are presented in Fig. 6. For 10 12 14 16 18 20 10 -3 10 -2 Fig. 6: BER performance of FBMC/QAM system with and without IM FBMC/QAM-IM system, we have selected ( n, k ) as (4 , 3) , which means that the SE of FBMC/QAM-IM is the same as the conv entional FBMC/QAM system. It can be seen from the results that FBMC/QAM with IM has better performance compared to con ventional FBMC/QAM system due to the presence of relativ ely lower interference. Since, FBMC/QAM- IM reduces the ef fect of residual interference at the receiver and also the po wer from inactive subcarriers are reallocated to the active carriers, the system can provide improved BER performance compared to its con ventional counterpart. It can also be seen that the proposed LLR detector exhibit same performance as ML detector but with much lo wer complexity . In the light of all the results, the improved performance of FBMC/QAM with IM compared to con ventional FBMC/QAM systems makes it a suitable candidate for ne xt generation wireless applications. V I . C O N C L U S I O N W e hav e ev aluated the performance of IM based QAM- FBMC system to highlight the potential of combining an emerging 5G modulation technique with our proposed FBMC/QAM system in [17]. W e first deriv ed a mathemat- ical model of the IM based QAM-FBMC system along with the deriv ation of interference terms at the recei ver due to channel distortions and the intrinsic behavior of the transceiv er model. W e have shown that the interference power in FBMC/QAM-IM is smaller compared to that of conv en- tional FBMC/QAM system as some subcarriers are inacti ve in IM based FBMC/QAM system. W e then ev aluated the performance of FBMC/QAM-IM in term of MSE and SINR and the results are compared with that of the con ventional FBMC/QAM system. The results show that combining IM with FBMC/QAM can improve the system performance since the inactiv e subcarriers do not contribute to the ov erall inter- ference in the system. The SIR performance of a FBMC/QAM block with and without IM is also presented. The results show the ef fect of interference on each FBMC/QAM symbol in a block. It can seen that the interference is higher for symbols in the middle of the block. 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