Repeated index modulation for ofdm systems

ted symbol s from all transmitted RIM-OFDM-MRC/SC signals. The estimated index symbol vector and the M -ary modulated symbol are given by( λˆ, sˆ ) = arg min λ,s ‖y −Hλs‖2F . (2.6) 2.2. Performance analysis of RIM-OFDM-MRC/SC under per- fect CSI This section presents the analysis of the symbol error probability (SEP) of the proposed scheme, i.e. RIM-OFDM-MRC and RIM-OFDM- SC using the ML detection under the assumption of perfect CSI. SEP 28 is defined by the ratio of the number of erroneous symbols to the to- tal number of transmitted symbols. SEP, denoted by Ps, is separated into two parts: index symbol error probability PI and M -ary modulated symbol error probability PM . Their average values are denoted by P s, P I and PM , respectively. 2.2.1. Performance analysis for RIM-OFDM-MRC a) Index error probability Firstly, the index error probability (IEP) is calculated by applying the pairwise index error probability (PIEP) of the ML detector [64]. PIEP is the probability that the detector makes a wrong estimation on a transmitted i-th index vector by the j-th index vector. It is assumed that s is an M -ary PSK symbol with the envelope |s| = √ϕEs, the PIEP can be expressed as P (λi → λj) = Q √ϕEs‖Hλi −Hλj‖2F 2N0  , (2.7) where λi and λj respectively represent the transmitted and the esti- mated index vector. Q (x) = 1√ 2pi ∫∞ x exp ( −y2 2 ) dy is the tail distribution function of the standard normal distribution [64]. Based on distance of 2D between λi and λj, equation (2.7) can be rewritten as P (λi → λj) = Q √∑Dd=1 γMRCαd + γMRCα˜d 2  , (2.8) where αMRCd ∈ θi, α˜MRCd ∈ θj, αMRCd , α˜MRCd /∈ θi ∩ θj, α and α˜ denote the active and inactive sub-carrier; θi and θj represent corresponding index sets λi, λj, respectively. The instantaneous SNR per sub-carrier α is 29 given by γα = γ¯|h (α)|2. Then, by applying the union bound, the index error probability of λi can be attained as follows PMRCIi ≤ c∑ j=1,j 6=i P (λi → λj) . (2.9) Therefore, the instantaneous PIEP of RIM-OFDM-MRC is given by PMRCI = 1 c c∑ i=1 PMRCIi ≤ 1 c c∑ i=1 c∑ j=1,j 6=i P (λi → λj) . (2.10) Denote the set of indices j, (j 6= i) by Ωi such that λj satisfies the Hamming distance of 2 with λi, i.e. D = 1. Let ηi represent a set of sub-carriers that their corresponding indices belong to Ωi. From (2.8) and (2.9), with PMRCIi is bound by the condition P (λi → λj|j ∈ Ωi), i.e. c∑ j=1,j 6=i P (λi → λj) ≈ ∑ j∈Ωi P (λi → λj) . The approximated expression of the instantaneous PIEP in (2.10) is given by PMRCI ≤ 1 c c∑ i=1 ∑ j∈Ωi P (λi → λj) = 1 c c∑ i=1 ∑ j∈Ωi P (α→ α˜) , (2.11) where P (α→ α˜) = Q (√ γMRCα +γ MRC α˜ 2 ) = Q (√ γMRCΣ 2 ) presents PIEP of incorrectly estimating an active sub-carrier αMRC ∈ θi by an inactive sub-carrier α˜MRC ∈ θj, γMRCΣ = γMRCα + γMRCα˜ . Hence, the average PIEP of RIM-OFDM-MRC is given by PMRCI ≤ 1 c c∑ i=1 ∑ j∈Ωi EγMRCΣ Q √γMRCΣ 2  . (2.12) Utilizing the approximation of Q (x) ≈ 1 12 e− x2 2 + 1 4 e− 2 3x 2 [65], the average PIEP can be expressed as P MRC I ≈ EγMRCΣ { ϑ ( 1 12 e− γMRCΣ 4 + 1 4 e− γMRCΣ 3 )} , (2.13) 30 where ϑ = c∑ i=1 ηi c . Applying the definition and properties of MGF:Mγ (z) = Eγ {e−zγ} [64]. The MGF of γMRCΣ is given by MγMRCΣ (z) =M2Lγ (z) = (1− zγ¯) −2L . (2.14) Accordingly, the average PIEP of RIM-OFDM-MRC can be obtained as P MRC I ≈ ϑ 12 [ MγMRCΣ ( −1 4 ) + 3MγMRCΣ ( −1 3 )] ≈ ϑ 12 [ 42L (4 + γ¯) 2L + 32L+1 (3 + γ¯) 2L ] . (2.15) It can be seen from (2.15) that the average PIEP is only effected by N and K via γ¯ = NEs KN0 and c = 2blog2(C(N,K))c without being influenced by the modulation order M . Furthermore, for given N and K, the PIEP is only affected by the index symbol λ via ∑c i=1 ηi and the number of receive antennas L. b) M-ary modulated symbol error probability TheM -ary symbol error probability is the probability that the receiver mis-estimates an M -ary modulated symbol while the indices of active sub-carriers are detected correctly. The instantaneous SEP of the M - ary modulated symbol is given by [64] P MRC M ≈ 2Q (√ 2γMRCΣ,α sin (pi/M) ) , (2.16) where γMRCΣ,α = L∑ l=1 K∑ k=1 γl,αk and αk ∈ θi. Then, applying the approximation of Q-function [65], PM of the RIM-OFDM-MRC system is given by P MRC M ≈ 1 6 ( e−ργ MRC Σ,α + 3e− 4ργMRCΣ,α 3 ) , (2.17) 31 where ρ = sin2 (pi/M). Employing the MGF approach for a random variable γMRCΣ,α , the MGF of γ MRC Σ,α is given by MγMRCΣ,α (z) =MLKγ (z) = (1− γ¯z) −LK . (2.18) Equation (2.17) now can be rewritten as P MRC M ≈ 1 6 [ 1 (1 + ργ¯) LK + 3( 1 + 4ργ¯ 3 )LK ] . (2.19) In general, a symbol is erroneous when the index symbol and/or the M - ary modulated symbol are/is estimated incorrectly. The instantaneous SEP of RIM-OFDM-MRC and its average value are given by [49], [55] Ps ≈ 1 2c c∑ i=1 [ PM + ∑ j∈Ωi P (α→ α˜) ] , (2.20) P s ≈ P¯I + P¯M 2 . (2.21) From equation (2.15), (2.19) and (2.20), the average SEP of the RIM- OFDM-MRC system is given by Ps MRC ≤ ϑ 24 [ 16L (4 + γ¯) 2L + 32L+1 (3 + γ¯) 2L ] + 1 12 [ 1 (1 + ργ¯) LK + 3( 1 + 4ργ¯ 3 )LK ] . (2.22) Equation (2.22) indicates that for large γ¯, P¯s MRC is a function of γ¯−2L. This implies that RIM-OFDM-MRC can achieve diversity order of 2L. This conclusion will be proved in the asymptotic analysis. c) Asymptotic analysis From (2.22), at high SNR region, the approximated expression for SEP of RIM-OFDM-MRC in the case of the perfect CSI can be written 32 as follows Ps MRC ≈ ( K N )2L42L + 32L+1 24 ( ϑ+ 2ξ (4ρ) 2L )( 1 γ0 )2L , ≈ Θ ( (γ0) −2L) (2.23) where γ0 = Kγ¯/N is the average SNR per sub-carrier, and ξ = 1 when K = 2, ξ = 0 for K > 2. Equation (2.23) provides an insight into the dependence of SEP on the system parameters as in following remarks. Remark 1. For given N,K and γ0, RIM-OFDM-MRC attains the diversity order of 2L. The SEP is decreased when increasing L. For large L, the average SEP exponentially decreases with the reduction of K/N . In order to improve the error performance, for given L, we can choose the values of N and K such that K/N is small. Consequently, for given γ0, the best performance of RIM-OFDM-MRC can be achieved by jointly selecting large L and small value of K/N . Remark 2. For K > 2, we attain P MRC s ≈ P MRC I 2 . When K = 2, for large L and given γ0, N,K, the selection of large M will make the SEP exponentially increase through ρ = sin2 (pi/M). Choosing a small M , M = {2, 4} leads to PMRCs ≈ P MRC I 2 . Hence, when K > 2 or K = 2 and M is small, SEP at large SNR mostly depends on the index symbol estimation and slightly depends on estimation of the M -ary modulated symbol. Remark 3. For given N,L and low spectral efficiency, i.e. small M , in- creasing K will make the reliability of RIM-OFDM-MRC reduced. The best performance can be attained by selecting K = 2. Nevertheless, this observation is no longer true when M is high (M ≥ 16). In particu- 33 lar, the higher K will make the error performance better. Thus, these recommend that selecting K not greater than 2 when M is small and high K for large M will be the best system configuration. In the RIM- OFDM-SC scheme, we also have the same statement. It will be verified by the simulation in later section. 2.2.2. Performance analysis for RIM-OFDM-SC a) Index error probability The instantaneous SNR of RIM-OFDM-SC can be determined by em- ploying the probability density function (PDF) of the effective SNR for SC [6] fγ (γα) = L γ¯ L−1∑ l=0 ( L− 1 l ) (−1)le−γα l+1γ¯ . (2.24) It is remarkable that γSCα = max l=1,L γSCl,α , where the instantaneous SNR of the l-th antenna at sub-carrier α is described by γSCl,α . By conducting the inverse Laplace transform of the PDF in (2.24), the MGF of the random variable γSCα can be expressed as MγSCα (z) = L L∑ l=0 ( L− 1 l ) (−1)l l + 1− zγ¯ . (2.25) The MGF of γSCΣ = γ SC α + γ SC α˜ is given by MγSCΣ (z) =M2γSCα (z). Similar to (2.15), PIEP of RIM-OFDM-SC is given by P SC I ≤ ϑ 12 [ MγSCΣ ( −1 4 ) + 3MγSCΣ ( −1 3 )] , ≤ ϑ 12 L2 ( P¯ SCI1 + 3P¯ SC I2 ) , (2.26) 34 where P¯ SCI1 and P¯ SC I2 are given as follows P¯ SCI1 = [ L−1∑ l=0 ( L− 1 l ) 4(−1)l 4l + 4 + γ¯ ]2 , P¯ SCI2 = [ L−1∑ l=0 ( L− 1 l ) 3(−1)l 3l + 3 + γ¯ ]2 . (2.27) b) M-ary modulated symbol error probability Similar to (2.17), the instantaneous SEP of the M -ary modulated symbol of the RIM-OFDM-SC system is given by P SC M ≈ LK 6 (P¯ SCM1 + 3P¯ SC M2 ), (2.28) where P SC M1 and P SC M2 are defined by P SC M1 = [ L−1∑ l=0 ( L− 1 l ) (−1)l l + 1 + ργ¯ ]K , P SC M2 = [ L−1∑ l=0 ( L− 1 l ) 3(−1)l 3l + 3 + 4ργ¯ ]K . (2.29) As a result, the closed-form expression for the average SEP of RIM- OFDM-SC is obtained as follows P¯s SC ≈ ϑL 2 24 ( P SC I1 + 3P SC I2 ) + LK 12 ( P SC M1 + 3P SC M2 ) . (2.30) Where P SC I1 , P SC I2 , P SC M1 , P SC M2 are determined in (2.27), (2.29), respectively. 2.3. Performance analysis of RIM-OFDM-MRC/SC under im- perfect CSI 2.3.1. Performance analysis for RIM-OFDM-MRC Practically, channel estimation errors can occur at the receiver. In this section, the evaluation of the SEP performance of RIM-OFM-MRC/SC 35 in the presence of channel estimation error at the receiver is conducted. The receiver utilizes the actually estimated channel matrix in place of the perfect H in (2.3) to detect the transmitted signals. It is assumed that the estimated channel matrix satisfies: H = H˜+E, where H˜ = [ H˜T1 , . . . , H˜ T L ]T , for H˜l = diag{h˜l (1) , . . . , h˜l (N)}, is the channel matrix when the CSI is imperfect, and E = [ET1 , . . . ,E T L] T , where El = [el (1) , . . . , el (N)], denotes the channel estimation error matrix which is independent to H. Their distributions is presented by el (α) ∼ CN (0, 2), h˜l (α) ∼ CN (0, 1− 2), where α = 1, 2, . . . , N , 2 ∈ [0, 1] is the error variance [29]. Under imperfect CSI condition, the received signal y is rewritten as y = H˜λs+ n˜, (2.31) where n˜ = ( H− H˜ ) λs + n; n˜ = [n˜ (1) , . . . , n˜ (N)] T and n˜ (α) ∼ CN (0, N0) for α ∈ θi and n˜ (α) = e (α) s + n (α) with the distribution n˜ (α) ∼ CN (0, (1 + γ¯2)N0) for α /∈ θi. a) Index error probability PIEP under the channel H˜ is now given by P (λi → λj) = P ( ‖y − H˜λis‖2F > ‖y − H˜λjs‖2F ) = P ( ‖n˜‖2F > ∥∥∥H˜ (λi − λj) s+ n˜∥∥∥2 F ) = P ( −2< { n˜HH˜λijs } > ∥∥∥H˜λijs∥∥∥2 F ) , (2.32) where λij = λi − λj. Assume that θiij = {α |α ∈ θi,α /∈ θj }, θjji = {α |α ∈ θj,α /∈ θi} and θij = θiij ∪ θjji for i 6= j = 1, 2, . . . , c. Follow- ing equation (2.32) and after manipulations, we have −2< { n˜HH˜λijs } ∼ 36 CN ( 0, (∑ α∈θjji γ˜α + (1 + γ¯ 2) ∑ α∈θiij γ˜α ) N0 ) and ∥∥∥H˜λijs∥∥∥2 F = N0 ∑ α∈θij γ˜α, where γ˜α = γ¯ ∣∣∣h˜ (α)∣∣∣2 is the instantaneous SNR per sub-carrier α under the imperfect CSI condition. PIEP in (2.32) now can be expressed as P (λi → λj) = Q  √√√√√ ∑ α∈θij γ˜α 2 ( 1 + ∑ α∈θi ij γ˜α∑ α∈θij γ˜α γ¯2 )  ≈ Q √∑α∈θij γ˜α 2 + γ¯2  . (2.33) For simplicity, ∑ α∈θiij γ˜α/ ∑ α∈θij γ˜α in (2.33) is approximated by 1/2 [49]. The instantaneous PIEP always depends on the conditional PIEP P (λi → λj), so that |θij| is minimized, i.e. |θij| = 2 since |θij| = 2D ≥ 2. Denote Ωi = {j} such that |θij| = 2 and its elements ηi = |Ωi|. Following (2.10), PIEP of RIM-OFDM-MRC under imperfect CSI condition is given by P˜MRCI ≤ 1 c c∑ i=1 ∑ j∈Ωi Eγ˜MRCΣ Q √ γ˜MRCΣ 2 + γ¯2  , (2.34) where γ˜MRCΣ = γ˜ MRC α + γ˜ MRC α˜ , α ∈ θiij, α˜ ∈ θjji. Then, applying the approximation of Q-function [65], the PIEP of RIM-OFDM-MRC under imperfect CSI condition is given by P˜MRCI ≈ Eγ˜MRCΣ { ϑ ( 1 12 e − γ˜ MRC Σ 2(2+γ¯2) + 1 4 e − γ MRC Σ 3(2+γ¯2) )} . (2.35) Based on the MGF definition in [64], the MGF of γ˜ can be expressed as Mγ˜ (z) = [1− γ¯ (1− 2) z]−1 . The MGF of γ˜MRCΣ can be given by Mγ˜MRCΣ (z) =M2Lγ˜ (z) = [ 1− γ¯ (1− 2) z]−2L. (2.36) 37 Then, the closed-form expression for the average PIEP of RIM-OFDM- MRC under the imperfect CSI condition is given by P˜MRCI ≤ ϑ 12 Eγ˜MRCΣ { e − γ˜ MRC Σ 2(2+γ¯2) + 3e − 2γ˜ MRC Σ 3(2+γ¯2) } ≤ ϑ 12 [ Mγ˜MRCΣ ( −1 4 + 2γ¯2 ) + 3MγMRCΣ ( −2 6 + 3γ¯2 )] ≤ ϑ 12 [( 4 + 2γ¯2 4 + γ¯ + γ¯2 )2L + 3 ( 6 + 3γ¯2 6 + 2γ¯ + γ¯2 )2L] . (2.37) b) M-ary modulated symbol error probability Similar to the case of perfect CSI, the average error probability of the M -ary modulated symbol under the imperfect CSI condition is given by P˜MRCM ≈ 2Q (√ 2γ˜MRCΣα sin (pi/M) ) , (2.38) where γ˜MRCΣα = L∑ l=1 K∑ k=1 γ˜l,αk. Since distribution of the noise caused by im- perfect CSI is presented by n˜ (α) ∼ CN (0, N0 (1 + γ¯ε2)), and the active sub-carriers transmit the same data symbol, the symbol s is estimated with an instantaneous SNR, γ˜MRCΣα = ∑ α∈θ γ˜ MRC α / (N0 (1 + γ¯ε 2)) [49]. Based on the approximation of Q-fuction, we have P˜MRCM ≈ 1 6 ( e−ργ˜ MRC Σα + 3e− 4ργ˜MRCΣα 3 ) . (2.39) Applying the MGF of γ˜, the MGF of γ˜MRCΣα is given as follows Mγ˜MRCΣα (s) =MLKγ˜ (s) = ( 1− γ¯ (1− 2) s)−LK. (2.40) TheM -ary modulated symbol error probability of the RIM-OFDM-MRC system is given by P˜MRCM ≈ 1 6  1( 1 + (1− 2)γ¯ρ 1+γ¯2 )LK + 3( 1 + 4(1− 2)γ¯ρ 3(1+γ¯2) )LK  . (2.41) 38 Accordingly, the average SEP of RIM-OFDM-MRC under the imperfect CSI condition is given by P˜s MRC≈ ϑ 24 [( 4 + 2γ¯ε2 4 + γ¯ + γ¯ε2 )2L + 3 ( 6 + 3γ¯ε2 6 + 2γ¯ + γ¯ε2 )2L] + 1 12  1( 1 + (1−ε 2)γ¯ρ 1+γ¯ε2 )LK + 3( 1 + 4(1−ε 2)γ¯ρ 3(1+γ¯ε2) )LK  . (2.42) It can be realized that when 2 = 0, P˜s MRC in (2.42) is equal to Ps MRC in (2.22). Especially, when 2 > 0, the SEP of RIM-OFDM-MRC is higher than that in the perfect CSI case, the reliability of the system considerably decreased in comparison with the certain CSI case. c) Asymptotic analysis The asymptotic analysis for SEP of RIM-OFDM-MRC under uncer- tain CSI provides an insight into the system behavior under the impact of the different CSIs. In high SNR region, for 2 > 0, we have P˜s MRC ≈ ϑ 24 [( 22 1 + 2 )2L + ( 32 2 + 2 )2L] + 1 12 [( 1 + 2 2 + ωρ )LK + 3 ( 32 32 + 4ωρ )LK] , (2.43) where ω = 1− 2. It can be seen from (2.43) that, for large SNRs, P˜sMRC only depends on 2, N,K and M , without be affected by γ¯. The SEP increases when increasing 2. An irreducible error floor occurs at high 2 and the system does not achieve the diversity gain. 39 2.3.2. Performance analysis for RIM-OFDM-SC a) Index error probability Similar to above analysis for RIM-OFDM-SC under perfect CSI con- dition, the MGF of γ˜SCα is given by Mγ˜SCα (z) = L L∑ l=0 ( L− 1 l ) (−1)l l + 1− z (1− 2) γ¯ . (2.44) Following (2.27), the approximated PIEP of RIM-OFDM-SC under the imperfect CSI condition is given by P˜I SC ≤ ϑ 12 [ Mγ˜SCΣ ( −1 4 + 2γ¯2 ) + 3Mγ˜SCΣ ( −2 6 + 3γ¯2 )] ≤ ϑ 12 L2 ( P˜ SCI1 + 3P˜ SC I2 ) , (2.45) where P˜ SCI1 , P˜ SC I2 is given as follows P˜ SCI1 = [ L−1∑ l=0 ( L− 1 l ) (4 + 2γ¯2) (−1)l (4 + 2γ¯ε2) l + 4 + γ¯ + γ¯2 ]2 , P˜ SCI2 = [ L−1∑ l=0 ( L− 1 l ) (6 + 3γ¯2) (−1)l (6 + 3γ¯2) l + 6 + 2γ¯ + γ¯2 ]2 . (2.46) b) M-ary modulated symbol error probability Similar to (2.28), the M -ary modulated SEP for RIM-OFDM-SC in the case of imperfect CSI can be approximated by P˜ SCM ≈ LK 6 (P˜ SCM1 + 3P˜ SC M2 ), (2.47) where P˜ SCM1 and P˜ SC M2 are respectively given by P˜ SCM1 = [ L−1∑ l=0 ( L− 1 l ) (−1)l l + 1 + ρ(1− 2)γ¯ 1+γ¯2 ]K , P˜ SCM2 = [ L−1∑ l=0 ( L− 1 l ) 3(−1)l 3l + 3 + 4ρ(1− 2)γ¯ 1+γ¯2 ]K . (2.48) 40 Accordingly, from (2.20), (2.45) and (2.47), the average SEP of RIM- OFDM-SC under imperfect CSI condition is given by P˜s SC ≈ L2 c∑ i=1 ηi 24c (P˜ SCI1 + 3P˜ SC I2 ) + LK 12 (P˜ SCM1 + 3P˜ SC M2 ), (2.49) where P˜ SCI1 , P˜ SC I2 , and P˜ SCM1 , P˜ SC M2 are given in (2.46) and (2.48), respectively. 2.4. Performance evaluation and discussion This section presents the analytical and Monte-Carlo simulation re- sults to prove the performance of RIM-OFDM-MRC/SC system in the different conditions of CSI. The IM-OFDM-MRC/SC system is selected as the reference system. It is assumed that the channel over each sub- carrier suffers from flat Rayleigh fading. In addition, the ML detection and M -PSK modulation are utilized for all considered systems. The IM- OFDM system with the total of N sub-carriers, K active sub-carriers and modulation order M is referred to as (N,K,M). 2.4.1. Performance evaluation under perfect CSI Fig. 2.2 illustrates the comparison between SEP performance of RIM- OFDM-MRC and IM-OFDM-MRC [6] at the spectral efficiency of 1 bit/s/Hz and 1.25 bits/s/Hz. As shown in Fig. 2.2, at the same and even the higher spectral efficiency, the transmission reliability of proposed system outperforms that of the reference system. It can be seen that at the spectral efficiency of 1.25 bits/s/Hz and SEP of 10−4, RIM-OFDM- MRC achieves SNR gain of about 5 dB over IM-OFDM-MRC. A possible 41 0 5 10 15 20 25 Es/No (dB) 10-6 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP IM-OFDM-MRC, (4,2,4) RIM-OFDM-MRC, (4,2,4) RIM-OFDM-MRC, (4,2,8) Theoretical Asymptotic 5 dB Figure 2.2: The SEP comparison between RIM-OFDM-MRC and the conventional IM-OFDM-MRC system when N = 4, K = 2, L = 2, M = {4, 8}. explanation for this might be that the RIM-OFDM-MRC uses L = 2 receive antennas, it can achieve the maximum diversity order of 2L = 4. The performance improvement is attained by jointly attaining the frequency and spatial diversity. The proposed system achieves double diversity gain compared to IM-OFDM-MRC which exploits the spatial diversity only. Analytical bounds tightly close to the simulation curves at high SNRs. The accuracy of expressions (2.22) and (2.23) is thus verified. The asymptotic analysis result in (2.23) has proved that the maximum diversity order of RIM-OFDM-MRC is limited to 2L and the same observation is attained for RIM-OFDM-SC as depicted in Fig. 2.3. Fig. 2.3 compares SEP performance of RIM-OFDM-SC and that of the 42 0 5 10 15 20 25 Es/No (dB) 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP IM-OFDM-SC, (4,2,4) RIM-OFDM-SC, (4,2,4) RIM-OFDM-SC, (4,2,8) Theoretical 2 dB Figure 2.3: The SEP performance of RIM-OFDM-SC in comparison with IM-OFDM- SC for N = 4, K = 2, L = 2, M = {4, 8}. IM-OFDM-SC system [6] when N = 4, K = 2, L = 2 and M = {4, 8}. It can be seen that at higher spectral efficiency, RIM-OFDM-SC still achieves substantially better SEP performance than IM-OFDM-SC. In particular, an improvement of 2 dB can be attained at the SEP of 10−4 and the spectral efficiency of 1 bit/s/Hz. As can be seen from Fig. 2.3, for different M , N , L and K, the curves obtained by the approximated expression (2.30) matches well with simulation results. This certainly verifies the theoretical analysis. The relationship between the index error probability (IEP) of RIM- OFDM-MRC/SC and the modulation order M in comparison with IM- OFDM-MRC/SC is presented in Fig. 2.4. As shown in this figure, no 43 0 5 10 15 20 Es/No (dB) 10-4 10-3 10-2 10-1 100 IE P IM-OFDM-MRC, (4,2,4) IM-OFDM-SC, (4,2,4) RIM-OFDM-MRC, (4,2,2) RIM-OFDM-MRC, (4,2,4) RIM-OFDM-MRC, (4,2,8) RIM-OFDM-MRC, (4,2,16) RIM-OFDM-SC, (4,2,2) RIM-OFDM-SC, (4,2,4) RIM-OFDM-SC, (4,2,8) RIM-OFDM-SC, (4,2,16) Theoretical Figure 2.4: The relationship between the index error probability of RIM-OFDM- MRC/SC and the modulation order M in comparison with IM-OFDM- MRC/SC for N = 4, K = 2, M = {2, 4, 8, 16}. differences were found in IEPs when increasing M . This result may be explained by the fact that the IEP slightly depends on the modulation order M , and mostly on the energy per symbol, i.e. ϕEs. The IEP performance of the proposed scheme outperforms that of IM-OFDM- MRC/SC with the gain of about 2 dB. Besides, the very tight IEP curves in the figure verifies the analysis in (2.15) and (2.26). The impact of the number of spatial diversity branches on SEP of the proposed system is shown in Fig. 2.5. It can be seen that the SEP per- formances of RIM-OFDM-MRC and RIM-OFDM-SC are significantly improved when increasing the number of spatial diversity branches. In addition, the SEP curves of the two schemes have the same slope, which 44 0 5 10 15 20 25 Es/No (dB) 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP RIM-OFDM-MRC, L=1 RIM-OFDM-SC, L=1 RIM-OFDM-MRC, L=2 RIM-OFDM-SC, L=2 RIM-OFDM-MRC, L=4 RIM-OFDM-SC, L=4 RIM-OFDM-MRC, L=6 RIM-OFDM-SC, L=6 Figure 2.5: The impact of L on the SEP performance of RIM-OFDM-MRC and RIM- OFDM-SC for M = 4, N = 4, K = 2 and L = {1, 2, 4, 6}. implicates that they have the same diversity order. Remark 1 is vali- dated. Fig. 2.6 and Fig. 2.7 illustrate the influence of the number of active sub-carriers on the SEP of RIM-OFDM-MRC and RIM-OFDM-SC. As can be seen that at the same spectral efficiency, the average SEP of the RIM-OFDM-MRC system increases when increasing K. The best SEP performance can be achieved when K = 2. However, this statement is no longer true when the spectral efficiency is not the same as illustrated in Fig. 2.6 and Fig. 2.7. The smallK is the best selection when system using low modulation size. However, at the large M , the best SEP performance is attained with large K. In conclusion, the best configuration should 45 0 5 10 15 20 25 Es/No (dB) 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP (N,K,M) = (8, 2, 8) (N,K,M) = (8, 3, 4) (N,K,M) = (8, 4, 2) (N,K,M) = (8, 5, 4) (N,K,M) = (5, 2, 4) (N,K,M) = (5, 3, 4) (N,K,M) = (5, 4, 4) (N,K,M) = (5, 2, 16) (N,K,M) = (5, 3, 16) (N,K,M) = (5, 4, 16) Figure 2.6: The SEP performance of RIM-OFDM-MRC under influence of K for M = {2, 4, 8, 16}, N = {5, 8}, K = {2, 3, 4, 5}. 0 5 10 15 20 25 Es/No (dB) 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP (N,K,M) = (8, 2, 8) (N,K,M) = (8, 3, 4) (N,K,M) = (8, 4, 2) (N,K,M) = (8, 5, 4) (N,K,M) = (5, 2, 4) (N,K,M) = (5, 3, 4) (N,K,M) = (5, 4, 4) (N,K,M) = (5, 2, 16) (N,K,M) = (5, 3, 16) (N,K,M) = (5, 4, 16) Figure 2.7: The SEP performance of RIM-OFDM-SC under influence of K when M = {2, 4, 8, 16}, N = {5, 8}, K = {2, 3, 4, 5}. 46 be selected with K not greater than 2 for small M , (M ≤ 8) and large K for large M , (M ≥ 16). This validates Remark 3. 0 5 10 15 20 25 Es/No (dB) 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP (N,K,M)=(5,4,2) (N,K,M)=(5,4,4) (N,K,M)=(5,4,8) (N,K,M)=(5,4,16) (N,K,M)=(5,4,32) RIM-OFDM-SC RIM-OFDM-MRC Figure 2.8: Influence of modulation size on the SEP of RIM-OFDM-MRC/SC for N = 5, K = 4, and M = {2, 4, 8, 16, 32}. The impact of M on SEP of RIM-OFDM-MRC/SC when N = 5, K = 4, and M = {2, 4, 8, 16, 32} is shown in Fig. 2.8. As can be seen from this figure that at high SNRs, there is very little difference in SEP of RIM-OFDM-MRC/SC when the value of M is small (M ≤ 8). The significant difference only occurs when M ≥ 16. This means that the SEP is slightly affected by modulation order when M is small, it is only affected by the index estimation. 47 0 5 10 15 20 25 30 35 40 Es/No (dB) 10-6 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP IM-OFDM-MRC, (4,2,8) RIM-OFDM-MRC, (4,2,8) Theoretical Asymptotic  2 =0.01  2 =0.05 Figure 2.9: The SEP performance of RIM-OFDM-MRC in comparison with IM- OFDM-MRC under imperfect CSI when N = 4, K = 2, M = {4, 8}, and 2 = {0.01, 0.05}. 2.4.2. SEP performance evaluation under imperfect CSI condition Fig. 2.9 and Fig. 2.10 depict SEP performance of RIM-OFDM-MRC and RIM-OFDM-SC when channel estimation errors occur at the re- ceiver with error variance 2 = {0.01, 0.05}. As can be seen that at the same spectral efficiency and imperfect CSI condition, RIM-OFDM- MRC/SC still achieves better SEP performance than that of the con- ventional IM-OFDM-MRC/SC. Since the value of 2 is fixed, the error floor occurs in both RIM-OFDM-MRC and RIM-OFDM-SC cases. The analytical results in equations (2.42) and (2.49) tightly close to the sim- ulation results. This validates the accuracy of theoretical analysis. 48 0 5 10 15 20 25 30 35 40 Es/No (dB) 10-6 10-5 10-4 10-3 10-2 10-1 100 A ve ra ge S EP IM-OFDM-SC, (4,2,4)