Luận án Repeated Index Modulation for OFDM Systems
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is given by 8 0s 19 c MRC MRC 1 X X < γΣ = PI ≤ EγMRC Q @ A : (2.12) c Σ 2 i=1 j2Ωi : ; x2 2 2 1 − 2 1 − 3 x Utilizing the approximation of Q (x) ≈ 12 e + 4 e [65], the average PIEP can be expressed as MRC MRC MRC 1 γΣ 1 γΣ − 4 − 3 P I ≈ EγMRC # e + e ; (2.13) Σ 12 4 30 c P ηi i=1 where # = c . Applying the definition and properties of MGF: Mγ (z) = −zγ MRC Eγ fe g [64]. The MGF of γΣ is given by 2L −2L M MRC (z) = M (z) = (1 − zγ¯) : (2.14) γΣ γ Accordingly, the average PIEP of RIM-OFDM-MRC can be obtained as MRC # 1 1 P ≈ MγMRC − + 3MγMRC − I 12 Σ 4 Σ 3 " # # 42L 32L+1 ≈ + : (2.15) 12 (4 +γ ¯)2L (3 +γ ¯)2L It can be seen from (2.15) that the average PIEP is only effected by N and K viaγ ¯ = NEs and c = 2blog2(C(N;K))c without being influenced by KN0 the modulation order M. Furthermore, for given N and K, the PIEP Pc is only affected by the index symbol λ via i=1 ηi and the number of receive antennas L. b) M-ary modulated symbol error probability The M-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] q MRC MRC P M ≈ 2Q 2γΣ,α sin (π=M) ; (2.16) L K MRC P P where γΣ,α = γl,αk and αk 2 θi. Then, applying the approximation l=1 k=1 of Q-function [65], P M of the RIM-OFDM-MRC system is given by 4ργMRC MRC 1 −ργMRC − Σ,α P ≈ e Σ,α + 3e 3 ; (2.17) M 6 31 where ρ = sin2 (π=M). Employing the MGF approach for a random MRC MRC variable γΣ,α , the MGF of γΣ,α is given by LK −LK M MRC (z) = M (z) = (1 − γz¯ ) : (2.18) γΣ,α γ Equation (2.17) now can be rewritten as " # MRC 1 1 3 P ≈ + : (2.19) M 6 (1 + ργ¯)LK 4ργ¯ LK 1 + 3 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] c " # 1 X X P ≈ P + P (α ! α~) ; (2.20) s 2c M i=1 j2Ωi P¯ + P¯ P ≈ I M : (2.21) s 2 From equation (2.15), (2.19) and (2.20), the average SEP of the RIM- OFDM-MRC system is given by " L 2L+1 # MRC # 16 3 Ps ≤ + 24 (4 +γ ¯)2L (3 +γ ¯)2L " # 1 1 3 + + : (2.22) 12 (1 + ργ¯)LK 4ργ¯ LK 1 + 3 ¯ MRC −2L Equation (2.22) indicates that for largeγ ¯, Ps is a function ofγ ¯ . 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 2L 2L 2L+1 ! 2L MRC K 4 + 3 2ξ 1 Ps ≈ # + 2L ; N 24 (4ρ) γ0 −2L ≈ Θ (γ0) (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. MRC MRC P I Remark 2. For K > 2, we attain P s ≈ 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 (π=M). Choosing a small M, MRC MRC P I M = f2; 4g leads to P s ≈ 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] L−1 ! L X L − 1 l −γ l+1 f (γ ) = (−1) e α γ¯ : (2.24) γ α γ¯ l l=0 SC SC It is remarkable that γα = max γl,α , where the instantaneous SNR of l=1;L SC the l-th antenna at sub-carrier α is described by γl,α . By conducting the inverse Laplace transform of the PDF in (2.24), the MGF of the random SC variable γα can be expressed as L ! l X L − 1 (−1) M SC (z) = L : (2.25) γα l l + 1 − zγ¯ l=0 SC SC SC 2 The MGF of γ = γ + γ is given by M SC (z) = M (z). Σ α α~ γ SC Σ γα Similar to (2.15), PIEP of RIM-OFDM-SC is given by SC # 1 SC 1 P ≤ MγSC − + 3Mγ − ; I 12 Σ 4 Σ 3 # ≤ L2 P¯SC + 3P¯SC ; (2.26) 12 I1 I2 34 where P¯SC and P¯SC are given as follows I1 I2 "L−1 ! l #2 X L − 1 4(−1) P¯SC = ; I1 l 4l + 4 +γ ¯ l=0 "L−1 ! l #2 X L − 1 3(−1) P¯SC = : (2.27) I2 l 3l + 3 +γ ¯ l=0 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 K SC L P ≈ (P¯SC + 3P¯SC); (2.28) M 6 M1 M2 SC SC where P and P are defined by M1 M2 "L−1 ! l #K SC X L − 1 (−1) P = ; M1 l l + 1 + ργ¯ l=0 "L−1 ! l #K SC X L − 1 3(−1) P = : (2.29) M2 l 3l + 3 + 4ργ¯ l=0 As a result, the closed-form expression for the average SEP of RIM- OFDM-SC is obtained as follows 2 K #L SC SC L SC SC P¯ SC ≈ P + 3P + P + 3P : (2.30) s 24 I1 I2 12 M1 M2 SC SC SC SC Where P , P , P , P are determined in (2.27), (2.29), respectively. I1 I2 M1 M2 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, h iT ~ ~ T ~ T ~ ~ ~ where H = H1 ;:::; HL , for Hl = diagfhl (1) ;:::; hl (N)g, is the T T T channel matrix when the CSI is imperfect, and E = [E1 ;:::; EL] , where El = [el (1) ; : : : ; el (N)], denotes the channel estimation error matrix which is independent to H. Their distributions is presented by el (α) ∼ 2 ~ 2 2 CN (0; ), hl (α) ∼ CN (0; 1 − ), 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 andn ~ (α) ∼ CN (0;N0) for α 2 θi andn ~ (α) = e (α) s + n (α) with the distribution 2 n~ (α) ∼ CN (0; (1 +γ ¯ ) N0) for α2 = θi. a) Index error probability PIEP under the channel H~ is now given by ~ 2 ~ 2 P (λi ! λj) = P ky − HλiskF > ky − HλjskF 2 2 ~ = P k~nkF > H (λi − λj) s + ~n F n o 2 H ~ ~ = P −2 Hλijs ; (2.32) F i j where λij = λi − λj. Assume that θij = fα jα 2 θi,α2 = θj g, θji = i j fα jα 2 θj,α2 = θi g and θij = θij [ θji for i 6= j = 1; 2; : : : ; c. Follow- n o H ~ ing equation (2.32) and after manipulations, we have −2< ~n Hλijs ∼ 36 2 P 2 P ~ P CN 0; α2θj γ~α + (1 +γ ¯ ) α2θi γ~α N0 and Hλijs = N0 α2θ γ~α, ji ij F ij 2 ~ whereγ ~α =γ ¯h (α) is the instantaneous SNR per sub-carrier α under the imperfect CSI condition. PIEP in (2.32) now can be expressed as 0 1 v P Bu γ~α C Bu α2θij C P (λi ! λj) = Q P γ~ Bu α2θi α C @t ij 2 A 2 1 + P γ¯ γ~α α2θij 0s 1 P γ~ α2θij α ≈ Q @ A : (2.33) 2 +γ ¯ 2 P P For simplicity, i γ~α= γ~α in (2.33) is approximated by 1=2 [49]. α2θij α2θij The instantaneous PIEP always depends on the conditional PIEP P (λi ! λj), so that jθijj is minimized, i.e. jθijj = 2 since jθijj = 2D ≥ 2. Denote Ωi = fjg such that jθijj = 2 and its elements ηi = jΩij. Following (2.10), PIEP of RIM-OFDM-MRC under imperfect CSI condition is given by 8 2s 39 c < MRC = ~MRC 1 X X γ~Σ PI ≤ Eγ~MRC Q 4 5 ; (2.34) c Σ 2 +γ ¯ 2 i=1 j2Ωi : ; MRC MRC MRC i j whereγ ~Σ =γ ~α +γ ~α~ , α 2 θij,α ~ 2 θji. Then, applying the approximation of Q-function [65], the PIEP of RIM-OFDM-MRC under imperfect CSI condition is given by ( γ~MRC γMRC !) 1 − Σ 1 − Σ ~MRC 2(2+¯γ2) 3(2+¯γ2) PI ≈ Eγ~MRC # e + e : (2.35) Σ 12 4 Based on the MGF definition in [64], the MGF ofγ ~ can be expressed as 2 −1 Mγ~ (z) = [1 − γ¯ (1 − ) z] : MRC The MGF ofγ ~Σ can be given by 2L 2 −2L M MRC (z) = M (z) = 1 − γ¯ 1 − z : (2.36) γ~Σ γ~ 37 Then, the closed-form expression for the average PIEP of RIM-OFDM- MRC under the imperfect CSI condition is given by ( γ~MRC 2~γMRC ) # − Σ − Σ ~MRC 2(2+¯γ2) 3(2+¯γ2) PI ≤ Eγ~MRC e + 3e 12 Σ # −1 −2 ≤ Mγ~MRC + 3MγMRC 12 Σ 4 + 2¯γ2 Σ 6 + 3¯γ2 " 2L 2L# # 4 + 2¯γ2 6 + 3¯γ2 ≤ + 3 : (2.37) 12 4 +γ ¯ +γ ¯ 2 6 + 2¯γ +γ ¯ 2 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 q ~MRC MRC PM ≈ 2Q 2~γΣα sin (π=M) ; (2.38) L K MRC P P whereγ ~Σα = γ~l,αk . Since distribution of the noise caused by im- l=1 k=1 2 perfect CSI is presented byn ~ (α) ∼ CN (0;N0 (1 +γ" ¯ )), and the active sub-carriers transmit the same data symbol, the symbol s is estimated MRC P MRC 2 with an instantaneous SNR,γ ~Σα = α2θ γ~α = (N0 (1 +γ" ¯ )) [49]. Based on the approximation of Q-fuction, we have 4ργ~MRC MRC 1 −ργ~MRC − Σα P~ ≈ e Σα + 3e 3 : (2.39) M 6 MRC Applying the MGF ofγ ~, the MGF ofγ ~Σα is given as follows LK 2 −LK M MRC (s) = M (s) = 1 − γ¯ 1 − s : (2.40) γ~Σα γ~ The M-ary modulated symbol error probability of the RIM-OFDM-MRC system is given by 2 3 1 1 3 ~MRC 6 7 PM ≈ LK + LK : (2.41) 6 4 (1−2)¯γρ 4(1−2)¯γρ 5 1 + 1+¯γ2 1 + 3(1+¯γ2) 38 Accordingly, the average SEP of RIM-OFDM-MRC under the imperfect CSI condition is given by " 2 2L 2 2L# MRC # 4 + 2¯γ" 6 + 3¯γ" P~ ≈ + 3 s 24 4 +γ ¯ +γ" ¯ 2 6 + 2¯γ +γ" ¯ 2 2 3 1 1 3 6 7 + LK + LK : (2.42) 12 4 (1−"2)¯γρ 4(1−"2)¯γρ 5 1 + 1+¯γ"2 1 + 3(1+¯γ"2) MRC 2 ~ MRC It can be realized that when = 0, Ps in (2.42) is equal to Ps 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 " 2 2L 2 2L# MRC # 2 3 P~ ≈ + s 24 1 + 2 2 + 2 " LK LK# 1 1 + 2 32 + + 3 ; (2.43) 12 2 + !ρ 32 + 4!ρ MRC 2 ~ where ! = 1 − . It can be seen from (2.43) that, for large SNRs, Ps only depends on 2; N; K and M, without be affected byγ ¯. 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- SC dition, the MGF ofγ ~α is given by L ! l X L − 1 (−1) M SC (z) = L : (2.44) γ~α l l + 1 − z (1 − 2)γ ¯ l=0 Following (2.27), the approximated PIEP of RIM-OFDM-SC under the imperfect CSI condition is given by ~ SC # −1 SC −2 PI ≤ Mγ~SC + 3Mγ~ 12 Σ 4 + 2¯γ2 Σ 6 + 3¯γ2 # ≤ L2 P~SC + 3P~SC ; (2.45) 12 I1 I2 where P~SC, P~SC is given as follows I1 I2 "L−1 ! l #2 X L − 1 (4 + 2¯γ2)(−1) P~SC = ; I1 l (4 + 2¯γ"2) l + 4 +γ ¯ +γ ¯ 2 l=0 "L−1 ! l #2 X L − 1 (6 + 3¯γ2)(−1) P~SC = : (2.46) I2 l (6 + 3¯γ2) l + 6 + 2¯γ +γ ¯ 2 l=0 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 LK P~SC ≈ (P~SC + 3P~SC); (2.47) M 6 M1 M2 where P~SC and P~SC are respectively given by M1 M2 "L−1 ! l #K X L − 1 (−1) P~SC = ; M1 l ρ(1−2)¯γ l=0 l + 1 + 1+¯γ2 "L−1 ! l #K X L − 1 3(−1) P~SC = : (2.48) M2 l 4ρ(1−2)¯γ l=0 3l + 3 + 1+¯γ2 40 Accordingly, from (2.20), (2.45) and (2.47), the average SEP of RIM- OFDM-SC under imperfect CSI condition is given by c 2 P L ηi SC P~ ≈ i=1 (P~SC + 3P~SC) s 24c I1 I2 LK + (P~SC + 3P~SC); (2.49) 12 M1 M2 where P~SC, P~SC, and P~SC, P~SC are given in (2.46) and (2.48), respectively. I1 I2 M1 M2 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 41 100 IM-OFDM-MRC, (4,2,4) RIM-OFDM-MRC, (4,2,4) 10-1 RIM-OFDM-MRC, (4,2,8) Theoretical Asymptotic 10-2 10-3 Average SEP 10-4 5 dB 10-5 10-6 0 5 10 15 20 25 Es/No (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 = f4; 8g. 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 42 100 IM-OFDM-SC, (4,2,4) RIM-OFDM-SC, (4,2,4) RIM-OFDM-SC, (4,2,8) 10-1 Theoretical 10-2 Average SEP 10-3 2 dB 10-4 10-5 0 5 10 15 20 25 Es/No (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 = f4; 8g. IM-OFDM-SC system [6] when N = 4, K = 2, L = 2 and M = f4; 8g. 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 43 100 IM-OFDM-MRC, (4,2,4) IM-OFDM-SC, (4,2,4) RIM-OFDM-MRC, (4,2,2) 10-1 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) -2 10 RIM-OFDM-SC, (4,2,8) RIM-OFDM-SC, (4,2,16) IEP Theoretical 10-3 10-4 0 5 10 15 20 Es/No (dB) 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 = f2; 4; 8; 16g. 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. 'Es. 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 44 100 RIM-OFDM-MRC, L=1 RIM-OFDM-SC, L=1 RIM-OFDM-MRC, L=2 RIM-OFDM-SC, L=2 10-1 RIM-OFDM-MRC, L=4 RIM-OFDM-SC, L=4 RIM-OFDM-MRC, L=6 RIM-OFDM-SC, L=6 10-2 Average SEP 10-3 10-4 10-5 0 5 10 15 20 25 Es/No (dB) 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 = f1; 2; 4; 6g. 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 small K 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 45 100 (N,K,M) = (8, 2, 8) (N,K,M) = (8, 3, 4) (N,K,M) = (8, 4, 2) 10-1 (N,K,M) = (8, 5, 4) (N,K,M) = (5, 2, 4) (N,K,M) = (5, 3, 4) (N,K,M) = (5, 4, 4) -2 (N,K,M) = (5, 2, 16) 10 (N,K,M) = (5, 3, 16) (N,K,M) = (5, 4, 16) Average SEP 10-3 10-4 10-5 0 5 10 15 20 25 Es/No (dB) Figure 2.6: The SEP performance of RIM-OFDM-MRC under influence of K for M = f2; 4; 8; 16g, N = f5; 8g, K = f2; 3; 4; 5g. 100 (N,K,M) = (8, 2, 8) (N,K,M) = (8, 3, 4) (N,K,M) = (8, 4, 2) (N,K,M) = (8, 5, 4) -1 10 (N,K,M) = (5, 2, 4) (N,K,M) = (5, 3, 4) (N,K,M) = (5, 4, 4) (N,K,M) = (5, 2, 16) -2 (N,K,M) = (5, 3, 16) 10 (N,K,M) = (5, 4, 16) Average SEP 10-3 10-4 10-5 0 5 10 15 20 25 Es/No (dB
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