Luận án Repeated Index Modulation for OFDM Systems

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