Chapter 4
Average Bit Error Probability Analysis for SMTs

In this chapter, average bit error ratio (ABER) performance analysis for the different space modulation techniques (SMTs) discussed in previous chapter is presented. First, the ABER is computed for space shift keying (SSK), spatial modulation (SM), quadrature space shift keying (QSSK), and quadrature spatial modulation (QSM) over Rayleigh flat fading channels. Detailed derivation of the error probability is presented. As well, asymptotic analysis at high, but pragmatic, signal‐to‐noise‐ratio (SNR) values are also given for all schemes. Second, the average and asymptotic error probabilities for these schemes in the presence of Gaussian imperfect channel estimation and over Rayleigh fading channels are discussed and derived [52, 65, 68, 188]. Finally, a general framework for the performance analysis of the different SMTs over arbitrary fading channels, in the presence of spatial correlation and imperfect channel estimation, is derived and thoroughly discussed [46, 48, 7275, 189].

4.1 Average Error Probability over Rayleigh Fading Channels

The derivation of the average error probability for SMTs can be computed by deriving the pairwise error probability (PEP). The PEP is defined as the error probability that a transmitted spatial and signal symbols, c04-i0001 and c04-i0002, are received as another spatial and signal symbols, c04-i0003 and c04-i0004, respectively, and is given by PEPc04-i0005 [120].

4.1.1 SM and SSK with Perfect Channel Knowledge at the Receiver

The derivation of the error probability for SM and SSK is almost the same. SSK scheme can be treated as a special case from SM, and the probability of error for SSK scheme can be obtained as such.

4.1.1.1 Single Receive Antenna c04-i0006

In what follows, the derivation is conducted for the special case where the receiver has single receive antenna. Later on, this is generalized to an arbitrary number of receive antennas.

For SM over multiple‐input single‐output (MISO) channel, the spatial and signal symbols are c04-i0007 and c04-i0008, respectively, with c04-i0009 being the c04-i0010th element of the c04-i0011‐length channel vector c04-i0012 and c04-i0013 is the c04-i0014th modulation symbol. Note, SNR c04-i0015, where for simplicity, c04-i0016 is assumed.

Assuming the maximum‐likelihood (ML)‐optimum receiver in (3.2) is used, the PEP is calculated as

(4.1)images

where c04-i0017 denotes the conjugate, c04-i0018 is the c04-i0019‐function, and

(4.2)images

is an exponential random variable (RV) with a probability distribution function (PDF) given by

(4.3)images

with c04-i0020 denoting the mean value and is given by

Therefore, the average PEP is

The formula in (4.5) is obtained for an exponential RV from [123, 190] as

(4.6)images

Thus, the ABER of SM over MISO Rayleigh fading channels can be computed using the union bounding technique as [120]

with c04-i0021 being the number of bits in error associated with the PEP event. The derived analytical ABER in (4.7) is compared to the simulated ABER for c04-i0022, and c04-i0023 in Figure 4.1. The results in the figure validate the derived bound in (4.7).

Graphical representation of the derived analytical ABER of SM for MISO Rayleigh fading channels compared to the simulated ABER for η = 4, Nt = 4 and η = 6, Nt = 8.

Figure 4.1 The derived analytical ABER of SM for MISO Rayleigh fading channels in (4.7) compared simulated ABER for c04-i0024 and c04-i0025.

4.1.1.2 Arbitrary Number of Receive Antennas c04-i0026

The previous derivations consider the special case of c04-i0027. For an arbitrary number of receive antennas, c04-i0028, the PEP is given by

(4.8)images

where c04-i0029 is the c04-i0030th column of the c04-i0031 channel matrix c04-i0032

(4.9)images

and c04-i0033 is a chi‐squared RV with a PDF given by

(4.10)images

where c04-i0034 is the c04-i0035th element of the c04-i0036‐length channel vector c04-i0037. Note, c04-i0038 is given in (4.4).

Thus, the average PEP is given by [123, 191],

(4.11)images

where c04-i0039. Note, from [123, 190] if c04-i0040 is a chi‐squared RV then,

(4.12)images

where c04-i0041.

Finally, the ABER for SM over Rayleigh fading multiple‐input multiple‐output (MIMO) channel is derived as

Figure 4.2 compares the derived analytical ABER bound in (4.13) to the simulated ABER for c04-i0042, c04-i0043, and c04-i0044. The accuracy of the bound is obvious in the figure where a close‐match between analytical and simulation curves can be clearly seen for all depicted SNR values greater than 5 dB.

Graphical representation of the derived analytical ABER of SM over MISO Rayleigh fading channels compared to the simulated ABER for Nr = 2 and 4, η = 4, and Nt = 4.

Figure 4.2 The derived analytical ABER of SM over MISO Rayleigh fading channels in (4.13) compared to the simulated ABER for c04-i0045, c04-i0046, and c04-i0047.

4.1.1.3 Asymptotic Analysis

Another very useful formula to obtain for such systems is the approximate average error probability, generally called asymptotic error probability, at high SNR values. Such formula clearly shows the diversity gain for such systems and can be utilized for optimization studies. The asymptotic error probability is derived based on the behavior of the PDF of c04-i0048 around the origin. This can be obtained by taking Taylor series of c04-i0049 leading to

(4.14)images

where c04-i0050 denotes higher‐order terms that are ignored. The average PEP is then computed as [123, 191]

It is evident that SM achieves a diversity gain of c04-i0051. This can be seen as well in Figures 4.1 and 4.2, where for MISO system in Figure 4.1, the diversity gain is 1 as there is only one receive antenna. For the MIMO system in Figure 4.2, the diversity gain is 2, which is equal to the number of receive antennas, c04-i0052.

For SSK, the ABER and asymptotic PEP can be computed by (4.13) and (4.15), respectively, by letting c04-i0053.

4.1.2 SM and SSK in the Presence of Imperfect Channel Estimation

The assumption of perfect channel knowledge at the receiver is impractical. In practical wireless systems, pilots are transmitted to estimate the channel at the receiver input. However and similar to transmitted data, transmitted pilots are corrupted by the additive white Gaussian noise (AWGN) at the receiver input, which leads to a mismatch between the exact channel and the estimated channel. The difference between both is generally called channel estimation errors (CSE). Generally, it is assumed that the true channel and the estimated channel are jointly ergodic and stationary processes. Also, the estimated channel and the estimation error are assumed to be orthogonal. Hence [192],

(4.16)images

where c04-i0054 denotes the c04-i0055 estimated channel matrix, and c04-i0056 denotes the CSE, which is a complex Gaussian random variable with zero mean and c04-i0057 variance, c04-i0058. Note that c04-i0059 is a parameter that captures the quality of the channel estimation and can be appropriately chosen depending on the channel dynamics and estimation schemes.

The derivation of the instantaneous PEP in the presence of imperfect channel knowledge at the receiver is similar to the derivation of perfect channel knowledge and is computed in what follows.

4.1.2.1 Single Receive Antenna c04-i0060

In the presence of CSE, the ML‐optimum receiver for SM is given by

(4.17)images

where c04-i0061 is the c04-i0062th element of the c04-i0063‐length estimated MISO channel vector c04-i0064.

Thus, the PEP for SM in the presence of CSE is given by

(4.18)images

where c04-i0065 is a complex Gaussian RV with zero mean and variance c04-i0066. Moreover,

(4.19)images

is an exponential RV with a mean value

where

Using the same method as discussed in Section 4.1.1.1 for perfect channel knowledge at the receiver, the ABER for SM over MISO Rayleigh fading channel with imperfect channel knowledge is given by [120, 123]

4.1.2.2 Arbitrary Number of Receive Antennas c04-i0067

For an arbitrary number of receive antennas c04-i0068, the PEP is given by

(4.23)images

where c04-i0069 is a chi‐squared RV with a PDF

(4.24)images

and

(4.25)images

where c04-i0070 is the c04-i0071th element of the c04-i0072‐length channel vector c04-i0073, and c04-i0074 is the c04-i0075th vector of the c04-i0076 estimated channel matrix c04-i0077. Note, c04-i0078 is given in (4.20).

Thus, and using similar procedure as in Section 4.1.1.2, the average PEP is given by [123, 191]

(4.26)images

where c04-i0079.

Finally, the ABER for SM over MIMO Rayleigh fading channels in the presence of CSE is given by

The derived ABER analytical bounds in (4.22) for MISO and in (4.27) for MIMO are depicted in Figure 4.3 and compared with the simulated ABER, for c04-i0080, c04-i0081, c04-i0082, and c04-i0083. The depicted results validate the accuracy of the derived formulas for the ABER.

Graphical representation of the derived analytical ABER of SM over Rayleigh fading channels in the presence of CSE for MISO and for MIMO compared to the simulated ABER for σ2e = σ2n, Nr = 1 and 2, η = 4, and Nt = 2.

Figure 4.3 The derived analytical ABER of SM over Rayleigh fading channels in the presence of CSE in (4.22) for MISO and in (4.27) for MIMO compared to the simulated ABER for c04-i0084, c04-i0085, c04-i0086, and c04-i0087.

4.1.2.3 Asymptotic Analysis

The asymptotic PEP in the presence of CSE can be computed as

The obtained asymptotic formula allows deep investigations of several cases.

  1. Case I – Fixedc04-i0088: Assuming fixed CSE, i.e. the CSE is not a function of the SNR, the asymptotic PEP is given by

    where c04-i0089. It is clear from (4.29) that increasing the SNR has no impact on the average PEP, which leads to zero diversity order and an error floor will occur.

  2. Case II – CSE is a function of SNR: In this case, the pilot symbols are assumed to be transmitted with the same energy as the symbols and the channel estimation error decreases as the SNR increases. Hence, the average asymptotic PEP is given by
    (4.30)images

    The diversity order in this case is c04-i0090, and increasing the SNR enhances the performance. This gain can be seen in Figure 4.3 where the ABER results at high SNR have diversity gains of 1 and 2 for c04-i0091 and c04-i0092, respectively.

In the case of SSK scheme, the ABER and the asymptotic PEP can be calculated using (4.27) and (4.28), respectively, by letting c04-i0093.

4.1.3 QSM with Perfect Channel Knowledge at the Receiver

The average error probability of QSM and QSSK can be derived following similar steps as discussed before for SM and SSK systems [65, 70].

Assuming the ML‐optimum receiver in (3.2) is used, the PEP is given by

(4.31)images

where c04-i0094 and c04-i0095 denote the real and imaginary parts of the complex number c04-i0096, respectively; c04-i0097, and c04-i0098 is an exponential RV with a mean value given by

(4.32)images

Therefore, from [123], and as discussed in Section 4.1.1.1, the average PEP is

(4.33)images

Thus, the ABER of QSM is

where c04-i0099.

For MIMO system with c04-i0100 receive antennas, using the same methodology as discussed earlier, the ABER for QSM is

where c04-i0101.

Taking the Taylor series and ignoring higher‐order terms give the asymptotic average PEP of QSM as

where a diversity gain of c04-i0105 is obtained. Figure 4.4 depicts the derived ABER analytical bounds (4.34) for MISO and in (4.35) for MIMO systems and compare it to the simulated ABER for QSM with c04-i0106, c04-i0107, and c04-i0108. From the figure, simulation and analytical results demonstrate close‐match, and the diversity gains equal the number of receive antennas.

Graphical representation of the derived analytical ABER of QSM over Rayleigh fading channels for MISO and for MIMO compared to the simulated ABER for Nr = 1 and 2, η = 4, and Nt = 2.

Figure 4.4 The derived analytical ABER of QSM over Rayleigh fading channels in (4.34) for MISO and in (4.35) for MIMO compared to the simulated ABER for c04-i0102, c04-i0103, and c04-i0104.

For QSSK, the ABER and PEP can be calculated using (4.34) and (4.36), respectively, by letting c04-i0109.

4.1.4 QSM in the Presence of Imperfect Channel Estimation

Following similar steps as discussed in Section 4.1.2.2 for SM, the ABER of QSM over MIMO Rayleigh fading channel in the presence of CSE is

where c04-i0110 and

(4.38)images

Note, c04-i0111 is given in (4.21). Figure 4.5 shows the derived bound and compares it to the simulated ABER for c04-i0112, c04-i0113, and c04-i0114. The figure shows that the simulated ABER is bounded by (4.37). Furthermore and from the figure, QSM offers a diversity gain equal to the number of receive antennas.

Graphical representation of the derived analytical ABER of QSM over Rayleigh fading channels in the presence of CSE compared to the simulated ABER for Nr = 1 and 2, η = 4, and Nt = 2.

Figure 4.5 The derived analytical ABER of QSM over Rayleigh fading channels in the presence of CSE in (4.37) compared to the simulated ABER for c04-i0115, c04-i0116, and c04-i0117.

The asymptotic error probability for QSM in the presence of CSE can be computed using similar steps as discussed earlier and is given by

(4.39)images

The asymptotic error probability can be analyzed for different cases based on the value of c04-i0118:

  • Case I –c04-i0119: Assuming a fixed channel estimation error that remains constant even if the SNR changes, the average asymptotic PEP is given by where c04-i0120. From (4.40), it can be seen that the diversity gain is zero, where changing the SNR would not change the PEP performance.
  • Case II –c04-i0121: this is the general case, where pilot symbols are transmitted with the same energy as the data symbols. The asymptotic PEP is given by
    (4.41)images
    Similar to the assumption of perfect channel knowledge, a diversity gain of c04-i0122 is achieved here as well.

4.2 A General Framework for SMTs Average Error Probability over Generalized Fading Channels and in the Presence of Spatial Correlation and Imperfect Channel Estimation

The previous analysis for SM, SSK, QSSK, and QSM systems is valid only over Rayleigh flat fading channels where the channel phase distribution is uniform. The performance of SMTs over generalized fading channels, such as Nakagami‐c04-i0123, Rician, c04-i0124c04-i0125, and others, attracted significant attention in literature [45, 84, 170, 193197]. However, in all these analyses, the phase distribution of the generalized channel fading distribution is assumed to be uniform, which is needed for mathematical tractability and simplified analysis. Though, this assumption leads to inaccurate conclusions. It was shown in [195, 196] that increasing the c04-i0126 value of the Nakagami‐c04-i0127 distribution enhances the performance of SMTs. This conclusion means that SMTs performance will be significantly enhanced for very large values of c04-i0128. However, for c04-i0129, the Nakagami‐c04-i0130 channel becomes Gaussian, and MIMO communication over Gaussian channels is impossible since it is not possible to resolve the different channel paths. This conclusion was reported recently, and a general framework for SMTs performance analysis over generalized fading channels is presented in [48, 73, 75]. In this section, the general framework for SMTs performance analysis over generalized fading channels and in the presence of spatial correlation and imperfect channel estimation is discussed in detail.

Consider a general MIMO system where the transmitter is equipped with c04-i0131 transmit antennas and the receiver has c04-i0132 receive antennas. The received signal at any particular time is given by

(4.42)images

where c04-i0133 denotes a correlated MIMO channel matrix given by

(4.43)images

where c04-i0134 and c04-i0135 being, respectively, the receiver and transmitter spatial correlation matrices as defined in Chapter 2, and c04-i0136 is the MIMO channel matrix. Assuming CSE at the receiver, the ML‐optimum receiver is written as

(4.44)images

where c04-i0137 is c04-i0138 size space containing all possible transmitted vectors c04-i0139, and c04-i0140 is the estimated MIMO channel matrix

(4.45)images

Thus, the PEP of SMTs over generalized and correlated MIMO channel matrix in the presence of CSE is given by

where c04-i0141, c04-i0142, c04-i0143, c04-i0144, and from [198], the alternative integral expression of the c04-i0145‐function is c04-i0146.

The average PEP is computed by taking the expectation of (4.46) as

where c04-i0147 denotes the moment‐generation function (MGF) of c04-i0148.

From [199], the Frobenius norm of c04-i0149 can be expanded as

where c04-i0150, c04-i0151 is the vectorization operator, where the columns of the matrix c04-i0152 are stacked in column vector, and c04-i0153 is the trace function.

Let c04-i0154 be an identical and independently distributed (i.i.d.) complex random vector with mean c04-i0155, covariance matrix c04-i0156, and real and imaginary components with equal mean and variance. From [200] and for any Hermitian matrix c04-i0157, the MGF of c04-i0158 is

Hence, from (4.49) and (4.48), the MGF in (4.47) can be written as

where c04-i0159 and

(4.51)images
(4.52)images

where c04-i0160 and c04-i0161 are the mean and variance of the channel c04-i0162, respectively, c04-i0163, with c04-i0164 being the Kronecker product, and c04-i0165 is an c04-i0166‐length all ones vector.

Substituting (4.50) in (4.47) gives

Finally, using (4.53), the ABER performance of SMTs over generalized correlated channels in the presence of CSE can be bounded by

Note, the ABER bound in (4.54) even though derived for SMTs is also valid for classical spatial multiplexing (SMX) MIMO systems as shown in the case of Rician fading channels in [199] and generalized fading channels in [54, 73, 74, 201]. Figure 4.6 validates the derived bound in (4.54), where it compares it to the simulated ABER of two SMT systems, SM and QSM, over correlated Rayleigh and Nakagami‐c04-i0167 fading channels, and in the presence of CSE, where c04-i0168, c04-i0169, and c04-i0170. As can be seen from the figure, analytical and simulation results demonstrate close match for a wide and pragmatic range of SNR values and for the different channel conditions.

Graphical representations of the derived analytical ABER of SMTs compared with the simulated ABER of SM and QSM over correlated Rayleigh and Nakagami-m = 4 fading channels in the presence of CSE, where σ2e = σ2n, η = 6, and Nt = Nr = 4.

Figure 4.6 The derived analytical ABER of SMTs in (4.54) compared with the simulated ABER of SM and QSM over correlated Rayleigh and Nakagami‐c04-i0171 fading channels in the presence of CSE, where c04-i0172, c04-i0173, and c04-i0174.

4.3 Average Error Probability Analysis of Differential SMTs

Performance analysis of differential space modulation techniques (DSMT) is presented hereinafter. The PEP for a DSMT can be formulated as follows:

where c04-i0175, and c04-i0176.

Now, the left‐hand side of the inequality in (4.55) can be written as

Moreover, the right‐hand side can be written as

From (4.56) and (4.57), the PEP in (4.55) can be simplified to

Note that c04-i0177 because c04-i0178.

Unfortunately, no closed‐form expression is available for the PEP in (4.58). Therefore, an approximate expression will be targeted in what follows. The approximation is based on assuming that c04-i0179 for high SNR values [[120], p. 274]. Notice that both c04-i0180 and c04-i0181 are Gaussian random variables with zero mean and c04-i0182 variance. Therefore, the mean of their product is also zero, and the variance is c04-i0183. Consequently, as SNR increases, c04-i0184 approaches zero and so the variance of c04-i0185.

Based on the previous approximation, (4.58) can be rewritten as

The average PEP is computed by taking the expectation of the PEP in (4.59)

(4.60)images

where c04-i0186 denotes the MGF of c04-i0187.

Based on [123], the variable c04-i0188 can be expanded as

(4.61)images

As in the previous section, the MGF of the variable c04-i0189 can be expressed using the quadratic form expression as [48, 73, 200]

where c04-i0190 and c04-i0191 are the mean vector and the covariance matrix of the vector c04-i0192, respectively, and c04-i0193.

Using (4.62), the average PEP can be upper bounded by

(4.63)images

Finally, the ABER performance of DSMT can be bounded as

where c04-i0194 is the number of bits error associated with the corresponding PEP event. In Figure 4.7, the derived bound in (4.64) is compared to the simulated ABER of differential spatial modulation (DSM) with c04-i0195, and differential quadrature spatial modulation (DQSM) with c04-i0196, where c04-i0197, and c04-i0198‐quadrature amplitude modulation (QAM) is used. From Figure 4.7, the simulated ABER validates the bound where it closely follows the bound for a wide range of SNR values.

Graphical representation of the derived analytical ABER of DSMTs compared with the simulated ABER of DSM and DQSM for M = 4-QAM, Nt = 3, Nr = 1, 2, 3, and 4.

Figure 4.7 The derived analytical ABER of DSMTs in (4.64) compared with the simulated ABER of DSM and DQSM for c04-i0199‐QAM, c04-i0200, c04-i0201.

4.4 Comparative Average Bit Error Rate Results

In this section, Monte Carlo simulation results are presented to study the ABER performance of SMTs, generalized space modulation techniques (GSMTs), quadrature space modulation techniques (QSMTs), and DSMTs system with different configurations.

4.4.1 SMTs, GSMTs, and QSMTs ABER Comparisons

In Figure 4.9, the ABER of SMTs is presented and compared with SMX for c04-i0202 bits. From the figure, it can be seen that QSSK offers the best performance compared to the rest of the SMTs, 2.5, 3, 3.7, and 7.6 dB better than SSK, QSM, SMX, and SM, respectively. This is because in QSSK all information bits are modulated in the spatial domain which is in Rayleigh fading channels more robust that the signal domain. This can also be seen in SSK where it performs better than QSM, SMX, and SM, 0.5, 1.2, and 5.1 dB, respectively. QSM and SM modulate part of the information bits in the signal domain as will as the spatial domain. However and even though QSSK and SSK modulate all information bits in the spatial domain, SSK performs 2.5 dB less than QSSK. This can be attributed to the need of much more transmit antennas in SSK than QSSK. In Figure 4.9, QSSK needs only c04-i0203, while SSK requires c04-i0204. Comparing QSM to SM, QSM modulates more bits in the spatial domain than SM, and it offers 4.6 dB better performance than SM. With c04-i0205, QSM uses 4‐QAM to achieve c04-i0206 bits. However, SM uses 32‐QAM to achieve the same spectral efficiency assuming similar number of transmit antennas. Finally, it can be seen that SMTs that modulate large bits in the spatial domain performs better than SMX. Yet, SMTs like SM that uses large signal constellation diagram perform worse than SMX.

Graphical representation of ABER performance comparison between the SMTs, QSSK, SSK, QSM and SM, and SMX systems over Nakagami-m = 4 fading channels for different number of transmit antennas and modulation orders achieving η = 8 bits with Nr = 4 antennas.

Figure 4.8 ABER performance comparison between the SMTs, QSSK, SSK, QSM and SM, and SMX systems over Nakagami‐c04-i0209 fading channels for different number of transmit antennas and modulation orders achieving c04-i0210 bits with c04-i0211 antennas.

Graphical representation of ABER performance comparison between the SMTs, QSSK, SSK, QSM and SM, and SMX systems over Rayleigh fading channels for different number of transmit antennas and modulation orders achieving η = 8 bits with Nr = 4 antennas.

Figure 4.9 ABER performance comparison between the SMTs, QSSK, SSK, QSM and SM, and SMX systems over Rayleigh fading channels for different number of transmit antennas and modulation orders achieving c04-i0207 bits with c04-i0208 antennas.

The ABER results of SMTs and SMX over Nakagami‐c04-i0212 fading channels for c04-i0213 and c04-i0214 are presented in Figure 4.8, where c04-i0215 for QSSK, c04-i0216 for SSK, and c04-i0217 for SM, QSM, and SMX. As in the case for Rayleigh fading channels in Figure 4.9, QSSK over Nakagami‐c04-i0218 fading channels offers the best performance with 1.3, 2.5, and 4 dB better performance than SMX and QSM, SSK, and SM, respectively. This can be attributed to the same reason as before where there exist no signal constellations and all the information bits are modulated in the spatial domain. Compared to Rayleigh, the performance gap between QSSK and the other systems is smaller, because it is harder to distinguish the different transmit antennas in Nakagami‐c04-i0219 fading channel compared to Rayleigh fading channels. This can also be seen in SSK, where it performs 1 dB worse than QSM even though it outperforms QSM performance over Rayleigh fading channels as shown in Figure 4.9. Furthermore, SM still demonstrates the worst performance as it uses large signal constellation diagram compared to all other systems. Finally, QSM is shown to offer nearly the same performance as SMX.

Graphical representation of ABER performance comparison between GSMTs, GQSSK, GSSK, GQSM, and GSM, and SMX systems over Rayleigh fading channels for different number of transmit antennas and modulation orders achieving η = 8 bits with Nr = 4 antennas.

Figure 4.10 ABER performance comparison between GSMTs, GQSSK, GSSK, GQSM and GSM, and SMX systems over Rayleigh fading channels for different number of transmit antennas and modulation orders achieving c04-i0220 bits with c04-i0221 antennas.

Figure 4.10 depicts the ABER performance of generalized quadrature space shift keying (GQSSK) with c04-i0233, generalized space shift keying (GSSK) with c04-i0234, generalized quadrature spatial modulation (GQSM) with c04-i0235, and generalized spatial modulation (GSM) with c04-i0236, and compares their performance to SMX with c04-i0237 over Rayleigh fading channel. All systems achieve c04-i0238, and c04-i0239 is considered in all systems. From the figure, GQSSK offers the best performance with 1.2, 2, and 4.6 dB better performance than GSSK and SMX, GQSM, and GSM, respectively. As in QSSK, GQSSK offers this performance because all information bits are modulated in the spatial domain, while using small number of transmit antennas. This also can be seen for GSSK in Figure 4.10, where it offers the same performance as SMX and better performance than both GQSM and GSM. However, GQSSK performs better than GSSK as it uses less number of transmit antennas. Comparing the performance of QSSK in Figure 4.9 to GQSSK in Figure 4.10, it can be observed that QSSK achieves an ABER of c04-i0240 at an SNR c04-i0241 dB while GQSSK achieves the same ABER at an SNR c04-i0242 dB, i.e. GQSSK even though it uses less number of transmit antennas it performs 2.1 dB worse than QSSK. Even though GSMTs in general reduce the number of transmit antennas, they create correlation between the different spatial symbols, which increases the ABER. That is unlike QSMTs, where the reduction in the number of transmit antennas and the increase in the number of bits modulated in the spatial domain are attained while keeping the spatial symbols orthogonal.

In summary, the more bits modulated in the spatial domain, the better is the performance of SMTs. Increasing the number of bits modulated in the spatial domain is best achieved through QSMTs. To increase the data rate, it is better to increase the number of transmit antennas, which as discussed in previous chapter comes at no extra hardware or computational complexity, nor energy consumption. Another way is to use GSMTs and generalized quadrature space modulation techniques (GQSMTs), which need less number of transmit antennas. However, they offer slightly worse performance than SMTs and QSMTs. Finally, SMTs offer the same performance as SMX or better, if the number of bits modulated in the signal domain is relatively similar to SMX. Though, as the number of bits modulated in the signal domain for SMTs increases, as compared to SMX, the worse the performance will be.

4.4.2 Differential SMTs Results

A comparison between DQSM with c04-i0243 and 4‐QAM and DSM with c04-i0244 and 8‐phase shift keying (PSK) is illustrated in Figure 4.11. Results for QSM with c04-i0245 and 4‐QAM, SM with c04-i0246 and 4‐QAM, and SMX over Rayleigh fading channels and assuming c04-i0247 are depicted as well. Note, DSM and DQSM do not need channel state information (CSI) at the receiver. However, SM, QSM, and SMX assume full CSI knowledge at the receiver.

Interesting conclusions can be obtained from the depicted results in Figure 4.11. First, the ABER curves of DSM and DQSM intersect at a SNR c04-i0248 dB, where DSM demonstrates slightly better performance for SNR c04-i0249 dB, whereas DQSM outperforms DSM for SNR c04-i0250 dB. At pragmatic ABER values of about c04-i0251, DQSM outperforms DSM by around 2.3 dB. In addition, around the same ABER, SM is shown in Figure 4.11 to outperform DQSM and DSM by about 2.5 dB and 5 dB, respectively. Furthermore, QSM outperforms DQSM and DSM by about 1.2 and 2.5 dB, respectively.

Finally, it can be seen from Figure 4.11 that SMX offers better performance than both DSMTs, while offering nearly the same performance as QSM, and 2.5 dB worse performance than SM. It should be noted, though, that different configurations might lead to different performances, but the trend is likely to remain the same.

In summary, DSMTs offer a relatively good performance without the need for a full CSI at the receiver, while retaining all advantages of SMTs since one antenna is active at a time and transmitter deployment through single radio frequency (RF)‐chain is anticipated.

Graphical representation of ABER performance comparison between DQSM with Nt = 2 and 4-QAM, DSM with Nt = 4 and 8-PSK, QSM with Nt = 4 and 4-QAM, and SMX with Nt = 4 and 4-QAM, and SMX with Nt = 4 and BPSK over Rayleigh fading channels for η = 4 bits and Nr = 4 antennas.

Figure 4.11 ABER performance comparison between DQSM with c04-i0222 and c04-i0223‐QAM, DSM with c04-i0224 and c04-i0225‐PSK, QSM with c04-i0226 and c04-i0227‐QAM, SM with c04-i0228 and c04-i0229‐QAM, and SMX with c04-i0230 and BPSK over Rayleigh fading channels for c04-i0231 bits and c04-i0232 antennas.

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