From the above section, we propose a method of determining power loading factors by minimizing the average BER upper bound in (4.39), subject to the power normalization constraint. Instead of considering channel realization at our design, the average BER for the channel distribution is evaluated. Determining power loading factors by minimizing the upper bound of the error probability averaged with respect to
there do not seem to exist closed-form optimal solutions. Instead, the problem is solved via numerical search (e.g. by using fmincon function in Matlab Optimization Toolbox).
4.5 Computer Simulations
First, Figure 4.1 shows the upper bound of average BER compared with the average BER. It is obvious that the upper bound of average BER is indeed larger than the average BER and the upper bound is tighter for low SNR than high SNR.
10 12 14 16 18 20 22 24 26 28 30
10-6 10-5 10-4 10-3 10-2
SNR
Average BER
Error Free Case
Upper bound (Deduction) Formula (Analysis)
Figure 4.1: Upper bound of average BER performances
To illustrate the numerical performance of the proposed scheme, we compare the simulated average BER of the following receivers: linear MMSE equalizer, QR-based detectors with and without power loading, and the joint ML decoding; the results are
shown in Figure 4.2 (QPSK modulation is used) and the solutions at various SNR are listed in Table 4.1. As we can see, the QR-based solution without power allocation only slightly outperforms the linear MMSE equalizer. When combined with the proposed optimal power loading scheme, performance improvement up to about 2 dB is achieved in the medium-to-high SNR region; in particular, the BER is almost identical to that attained by the optimal ML decoding for SNR above 22.5 dB. From Table 4.1, we can see the relationship of power loading factors as follows: p1 =p2, p3 =p4, and p3 >p1 because R11 =R22, R33 =R44, and R11≥R33.
5 10 15 20 25 30
10-7 10-6 10-5 10-4 10-3 10-2 10-1
SNR
Average BER
LMMSE receiver
QR receiver without power loading QR receiver with optimal power loading ML decoding
Figure 4.2: Average BER performances of Q-OSTBC with different receivers
Table 4.1: Computed optimal power loading factors in Figure 4.2
Power loading factors SNR (dB)
p1 p2 p3 p4
5 0.9982 0.9982 1.0018 1.0018 7.5 0.9807 0.9807 1.0189 1.0189 10 0.9529 0.9529 1.0449 1.0449 12.5 0.9197 0.9197 1.0743 1.0743 15 0.7769 0.7769 1.1817 1.1817 17.5 0.7259 0.7259 1.2137 1.2137 20 0.6724 0.6724 1.2441 1.2441 22.5 0.6180 0.6180 1.2720 1.2720 25 0.5641 0.5641 1.2968 1.2968 27.5 0.5123 0.5123 1.3182 1.3182 30 0.4633 0.4632 1.3362 1.3362 32.5 0.4177 0.4177 1.3511 1.3511
35 0.3755 0.3755 1.3635 1.3635
In Figure 4.3, we show that when there is channel estimation error, our method can be still used. Considering that the channel estimation error covariance matrix is equal to 0.01I, the average BER performance is presented and it is found that the BER performance is dominated by the channel estimation error instead of SNR in the high SNR region. When the diagonal entries of the channel estimation error covariance matrix is much larger than the noise power, the BER performance exhibits slight saturation in the high SNR region.
5 10 15 20 25 30 10-6
10-5 10-4 10-3 10-2 10-1
SNR
Average BER
Cee=0.01
QR receiver without power loading QR receiver with upper power loading
Figure 4.3: Average BER performances of Q-OSTBC with power loading in the channel estimation error case
4.6 Summary
In Section 4.1, we introduce the closed-form formula toward the upper bound of mean BER averaged with respect to the channel distribution. Because the diagonal entries of the upper triangular matrix are related to the channel determinant, we find the bound of the channel determinant in Section 4.2. Then, we exploit the bound of the channel determinant to derive the upper bound of average BER. The upper bound is written as a quadratic form, so we can evaluate the upper bound of the mean BER with respect to the channel distribution and obtain the corresponding closed-form formula.
Considering perfect channel estimation, we determine our power loading factors by minimizing the corresponding closed-form formula. Furthermore, we consider
with respect to the channel distribution in closed-form. By minimizing this upper bound, we obtain the power loading factors. We compare the simulated average BER of the following receivers: linear MMSE equalizer, QR-based detectors with and without power loading, and the joint ML decoding. In Figure 4.2, we can see that when the QR-based solution is combined with the proposed optimal power loading scheme, performance improvement up to about 2 dB is achieved in the medium-to-high SNR region, in particular, and the BER is almost identical to that attained by the optimal ML decoding for SNR above 22.5 dB.
Chapter 5 Conclusion
In this thesis, we consider the transmission of the ABBA code over i.i.d. Rayleigh fading channels, and propose a symbol power allocation scheme for minimizing the average BER performance. In order to achieve a bit-error-rate (BER) performance compromise between linear equalization and joint maximum likelihood (ML) decoding, we propose to adopt QR-based successive detection with proper symbol power allocation. In Chapter 2, we introduce OSTBC and Q-OSTBC; their corresponding decoding methods are introduced. The QR decomposition of the channel matrix for the ABBA code is derived in Section 2.3.2. By exploiting a distinctive channel matrix structure induced by the ABBA code, we derive an explicit formula of the associated QR-decomposition. Then, we detect the received signals with QR-based successive detection. In Chapter 3, it is shown that the average BER with errorless front-layer decision feedback, although being merely a lower bound of the true mean error rate, remains simple to characterize and, moreover, is closely related to an upper bound of the block error probability when error-propagation occurs [10]. Motivated by this fact and to guarantee a performance improvement, the optimal power allocation schemes are introduced under a fixed channel realization without considering error propagation.
Then, considering the case that when error propagation occurs, the corresponding
method is presented. The simulations show that the performance is improved by allocating transmit power via the minimum BER criterion.
In Chapter 4, the overall mean BER averaged with respect to the channel distribution is introduced first and the bound is derived for the channel determinant in Section 4.2. The exploitation of a symmetric channel matrix structure unique to the ABBA code leads to a closed-form upper bound of the overall mean BER (averaged over the channel distribution). The optimal power allocation factors obtained by minimizing this bound thus guarantee a universal performance regardless of the instantaneous channel characteristics. That is, we propose to determine the power loading weights toward minimizing the overall mean BER, averaged with respect to the channel distribution. Simulation results confirm the effectiveness of the proposed solution: the achievable BER result is almost identical to that of joint ML detection when SNR is high.
The study presented in the thesis has discussed a power allocation scheme by minimizing the average BER in the error propagation free case. In particular, we derive the upper bound of the mean BER averaged with respect to the channel distribution in closed-form. Instead of considering the channel realization, we only require to know SNR, the channel covariance matrix, and the estimation error covariance matrix. No error propagation is considered in the above discussions. We then take error propagation into account and derive the corresponding mean BER formula averaged with respect to the channel distribution. Considering error propagation, the multiplication of two Q-functions problem occurs. However, it is not easy for us to deal with the multiplication of two Q-functions problem averaged with respect to the channel distribution. The derivation of the corresponding upper bound is thus a problem worthy of investigation in the future.
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