2.2 ZF Method
2.2.5 Simulation Results
In this subsection, we report simulation results to demonstrate the effectiveness of the proposed method. We consider an OFDM system with and the CP size of 16. The modu-lation scheme is 16-QAM. The channel length is set to i X and the power delay profile is characterized by an exponential function, i.e., 9
index. Each channel tap is generated by Jakes’ model [70]. Here, we assume that the channel response is exactly known for the direct ZF method. For the proposed method, the parameters of the LTV channel model are obtained by least-squares (LS) fittings. Define the normalized Doppler frequency asH&I H&å oqp , where H&å is the maximum Doppler frequency andorp is the sampling period. Since the N-ZF method with ! have the similar performance to that with
5»
b
, we set ! for all simulations.
First, we evaluate the validity of the analytic output SINRs for the proposed method. Two cases are considered; case meets the convergence condition derived in Subsection 2.2.4, whereas case doesn’t. We let H&I T3KJL , SNR TRX dB, , and % T3 . Figure 2.3 shows the analytic subcarrier SINRs for case . Since the simulated SINRs are identical to the analytic SINRs, they are not shown in the figure. From this figure, we find that each subcarrier exhibits a different SINR due to the characteristic of the frequency-selective channel. Also, subcarrier SINRs are all increased as the number of iterations is increased. The performance of the proposed method with two iterations is similar to that with three iterations. We also can find that the output SINRs of the proposed method are very close to those of the direct ZF method.
Fig. 2.4 shows analytic subcarrier SINRs for case 2. We see that even in this divergence case, SINRs are still increased for the first two or three iterations. For the fourth iteration, subcarrier SINRs start to fall. The result for % > is similar to that for % Z3 except that the required number of iterations is reduced to one or two.
Next, we consider the performance comparison among the proposed and conventional meth-ods. Here, H&I is set to 3KJL . Specifically, the bit-error-rate (BER) is used as the performance measure. For the purpose of benchmark, we also show the result of the direct MMSE method, and that of the direct ZF method with H I 83 (ICI-free). From extensive simulations, we also find that the performance of the PSE method cannot be further enhanced when ¤ . For this reason, we only show the result for 8 . Fig. 2.5 shows the BER performance comparison for% 43 . As we can see, the performance of the PSE method is limited and has an error floor phenomenon. The N-ZF method outperforms the PSE method even with one iteration only.
As mentioned, there is a convergence condition for the PSE method. This condition is totally dependent on the channel. The N-ZF method also has its convergence condition. However, it depends on the initial matrix as well as the channel. It is then possible to reduce the probability of divergence through the determination of¶ o . This is the main reason why the N-ZF method can outperform the PSE method. The required complexity of the N-ZF method is lower than that of the PSE method (this can be seen later). Moreover, the performance of the N-ZF method
with three iterations can approach to that of the direct ZF method. Here, the performance of the direct ZF method is only slightly worse than that of the direct MMSE method. Figure 2.6 shows the BER performance comparison for % . It is obvious that the N-ZF method can quickly approach to the direct ZF method with one or two iterations. Note that the N-ZF method with two iterations can even perform slightly better than the direct ZF method. This behavior is interesting and it needs further discussions. Since the LTV model instead of the Jakes’ model is used, one may be curious if the result is due to the modeling error. We then conduct a simula-tion, in which the Jake’s model is used, to answer the question. Fig. 2.7 shows the result. From this figure, we see that the performance of the ZF and MMSE methods using the exact Jakes’
channel is almost the same as that of the ZF and MMSE methods using the LTV channel. This indicates the behavior observed in Fig. 2.6 is not due to the modeling error. The reason for the behavior is explained as follows. The N-ZF method only iterates Newton’s method two or three times, and it may not converge completely in all cases. As known, the direct ZF method has a noise enhancement problem. It is then possible that the noise enhancement caused by the N-ZF method is smaller. As a result, the performance of the N-ZF method can be better than that of the direct ZF method. If the convergence condition is met and the number of iterations is large enough, the performance of the N-ZF method and the direct ZF method will be the same. This phenomenon has been verified by simulations, but the result is not shown here. Compared to Fig. 2.5, we see that the N-ZF method with % ` has the better performance than that with
%
53 , and it can approach to the direct ZF method more quickly.
To test the limitation of all algorithms, we consider a severer case in whichHI 53KJc . Figure 2.8 shows the simulation result. From this figure, we see that the N-ZF method can still work, but its performance is degraded since ICI is much larger than that in the previous cases. Also, we can see that the degradation of the MMSE method is smaller, and the performance gap between the ZF and MMSE methods become larger. We also conduct simulations to evaluate the robustness of all algorithms to the variation of the normalized Doppler frequency. Figure 2.9 shows the results for H&I varying from 0 to 0.2. Here, the SNR is set to 30 dB. From this
figure, we can see that the performance of all methods is degraded as the normalized Doppler frequency is increased. Also, the MMSE method is the most robust method while the N-ZF method is the second.
Table 2.3 summarizes the required computational complexity of the direct ZF method, the PSE method, and the N-ZF method for the simulation setting shown above. In this table, the ratios in the parenthesis indicate the number of operations required for the N-ZF method divided by those for the direct ZF and PSE methods. The first ratio is for the direct ZF method and the second one is for the PSE method. From the above simulations, we can say that for % !3 , the required iteration number for the N-ZF method is two or three, whereas for % ò , it is one or two. From Table 2.3, we can see that the N-ZF method with % >3 can save tremendous computations compared to the direct ZF method. With two (three) iterations, its multiplication complexity is only 0.007 (0.015) times that of the direct ZF method. As for the case of %
, it also saves a lot of computations compared to the direct ZF method. We find that the multiplication complexity of the N-ZF method with one (two) iterations is only 0.008 (0.013) times that of the direct ZF method. As to additions, the complexity ratios are similar to those of multiplications. As to divisions, the ratios are 0.016 and 0.015 for % 3 and % , respectively. Compared to the PSE method ( ! ), the N-ZF method (Q and% 53 ) also has lower complexity and better performance. Its required multiplications (additions, divisions) is 0.85 (0.745, 0.5) times those of the PSE method. For various , we show the required computational complexity for the direct ZF method and the N-ZF method (Q ` ) R ) in Figs.
2.10 and 2.11, respectively. In these two figures, RM, RD, and RA denote real multiplication, real division, and real addition, respectively.
Another important property of the proposed N-ZF method is that it can trade the desired performance for the required complexity. However, the direct ZF method doesn’t have such a choice. This property will make the N-ZF method a more efficient method since it can adapt itself to various SNR environments. If the operated SNR is not high, the iteration number can be reduced. For example, in Fig. 2.5, it only requires one iteration to approach the direct ZF
method when SNR is less than X dB.
0 5 10 15 20 25 30 35 10−3
10−2 10−1 100
SNR (dB)
BER
One−tap FEQ Direct ZF Direct MMSE PSE (U=2) N−ZF (k=1) N−ZF (k=2) N−ZF (k=3) Direct ZF (ICI−free)
Figure 2.5: BER comparison among one-tap FEQ, PSE, N-ZF (% 83 and Z ), direct ZF, and direct MMSE methods in a SISO-OFDM system;H I = 0.1 and 16-QAM modulation.
0 5 10 15 20 25 30 35
10−3 10−2 10−1 100
SNR (dB)
BER
One−tap FEQ Direct ZF Direct MMSE N−ZF (k=0) N−ZF (k=1) N−ZF (k=2) Direct ZF (ICI−free)
Figure 2.6: BER comparison among one-tap FEQ, N-ZF (% " and ` ), direct ZF, and direct MMSE methods in a SISO-OFDM system;H I = 0.1 and 16-QAM modulation.
0 5 10 15 20 25 30 35 10−3
10−2 10−1 100
SNR (dB)
BER
One−tap ZF FEQ−ZF (Jakes) FEQ−ZF (LTV) FEQ−MMSE (Jakes) FEQ−MMSE (LTV)
Figure 2.7: BER comparison between the direct ZF and MMSE methods using the exact Jakes and LTV channels in a SISO-OFDM system;H I = 0.1 and 16-QAM modulation.
0 5 10 15 20 25 30 35
10−3 10−2 10−1 100
SNR (dB)
BER
One−tap FEQ Direct ZF Direct MMSE N−ZF (k=0) N−ZF (k=1) N−ZF (k=2) Direct ZF (ICI−free)
Figure 2.8: BER comparison among one-tap FEQ, N-ZF (% " and > ), direct ZF, and direct MMSE methods in a SISO-OFDM system;H I = 0.2 and 16-QAM modulation.
0 0.05 0.1 0.15 0.2 10−3
10−2 10−1
Normalized Doppler frequency (f
d)
BER
One−tap FEQ Direct ZF Direct MMSE N−ZF (k=2)
Figure 2.9: BER comparison among one-tap FEQ, N-ZF (% " and ` ), direct ZF, and direct MMSE methods in a SISO-OFDM system;H I = 0 O 0.2, 16 QAM modulation, and SNR
= 30 dB.
0 1000 2000 3000 4000 5000 6000 7000 8000 10−2
100 102 104 106 108 1010 1012
Number of subcarriers
Number of mathematical operations
RM of N−ZF (k=2) RD of N−ZF (k=2) RA of N−ZF (k=2) RM of direct ZF RD of direct ZF RA of direct ZF
Figure 2.10: Complexity comparison between N-ZF (% M , # , and Q ) and direct ZF methods in a SISO-OFDM system for various .
0 1000 2000 3000 4000 5000 6000 7000 8000
Number of mathematical operations
RM of N−ZF (k=3) RD of N−ZF (k=3) RA of N−ZF (k=3) RM of direct ZF RD of direct ZF RA of direct ZF
Figure 2.11: Complexity comparison between N-ZF (% M , # , andQ GR ) and direct ZF methods in a SISO-OFDM system for various .