In this section, we provide numerical simulations to verify the theoretical results developed in this chapter.
The first example is to plot the subchannel gains of the ZF-BD-GMD and the ZF-Optimal sys- tems for a MIMO (NT = 2, andNR = 3) frequency selective channel Ha(z)when K = 5 and K = 10. HereHa(z) = H0+H1z−1+H2z−2+H3z−3, and the coefficients of these matrices are shown below:
H0=
0.0476−0.4556i −1.1298 + 1.2318i 0.5694 + 0.9440i 0.4338−0.9422i
−1.0402 + 0.2657i 0.7277 + 0.2035i
H1=
0.7978 + 1.2002i 0.2938−0.2975i
−0.7598 + 0.4878i −1.0624 + 0.2195i
−0.1242 + 0.9503i 0.6558−1.0261i
H2=
−0.4325 + 0.6604i −0.9536 + 1.0979i 0.0105−0.2053i −0.2412−1.0218i 0.2921−0.2946i 0.8198 + 0.9613i
H3=
−0.0030 + 0.0512i −0.2331 + 0.3128i
−0.7642−0.2375i −1.1634−1.5280i 1.0184 + 0.2687i −0.2281 + 0.8638i
Fig. 4.3 shows the subchannel gains. For the ZF-BD-GMD systems, the first10subchannel gains whenK = 5are the same as that whenK = 10. This can be seen from (4.10). The11th to the20th subchannel gains whenK= 10are smaller than the first10subchannel gains. Also, the subchannel gains are non-increasing with the subchannel index. This is consistent with Theorem 6.4.1. For ZF- Optimal systems, all the subchannel gains are identical and equal (see Eq.(4.18)) to the geometric mean of the subchannel gains in the ZF-BD-GMD system. For example, all the subchannel gains equal5.08dB inK = 5case. It can be seen that the subchannel gains forK = 10equal4.92dB, which is less than that forK = 5. This is consistent with Theorem 4.5.1, where the tradeoff exists in the system BER performance and the BW efficiency. We also calculated the value ofrdefined in
(4.16) by numeric integration. The result isr= 4.787dB and is shown in Fig. 4.3. From this figure, we can also see the trend of the subchannel gains asKincreases. AsKincreases, the subchannel gains of the ZF-Optimal system will be lower and closer to the value ofr. ForK →infinity, these channel gains should converge tor. This fact was also predicted by Theorem 4.3.4, which states that both the ZF-BD-GMD systems and the ZF-Optimal systems will have BER close toP(r)for very largeK.
0 5 10 15 20
4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5
sub−channel index
effective channel gain(dB)
ZF−BD−GMD,K=5 ZF−Optimal,K=5 ZF−BD−GMD,K=10 ZF−Optimal,K=10 r, as in Eq.(16)
Figure 4.3: The effective channel gain of ZF-BD-GMD and ZF-Optimal transceivers for channel Ha(z).
In the following simulations, symbols are generated using gray encoded QPSK constellations with symbol powerσ2s. In each case,103different channels are used for the Monte Carlo simula- tions. These channels have the entries coming from i.i.d. complex zero-mean Gaussian distribu- tions with unit variance. The additive channel noise has covariance matrixRn=I.
In Fig. 4.4 and Fig. 4.5 we show the average BER simulation results with respect to different values ofσ2sfor the MIMO systems withNT =NR = 2, for ZF-DFE and MMSE-DFE case, respec- tively. The MIMO channels are with orderL= 2. The zero-forcing system performances forK= 3, K= 10, andK= 20are shown in Fig. 4.4. The ZF-Optimal system appears to have the best perfor- mance for allK. The ZF-DFE system with lazy precoder has about2dB loss at BER= 10−5when K= 3compared to ZF-Optimal. However, the ZF-BD-GMD only has about0.3dB loss. We can see that the ZF-BD-GMD system indeed has similar performance as the ZF-Optimal system. For larger K, the performance difference between the two systems becomes even smaller.
The MMSE system performance with the same channel setting is shown in Fig. 4.5. Similar conclusions as the ZF case can be made here for MMSE case. From these results, we see that the BD-GMD systems have nearly optimal uncoded BER performance for both ZF and MMSE case, and are much better than the lazy precoder systems. Compared to these two figures, we can see that by using the MMSE-DFE systems, the average BER performance can be a little better than using the ZF-DFE systems. It is important to note that the MMSE-BD-GMD systems have the same implementation cost as the ZF-BD-GMD systems, thus the MMSE-BD-GMD systems is an even better candidate in practice.
−8 −6 −4 −2 0 2 4 6 8
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100
σ s 2 (dB)
BER
ZFBDG,K=3 ZFBDG,K=10 ZFBDG,K=20 ZFOPT,K=3 ZFOPT,K=10 ZFOPT,K=20 ZF−Lazy,K=3 ZF−Lazy,K=10 ZF−Lazy,K=20
Figure 4.4: The BER performance of the zero-forcing systems for MIMO (NT =NR = 2) Rayleigh channels of order3, withK = 3,K = 10, andK = 20. “ZFBDG” represents the ZF-BD-GMD system; “ZFOPT” represents the ZF-Optimal system; and “ZF-Lazy” represents the lazy precoder with zero-forcing DFE.
In Fig. 4.6 and Fig. 4.7 we show the simulation results for case of single transmitting antenna and single receiving antenna. The SISO channels haveL = 2. The zero-forcing system perfor- mances forK= 3,K= 10, andK= 20are shown in Fig. 4.6. The MMSE system performances for the same channel settings is shown in Fig. 4.7. For largeK, lazy precoder case has BER performance almost identical to that of the optimal systems. The simulation results confirm the discussions in 4.6. Also, whenKis larger, the BER performance is worse. This confirms the tradeoff between BW efficiency and BER.
−8 −6 −4 −2 0 2 4 6 8 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100
σ s 2 (dB)
BER
MSBDG,K=3 MSBDG,K=10 MSBDG,K=20 MSOPT,K=3 MSOPT,K=10 MSOPT,K=20 MS−Lazy,K=3 MS−Lazy,K=10 MS−Lazy,K=20
Figure 4.5: The BER performance of the MMSE systems for MIMO (NT = NR = 2) Rayleigh channels of order 3, with K = 3, K = 10, andK = 20. “MSBDG” represents the MMSE-BD- GMD system; “MSOPT” represents the MMSE-Optimal system; and “MS-Lazy” represents the lazy precoder with MMSE-DFE.
0 2 4 6 8 10 12
10−6 10−5 10−4 10−3 10−2 10−1
σ s 2 (dB)
BER
ZF−Lazy,K=3 ZF−Lazy,K=10 ZF−Lazy,K=20 ZFOPT,K=3 ZFOPT,K=10 ZFOPT,K=20
Figure 4.6: The BER performance of the zero-forcing systems for SISO (NT = NR = 1) Rayleigh channels of order3, withK= 3,K= 10, andK= 20. “ZFOPT” represents the ZF-Optimal system;
and “ZF-Lazy” represents the lazy precoder with zero-forcing DFE.
0 2 4 6 8 10 12 10−6
10−5 10−4 10−3 10−2 10−1
σs 2 (dB)
BER
MS−Lazy,K=3 MS−Lazy,K=10 MS−Lazy,K=20 MSOPT,K=3 MSOPT,K=10 MSOPT,K=20
Figure 4.7: The BER performance of the MMSE systems for SISO (NT =NR= 1) Rayleigh channels of order3, withK= 3,K= 10, andK= 20. “MSOPT” represents the MMSE-Optimal system; and
“MS-Lazy” represents the lazy precoder with MMSE-DFE.