Computer Simulations
5.1 Channel Estimate Error Performance
5.1.1 Subspace-based Method
T K a b
I of the matlab form IK( : ,:)a b is the submatrix of I which contains from K
the ath to the bth row of I . K , , , and are all used in four-antenna STBC OFDM systems. Precoders in these forms can keep the algorithm in section 4.2.2 works successfully [17].
θ1 θ2 θ3 θ4
5.1 Channel Estimate Error Performance
5.1.1 Subspace-based Method
In section 5.1 and 5.2, static channels are used in examining the estimator error. We want to see how the subspace method performs in different SNR. Several basic simulation setups are as follows:
■ N =100
■ M =32, K =24
■ Rayleigh fading channels ■ L=4 ( 5−raychannels)
■ BPSK or QPSK (in section 5.2) modulation
We first illustrate the tests of theoretical and simulated subspace method in four STBC models. The theoretical result follows Eq. (4.23). Fig.5.1 shows the theoretical result of subspace method in diagonally weighted RO and BD STBC OFDM. As SNR becomes higher, the performance becomes better. This property appears in all four STBCs and will be shown in the following figures. Fig.5.1 also shows that BD outperforms RO under the same k.
0 5 10 15 20 25 30 35
10-7 10-6 10-5 10-4 10-3 10-2
10-1 RO & BD, Theoretical Subspace, N=100
SNR(dB)
NMSCE
RO (k=1) RO (k=2) RO (k=3) BD (k=1) BD (k=2) BD (k=3)
Fig. 5.1 RO & BD, Theoretical Subspace NMSCE
From Fig.5.2 to Fig.5.5, theoretical and simulated subspace NMSCE are exhibited in diagonally weighted STBCs RO, SD, DD, and BD. =1 and 2 is set. In these figures, we can see that the simulated result is worse than the theoretical one.
However, as SNR increases, the simulated result approaches the theoretical one, which fits the assumption of Eq. (4.23) in high SNR condition.
k
0 5 10 15 20 25 30 35 10-5
10-4 10-3 10-2 10-1
RO
SNR(dB)
NMSCE
k=1, Theory k=1, Simulation k=2, Theory k=2, Simulation
Fig. 5.2 RO, Theoretical and Simulated Subspace NMSCE
0 5 10 15 20 25 30 35
10-6 10-5 10-4 10-3 10-2
S D
S NR(dB )
NMSCE
k=1, Theory k=1, S im ulation k=2, Theory k=2, S im ulation
Fig. 5.3 SD, Theoretical and Simulated Subspace NMSCE
0 5 10 15 20 25 30 35 10-6
10-5 10-4 10-3 10-2
DD
SNR(dB)
NMSCE
k=1, Theory k=1, Simulation k=2, Theory k=2, Simulation
Fig. 5.4 DD, Theoretical and Simulated Subspace NMSCE
0 5 10 15 20 25 30 35
10-6 10-5 10-4 10-3 10-2
10-1 BD
SNR(dB)
NMSCE
k=1, Theory k=1, Simulation k=2, Theory k=2, Simulation
Fig. 5.5 BD, Theoretical and Simulated Subspace NMSCE
The performance of Subspace method of four-antenna STBCs in BPSK is shown in Fig.5.6 (k=2, k =1 also for RO). It shows that RO is worse than three complex non-orthogonal models.
0 5 10 15 20 25 30 35
10-6 10-5 10-4 10-3 10-2 10-1
4 models (k=1(RO),2, BPSK) Subspace
SNR(dB)
NMSCE
RO (k=1) RO (k=2) SD (k=2) DD (k=2) BD (k=2)
Fig 5.6 Four models, k =2 (k =1 for RO), Subspace
Since the theoretical value of Eq. (4.23) derived in [17] is an approximation for the channel estimate MSE for high SNR, there is a difference gap between theoretical and simulated results in all four models, which will become smaller when SNR increases.
Another property of diagonally weighted STBCs shown in Fig.5.1~5.5 is that in the same kind of STBC, the estimated error performance becomes better when is larger. Let’s discuss about it from Eq. (4.23), (4.24), and (4.25), as the diagonal
elements of in Eq. (4.24) become larger, those of
k
~
∑
q ⎛⎜⎝∑
~ q⎞⎟⎠−1 in Eq. (4.25) will become smaller, and thus will cause the estimated mean square error in Eq. (4.23)smaller. We had observed that when k grows, diagonal elements in also
increase, which makes a better estimator. NMSCE in different when SNR = 15 dB in RO with BPSK is shown in Fig.5.7. The results of three non-orthogonal models are similar to that of RO.
~
∑
qk
0.1 1 10
10-4 10-3 10-2
10-1 RO, NMS vs k (SNR = 15 dB)
k
NMSCE
CE
Sim ulation Theory
Fig. 5.7 RO, NMSCE vs. k (SNR = 15 dB)
5.1.2 Performance of PD
The performance of the improved method on subspace method, PD, will be illustrated and compared with subspace method both in BPSK and QPSK systems.
First, we demonstrate the performance of PD of four-antenna STBCs in BPSK (Fig.5.8, k =2, k=1 also for RO) and in QPSK (Fig.5.9, k =2, no RO). Fig.5.8 shows that in BPSK, RO always has the worst performance, while DD has the best at low SNR. The second is SD and the third BD. At high SNR, however, the better and the worse performance orders of three non-orthogonal STBCs are different from that at low SNR. Such situation is the same in QPSK in Fig.5.9.
0 5 10 15 20 25 30 35 10-7
10-6 10-5 10-4 10-3 10-2
4 models (k=1(RO),2, BPSK) Subspace + PD
SNR(dB)
NMSCE
RO (k=1) RO (k=2) SD (k=2) DD (k=2) BD (k=2)
Fig. 5.8 Four models, k=2 (k=1 for RO), Subspace + PD BPSK
0 5 10 15 20 25 30 35
10-6 10-5 10-4 10-3 10-2
3 models (k=2, QPSK) Subspace + PD
SNR(dB)
NMSCE
SD (k=2) DD (k=2) BD (k=2)
Fig. 5.9 Three models, k=2, Subspace + PD in QPSK
Performance comparisons between subspace method and subspace with PD in four models are displayed in Fig.5.10~5.13. Note that in RO, only BPSK is used and , 2 is set in simulation. In other three non-orthogonal models,
1 k= 2
k = and both BPSK and QPSK are used
We can see that PD does improve the subspace method of all the four models in BPSK and SD, DD, BD in QPSK. Besides, the estimator in BPSK is better than that in QPSK in same conditions.
0 5 10 15 20 25 30 35
10-6 10-5 10-4 10-3 10-2 10-1
RO (k=1 & k=2, BPSK)
Subspace & Subspace + PD
SNR(dB)
NMSCE
RO, k=1, Subspace + PD RO, k=1, Subspace RO, k=2, Subspace + PD RO, k=2, Subspace
Fig. 5.10 RO, k =1, 2, Subspace & Subspace + PD in BPSK
0 5 10 15 20 25 30 35 10-7
10-6 10-5 10-4 10-3 10-2
SD (k=2, BPSK & QPSK) Subspace & Subspace + PD
SNR(dB)
NMSCE
SD, Subspace, BPSK SD, Subspace + PD, BPSK SD, Subspace, QPSK SD, Subspace + PD, QPSK
Fig. 5.11 SD, k =2, Subspace & Subspace + PD in BPSK & QPSK
0 5 10 15 20 25 30 35
10-7 10-6 10-5 10-4 10-3 10-2
DD (k=2, BPSK & QPSK) Subspace & Subspace + PD
SNR(dB)
NMSCE
DD, Subspace, BPSK DD, Subspace + PD, BPSK DD, Subspace, QPSK DD, Subspace + PD, QPSK
Fig. 5.12 DD, k =2, Subspace & Subspace + PD in BPSK & QPSK
0 5 10 15 20 25 30 35 10-7
10-6 10-5 10-4 10-3 10-2
BD (k=2, BPSK & QPSK) Subspace & Subspace + PD
SNR(dB)
NMSCE
BD, Subspace, BPSK BD, Subspace + PD, BPSK BD, Subspace, QPSK BD, Subspace + PD, QPSK
Fig. 5.13 BD, k =2, Subspace & Subspace + PD in BPSK & QPSK
We also show how PD acts here with different multipath lengths. and BPSK are used, here. With and
2 k = 4
L= L=5, subspace method performs in four models from Fig.5.14 to Fig.5.17. In the model of larger CP length, it will be less sensitive to channel noise variation, and will lead to a poorer performance. It is shown that PD is immune to multipath length.
0 5 10 15 20 25 30 35 10-6
10-5 10-4 10-3 10-2
10-1 RO (k=2, BPSK) with different multipath lengths
SNR(dB)
NMSCE
Subspace, L=5 Subspace, L=4 Subspace + PD, L=5 Subspace + PD, L=4
Fig. 5.14 RO, k =2, BPSK, Subspace & Subspace + PD with different multipath lengths
0 5 10 15 20 25 30 35
10-7 10-6 10-5 10-4 10-3 10-2
10-1 S D (k = 2, B P S K ) with different m ultipath lengths
S NR(dB )
NMSCE
S ubs pac e, L= 5 S ubs pac e, L= 4 S ubs pac e + P D, L= 5 S ubs pac e + P D, L= 4
Fig. 5.15 SD, k =2, BPSK, Subspace & Subspace + PD with different multipath lengths
0 5 10 15 20 25 30 35 10-7
10-6 10-5 10-4 10-3 10-2
10-1 DD (k = 2, B P S K ) with different m ultipath lengths
S NR(dB )
NMSCE
S ubs pac e, L= 5 S ubs pac e, L= 4 S ubs pac e + P D, L= 5 S ubs pac e + P D, L= 4
Fig. 5.16 DD, k =2, BPSK, Subspace & Subspace + PD with different multipath lengths
0 5 10 15 20 25 30 35
10-7 10-6 10-5 10-4 10-3 10-2
BD (k=2, BPSK) with different multipath lengths
SNR(dB)
NMSCE
Subspace, L=5 Subspace, L=4 Subspace + PD, L=5 Subspace + PD, L=4
Fig. 5.17 BD, k =2, BPSK, Subspace & Subspace + PD with different multipath lengths