Simulation Results and Analysis
4.4 Symbol Error Rate Results
As said before considering multipath channels, we do simulation for AWGN channel to validate the simulation model. We compare the theoretical SER curves with the SER curves resulting from simulation. For an even number of bits per symbol, the SER of rectangular QAM is given by
Ps = 4(1− 1
√M)Q(
√ 3
M − 1 Es
N0), (4.5)
where
• M = number of symbols in modulation constellation; for example, M = 4 for QPSK, 16 for 16QAM, and 64 for 64QAM,
• Es = average symbol energy,
0 5 10 15 20 25 30 35 40 45 50 10−6
10−5 10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
(c)
Figure 4.50: MSE performance of different channel separation methods equipped with FD GWD in 2× 2 MIMO over single-path Rayleigh fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−5
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
(c)
Figure 4.51: MSE performance of different channel separation methods equipped with FD GWD in 2× 2 MIMO over SUI1 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−4
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
(c)
Figure 4.52: MSE performance of different channel separation methods equipped with FD GWD in 2× 2 MIMO over SUI2 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−3
10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
(c)
Figure 4.53: MSE performance of different channel separation methods equipped with FD GWD in 2× 2 MIMO over SUI3 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−6
10−5 10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
sray
FDCS TDCS
(c)
Figure 4.54: MSE performance of different channel separation methods equipped with FD GWD in 4× 4 MIMO over single-path Rayleigh fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−5
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI1
FDCS TDCS
(c)
Figure 4.55: MSE performance of different channel separation methods equipped with FD GWD in 4× 4 MIMO over SUI1 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−4
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−4 10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI2
FDCS TDCS
(c)
Figure 4.56: MSE performance of different channel separation methods equipped with FD GWD in 4× 4 MIMO over SUI2 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 20 25 30 35 40 45 50 10−4
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
(a)(b)
0 5 10 15 20 25 30 35 40 45 50
10−3 10−2 10−1 100
SNR(dB)
MSE
SUI3
FDCS TDCS
(c)
Figure 4.57: MSE performance of different channel separation methods equipped with FD GWD in 4× 4 MIMO over SUI3 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 2 4 6 8 10 12 14 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100
Es/N0 (dB)
Symbol Error Rate
AWGN
Simulated−no estimation error Theory
Figure 4.58: SER performance under perfect channel state information (CSI) compared with theory in AWGN for QPSK.
• N0 = noise power spectral density (W/Hz), and
• Q(x) = 2π1 ∫∞
x e−t2/2dt.
Figure 4.58 plot the simulated SER values with uncoded QPSK in AWGN according to the block diagram in Figure 4.1 together with the theoretical values. We see that the simulation result agrees with the theoretical curve.
4.4.1 SISO
Figures 4.59–4.63 show the SER performance of different channel estimation methods in AWGN, single-path Rayleigh, SUI1, SUI2 and SUI3 channels, respectively. We can see that the SER performance of the different methods largely follows a similar ordering as their MSE performance, but the disparity is not as pronounced. Such a phenomenon has been observed in other simulations conducted previously [22]. We suspect that the reason may have to do with insufficient whiteness of the channel estimation error. But this is yet to be investigated.
0 5 10 15 10−5
10−4 10−3 10−2 10−1 100
Es/N 0 (dB)
Symbol Error Rate
AWGN
LS LMMSE GWD
Figure 4.59: SER performance of different channel estimation methods with SL = 12, SG = 12 in AWGN channel.
0 5 10 15 20 25 30 35 40 45 50
Symbol Error Rate
sray
Symbol Error Rate
sray
Symbol Error Rate
sray
LS LMMSE GWD
(c)
Figure 4.60: SER performance of different channel estimation methods with SL = 12, SG = 12 in signal path Rayleigh fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 10 20 30 40 50 60
Symbol Error Rate
SUI1
Symbol Error Rate
SUI1
Symbol Error Rate
SUI1
LS LMMSE GWD
(c)
Figure 4.61: SER performance of different channel estimation methods with SL = 12, SG = 12 in SUI1 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 10 20 30 40 50 60
Symbol Error Rate
SUI2
Symbol Error Rate
SUI2
Symbol Error Rate
SUI2
LS LMMSE GWD
(c)
Figure 4.62: SER performance of different channel estimation methods with SL = 12, SG = 12 in SUI2 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 10 20 30 40 50 60
Symbol Error Rate
SUI3
Symbol Error Rate
SUI3
Symbol Error Rate
SUI3
LS LMMSE GWD
(c)
Figure 4.63: SER performance of different channel estimation methods with SL = 12, SG = 12 in SUI3 fading channel at speeds (a) 3 km/h, (b) 60 km/h, and (c) 120 km/h.
0 5 10 15 10−5
10−4 10−3 10−2 10−1 100
Es/N 0 (dB)
Symbol Error Rate
AWGN
LS LMMSE size2 LMMSE size4 LMMSE size12 LMMSE size36
Figure 4.64: SER performance at different FD LMMSE submatrix sizes SL in AWGN chan-nel.
Since the SER and MSE performance have similar ordering, we now only concentrate on the 60 km/h case. Figures 4.64–4.66 show the SER performance of different FD LMMSE sub-matrix size SLin AWGN, single-path Rayleigh, SUI1, SUI2 and SUI3 channels, respectively, for a speed 60 km/h. The ordering of simulated SER performance is similar to the MSE performance as shown in Figures 4.10–4.14. We note that difference in SER performance with different SL is often quite.
Figure 4.67–4.69 show SER performance of different FD GWD bandwidth SGin AWGN, single-path Rayleigh, SUI1, SUI2 and SUI3 channels, respectively, for speed 60 km/h. The results in these figures are rather similar to LMMSE qualitatively. The ordering of simulated SER performance is similar to the MSE performance shown in Figures 4.15–4.19. We note again that the difference in SER performance with different SG is often small.
0 10 20 30 40 50 60 70 80 90 100 10−4
10−3 10−2 10−1 100
Es/N 0 (dB)
Symbol Error Rate
sray
LS FDLMSE size2 FDLMSE size4 FDLMSE size12 FDLMSE size36
Figure 4.65: SER performance at different FD LMMSE submatrix sizes SL in single-path Rayleigh channel at speed 60 km/h.
4.4.2 MIMO
In MIMO transmission, the SER performance depends highly on the multi-antenna receiving method, for which many techniques have been proposed by many people. As our interest is mainly in channel estimation and not in signal detection, we relegate the associated study on SER performance to potential future research.
0 10 20 30 40 50 60 70 80 90 100
Symbol Error Rate
SUI1
Symbol Error Rate
SUI2
Symbol Error Rate
SUI3
Figure 4.66: SER performance at different FD LMMSE submatrix sizes SL in (a) SUI1, (b) SUI2, and (c) SUI3 fading channels at speed 60 km/h.
0 5 10 15 10−5
10−4 10−3 10−2 10−1 100
Es/N 0 (dB)
Symbol Error Rate
AWGN
LS GWD length3 GWD length5 GWD length13 GWD length35
Figure 4.67: SER performance at different FD GWD bandwidth SG in AWGN channel.
0 10 20 30 40 50 60
10−5 10−4 10−3 10−2 10−1 100
Es/N 0 (dB)
Symbol Error Rate
sray
LS GWD length3 GWD length5 GWD length13 GWD length35
Figure 4.68: SER performance at different FD GWD bandwidth SG in single-path Rayleigh channel at speed 60 km/h.
0 10 20 30 40 50 60
Symbol Error Rate
SUI1
Symbol Error Rate
SUI2
Symbol Error Rate
SUI3
Figure 4.69: SER performance at different FD GWD bandwidth SG in (a) SUI1, (b) SUI2, and (c) SUI3 fading channel at speed 60 km/h.