Computer Simulations
4.5 Balanced step-size control method for LMS channel tracking tracking
better than LMS method. But, the main disadvantage of RLS method is its complexity.
Although algorithm of LMS method is simpler, it has several problems with convergence and tracking speed. Now, we will discuss these problems and propose an improved balanced step-size control method.
In subsection 3.4.4, we discussed the problem of step-size. In general, step-size will influence convergence rate and the minimum mean-square error Jmin in static channel. Next example can clearly explain the influence of step-size in LMS channel tracking. The parameters of example are η =67%, Ω=2 and Λ=1000. Two step-sizes respectively are 0.008 and 0.03. Learning curve is showed in Fig. 4.17.
Figure 4.17 Learning curves of LMS channel tracking in static channel, SNR=30dB
Fig 4.17 is the learning curve of channel at SNR=30dB. From this figure can see that convergence rate of a large step-size is faster and its Jmin is larger. Small step-size, its convergence is slower and its Jmin is smaller. Convergence rate and
Jmin is a trade-off in static channel. But, choice of step-size in time variant channel must consider the problem of slow tracking. Next, MSE of channel will be shown in Fig. 4.18
Figure 4.18 MSE of LMS channel tracking in static channel
This figure clearly shows that MSE of small step-size (circular point) is better than large step-size (triangular point). This implies in case of high SNR a small step-size is preferred in static channel, because its Jmin is smaller than that of a large step-size.
In subsection 3.4.4, we discussed problem of tracking rate in time variant channel. Large step-size can be easier to keeping up with variation of true channel, especially in short time, channel changes very fast. This implies a trade off of tracking rate and Jmin. In low SNR, problem of Jmin is more serious than problem of tracking. Therefore, small step-size can reduce channel estimation error in low SNR.
In high SNR, noise variance is very small. So, problem of Jmin is less serious than
step-size control method in subsection 3.4.4 can select proper step-size for all SNR.
Next the example can clearly explain problem of choice of step-size. Its simulation parameters are η =67% , Ω=2 , fD =50Hz and Λ=1000 . Parameters of balanced step-size control in Eq. (3.64) are γ =0.1,a1 =0.008 and a2 =0.03. Three choices of Step-sizes include 0.008, 0.03 and balanced step-size control method to simulate. MSE of channel is showed in Fig. 4.19.
Figure 4.19 MSE of LMS channel tracking in time variant channel
This figure shows that performance of small step-size (circular point) is best in low SNR. And in high SNR, performance of large step-size (triangular point) is best. The performance of balanced step-size control (diamond point) follows closely to the small step-size in low SNR. In high SNR, it will follow closely to the large step-size.
We will use Fig. 4.20 and 4.21 to explain this phenomenon.
Figure 4.20 Learning curves of LMS channel tracking in time variant channel, SNR=10dB
Figure 4.21 Learning curves of LMS channel tracking in time variant channel, SNR=25dB
respectively. In Fig. 4.19, convergence rate of small step-size is slowest. But, its Jmin is smallest. So, its MSE performance is best in low SNR. In Fig. 4.20, small step-size tracking rate is not enough fast. So, its MSE is worst in high SNR. On the contrary, tracking rate of a large step-size because is fast enough. Besides, Jmin is not influenced in the high SNR. So in high SNR, MSE of large step-size is best. For problem of slow tracking, we can explain it by Fig 4.22.
Figure 4.22 Instantaneous channel errors for LMS channel tracking,SNR=25dB
In Fig. 4.22, small step-size suffers slow tracking. Because a small step-size, estimated channel can not follow true channel. Large step-size and balanced step-size control do not have not problem of channel tracking. For these two methods, estimated channel can follow true channel, so their instantaneous channel errors are smaller than small step-size.
The balanced step-size control can solve the problem of large Jmin in low SNR
as well as the problem of slow tracking in high SNR. In Fig. 4.19, balanced step-size control method and small step-size have the same small Jmin. And in Fig. 4.20, tracking of balanced step-size control method and large step-size are nearly same fast.
We conclude that the performance of balanced step-size control is most suitable for all SNR in channel tracking.
Chapter 5
Conclusion
In wireless communication, the bandwidth is a very important resource. The conventional OFDM system cancels IBI and ICI problems on purpose, and therefore it needs to use extra guard interval. Thus, it will reduce the bandwidth usage efficiency. The BE-OFDM system solves the problem of efficiency of bandwidth usage and it can adjust efficiency of bandwidth usage. But, the channel equalizer is very complex.
In this thesis, we find out cause of complex in BE-OFDM system. Proper adjustment of oversampling factor will be effective to decrease complexity of channel equalizer. Consequently, it is shown by simulation that the system complexity is reduced.
As for channel estimation, we use combination of the training and tracking methods. For BE-OFDM system, we propose adaptive channel tracking with better method. The method estimates coefficients of channel equalizer, accuracy is higher than adaptive equalizer, and higher complexity. But, in high SNR, LMS algorithm suffers problem of slow tracking rate. We propose the balanced step-size control to solve problem of tracking.
In this thesis, there are many problems need to be improved, such as channel equalizer in BE-OFDM system is still very complex. In the future, simplify the complexity of channel equalizer in BE-OFDM system or using another more effective channel equalizer will be very important. Extending the concept of bandwidth efficiency OFDM system to another communication system, such as space-time technique or CDMA system, etc, also needs further investigation.