1. By Viput Subharngkasen
Channel/System identification using Total Least Mean Squares Algorithm (TLMS) By
Viput Subharngkasen

2. Outline Introduction/Motivation Review of TLMS algorithm Result
Discussion
Conclusion

3. Introduction/Motivation
The TLMS algorithm is unsupervised learning adaptive linear combiner based on the total least mean squares of the minimum Raleigh quotient which used to extract minor features from the training sequences.
In adaptive linear combiner, we use only input and output data to compute the transfer function of the analyzed channel/system.
What’s happen when the input and output data that we get are corrupted with interference or we can not access the real input and output ports and can get only partial part of them. ?????
How can we counter these problems ?????

4. Introduction/Motivation (cont.)
We use TLMS algorithm to solve this problem compare with well-known algorithm, LMS algorithm.
If interference only exists in the output of the analyzed system, the LMS algorithm can obtain the optimal solution. However, if the interference occur on both input and output end of the problem, the LMS algorithm is failed to achieve the optimal solution.
On the other hand, the TLMS algorithm can solve for the desired solution in case of interference existed in both input and output end.

5. TLMS algorithm
We have n-dimensional input vector X(k), the desire output d(k). Let Z(k) be the augmented vector of X(k) and d(k)
Z(k) = [XT(k) | d(k)]T
Assume that interference exists in both input and output called  X(k) and  d(k), and the corresponding set of adjustable weight w(k).
Then the estimated output y(k) is a linear combination of the input and weight vector as
y(k) = XT(k) W(k)
To find the optimal solution, we will minimized the effect of the interference, thus,
min E{||XT(k) | d(k)||2 }
Where || . ||2 is the Euclidean Distance.

6. TLMS algorithm (cont.) The minimization can be achieved through
min E{ | XT(k)WTLMS – d(k) |2 }
min E{ | ZT(k)W(k) |2 } subject to ||W(k)|| 2 = c
where X(k) = X(k) + X(k), d(k) = d(k) + d(k),
Z(k) = Z(k) + Z(k), and W(k) = [WT(k) | Wk+1 ]
and c is any positive constant
Let R = E{Z(k) ZT(k)}, then the problem become to
min E{ WT(k)RW(k) } subject to ||W(k)|| 2 = c
By LaGrange expansion, to optimize this problem is to choose the eigenvector corresponded to the smallest eigenvalue of the augmented correlation matrix R

7. TLMS Algorithm (cont.)
The search equation for eigenvector corresponded to the smallest eigenvalue is
W(k+1) = W(k) + ( W(k) - ||W(k)||22 RW(k) )
Where k is an iteration number and  is the positive constant that control the rate of convergence and stability of algorithm.
The equation can also be rewrite as
Y(k) = ZT(k)W(k)
W(k+1) = W(k) + ( W(k) - ||W(k)||22 Y(k)Z(k) )
The above equation is the one that I used to simulation in this study.

8. Result

9. The ideal transfer function = [-0.3 –0.9 0.8 –0.7 0.6]
Result (cont.)
The ideal transfer function = [-0.3 – – ]
SNR Weight from TLMS Weight from LMS
10dB [ , , , , ] [ , , , , ]
5 dB [ , , , , ] [ , , , , ]
2 dB [ , , , , ] [ , , , , ]
0 dB [ , , , , ] [ , , , , ]
-2dB [ , , , , ] [ , , , , ]
-5dB [ , , , , ] [ , , , , ]

10. Discussion
From the result, the TLMS algorithm performs better than the LMS algorithm.
In the error plot, after using 30,000 samples, the TLMS algorithm’s error plot are minimized whereas the error plot of LMS algorithm can not be minimized even though the large numbers of samples are employed.
The error resulted from the TLMS algorithm are averaged around –15dB which is considerably negligible, while the error resulted from the LMS algorithm are around 0-5dB.
In the plot, the errors of simulation still fluctuate due to the nature of Gaussian interference, and self adaptation of the algorithm.

11. Discussion (cont.)
In the table shown the computed transfer function between the TLMS and LMS algorithm, we can see that for the high SNR, both algorithms perform well, but the TLMS one will a bit better.
At the low SNR, the LMS algorithm can not find the solution whereas the TLMS algorithm can find the solution which is very close to the true transfer function of the system.
Moreover, the TLMS algorithm also has small variance in compute the error. (see in report.)

12. Conclusion
The TLMS algorithm is unsupervised learning adaptive linear combiner based on the total least mean squares of the minimum Raleigh quotient which used to extract minor features from the training sequences.
From the simulation result, we can see that the TLMS algorithm is out performance the LMS algorithm in case of there exist the interference in both input and output ends.
The advantages of using the TLMS algorithm are its ability to perform in a system, which is corrupted with noise, and the stability of its results.