Steady-state performance of the FT algorithm 120 3 The parameter A should be as large as possible to*' obtain a

fast convergence rate of the tap-length, but also much smaller than the estimate of so that the steady-state tap-length formulated in (6.2.16) will not be significantly biased from Lopt. For example, A « O.lLopt will be a good choice for a wide range of optimal tap- lengths.

4. The leakage parameter a should not be too large, so that it will not influence the initial tap-length convergence rate too much. The parameter a should not be too small either, so that once the tap- length is over estimated, a can make the tap-length converge close to the steady-state value as soon as possible. For example, a = 0.0001 is not a good choice, since it means that after 10,000 iterations, the leakage parameter a will reduce the tap-length by one tap, which is usually too slow. Generally, values between 0.001 and 0.01 are good choices for a.

5. The parameter 7 is the step size parameter which controls the adaptation process of the variable tap-length. Similar to the step size in the LMS algorithm, a large parameter 7 will speed up the convergence rate of the tap-length, but will result in a large fluctuation of the steady- state tap-length. On the other hand, a small parameter 7 can obtain a small fluctuation of the steady-state tap-length, but lead to a slow convergence rate. Thus 7 provides a trade-off between the convergence rate of the tap-length and the steady-state tap-length variance. The choice of this parameter is important in the FT algorithm. A detailed discussion for the choice of this parameter is as follows:

At first, to avoid under-modelling the optimal tap-length, the steady- state tap-length of the FT method, L(00), should not be less than L ^ .

Section 6.2. Steady-state performance of the FT algorithm 121

Considering the fluctuation of the steady-state tap-length, the param­ eter 7 should be set properly so that L(oo) > L^t + where k is a small positive integer and can be chosen according to the system requirement of the fluctuation of the steady-state tap-length. For ex­ ample, k — 2 is a reasonable choice. Substituting (6.2.16) into the inequality L(oo) > Lopt + K<$ the lower bound value 7* is obtained:

7i = --- a T -5 — ( 6 .2 .1 7 )

(A

Secondly, the parameter 7 should not be too large to avoid a large fluctuation of the steady-state tap-length. The update process for the steady-state fractional tap-length is formulated in (6.2.7). The fluctu­ ation of the steady-state fractional tap-length is brought about by the fluctuation of the term a -I- j ( A — B + C — D + E — F + G — H). It is straightforward to see in (6.2.7) that large variance of the term

a + j ( A — B + C — D + E — F + G — H ), which is denoted as r f, will

result in large fluctuations of the steady-state fractional tap-length. To avoid such a situation, a simple and intuitive approach is to make the standard deviation af much smaller than the parameter 5 in equation

(6.1.5), so that the probability of the steady-state fractional tap-length fluctuating outside the range (L(oo) — 5, L(00) -I- 5) can be very small. A simple criterion to satisfy such a requirement is

<Tf < p S (6.2.18)

where p is a small positive value and can be decided according to the system requirement of the fluctuation of the steady-state tap-length. The derivation of the variance a j is given in (6.7.18) in Appendix B.

Section 6.2. Steady-state performance of the FT algorithm 122

Using (6.7.18) in (6.2.18) and after rearrangement the uppier bound value 7U is obtained:

where K 2 and K$ are respectively formulated in Appendix B (6.7.20) and (6.7.21).

The parameter p should be chosen so that the possibility of the tap-length fluctuating under is nearly zero. In general, , for high noise condition, this parameter should be chosen small, and for low noise condition it can be chosen larger. Examples for the choice of

p can be seen in the simulations in the next subsection. With the

lower bound value given in (6.2.17) and the upper bound value given in (6.2.19), the parameter 7 can then he easily chosen. According to the motivation of the upper bound value 7„, the values close to this value are good choices to avoid a large fluctuation of the steady-state tap-length while retaining as quick as possible convergence rate; thus, in practice 7 should be chosen close to and larger than 7j.

Since in practice, all the parameters o\, o 2, a2, especially the pa­ rameter Lopt are unknown, approximate estimations of these parame­ ters can be used in the calculations. Next several simulations will be performed to confirm the above analysis and discussions.

6.2.3 Simulation

In this subsection two simulations are performed to support the analysis and discussions in the previous subsection. In the first simulation a low noise condition is used while a high noise environment is utilized in the

S ectio n 6 .2 . S t e a d y -sta te perform ance o f th e FT algorith m 123 400 350 300 250 2 2 0 200 150 100 50 0.2 04 0 6 0.8 1 2

S am ple num ber

1 .2

num ber 14

1.8 1.8

F ig u r e 6 .1 . T he evolution curves of th e tap-length w ith different step sizes under a low noise condition, SN R =20dB.

s

I

S am ple num ber

F ig u r e 6 .2. The evolution curves of th e EM SE w ith different step sizes under a low noise condition, SN R =20dB .

second simulation.

Low noise case: SNR = 20dB

The setup of this sim ulation is as follows. The im pulse response se­ quence of the unknown filter is a w hite G aussian sequence w ith zero mean and variance 0.01. T he tap-length L ^ t is set to 200. T he inp ut signal is another white G aussian sequence w ith zero m ean and u n it vari­

Section 6.2. Steady-state performance of the FT algorithm 124