A new VSSLMS algorithm with robustness to statistically nonstationary noise 1 0

tionary noise signal can be obtained, a more accurate criterion for the

practice, since the noise variance is not infinite, the parameter P can be a little larger than the value obtained from (5.4.7).

In the next subsection, all the above analysis and discussion will be supported by simulations in the context of a statistically nonstationary noise signal.

5.4.3 Simulation

In this subsection the performance of the sum method and the proposed algorithm is compared. The input signal x{n) is a pseudo-random, zero- mean unit-variance Gaussian signal with a length of 100,000 samples. The noise signal t(n) is the first 100,000 samples of a speech signal which is available from

http:/ /www.voiptroubleshooter.com/openjspeech/american.html, and the file name is “OSR-us_000_0016_8k.wav”. This noise signal is scaled to make the average SNR OdB over the entire observation. The noise signal and one representation of the input signal can be seen in Fig.

choice of P similar to (5.4.7) can be obtained according to (5.4.5). In

5.3.

The primary signal d(n) is obtained as follows:

d(n) = x(n) * h(n) + t(n) (5.4.8) where h(n) is the causal optimal filter obtained by

h(n) = i e 0 05(n 1)r(n), n = l,...,100 (5.4.9)

0 otherwise

S ectio n 5.4. A new V SSL M S algorithm w ith rob u stn ess t o sta tistica lly n on stationary n oise 105

(a) The noise signal

1 0 i---1---1---1--- r

,i--- 1--- 1--- 1---1---

0 2 4 6 8 10

Sample number (b) One representation of the input signal

Sample number x

Figure 5.3. T he noise signal (a) and one representation of th e input signal (b).

where r(n ) is draw n from a zero mean unit variance Gaussian sequence. One representation of th e nonzero term s of h(n) can be seen in Fig. 5.4(a).

In this sim ulation the proposed algorithm will be compared w ith the sum m ethod w ith different step sizes 0.1 and 0.02. The initial step sizes and adaptive filter vectors of th e proposed algorithm are set to zero. The param eter (3 for the proposed algorithm is set to 0.999 to perform a sufficient sm oothing operation. T he param eter P in the proposed algorithm is em pirically set to 80, and produces a steady-state EM SE as predicted by (5.4.6). T he param eter sets for the proposed algorithm are chosen to make its initial convergence rate approxim ately equal to th a t of th e sum m ethod w ith a step size 0.1. The estim ates <r2(n) and d2(n) used in the sum algorithm and the proposed algorithm are obtained by sm oothing the input and error signals as

S ectio n 5.4. A new V SSL M S algorithm w ith rob u stn ess t o sta tistica lly n o n station ary n oise 106

(a) One representation of the optimal filter h(n)

0.2 1 -0.4 100 0 40) so e Sample number 60 70 80 90 10 20 30

(b) The evolution curves of EMSE

20 10 0 - 1 0 w -2 0 -3 0 -4 0 0 1 2 3 4 5 6 7 8 9 10 Sample number x , 0«

Figure 5.4. One representation of the optim al filter (a) and the M onte Carlo averaged evolution curves of the EMSE for the sum m ethod and the proposed NSVSSLMS algorithm s (b).

and

&l(n) = 0.9 9 .7 l ( n - 1) + (1 - 0.99)x (5.4.11)

T he initial values of o f (n) and d f(n ) are set to zero and unity re- spectively. The evolutions of th e EMSE curves for all the experim ents are shown in Fig. 5.4(b). T he results are obtained over 200 M onte Carlo trials of the sam e experim ent.

It is clear to see in Fig. 5.4(b) th a t the proposed algorithm has an EM SE convergence rate sim ilar to th a t of the sum m ethod w ith a pa­ ram eter 0.1 a t th e early s ta te of th e process. The EMSE of bo th m eth­ ods converges to —20dB a t approxim ately 3,000 samples. However, the EM SE of th e sum m ethod w ith param eter 0.1 fluctuates greatly w ith the variation of the noise signal energy. The performance of the sum m ethod w ith param eter 0.02 has a small EMSE and slight fluctuation of th e EMSE, bu t the convergence rate is very slow. The proposed al­

Greenburg's. 0.1 Green burg's 0 02 ■ Proposed

Section 5.4. A new VSSLMS algorithm with robustness to statistically nonstationary noise 107

gorithm has a fast convergence rate which is similar to the sum method with parameter 0.1, and a small EMSE which is close to that of the sum method with parameter 0.02. Therefore, the proposed NSVSSLMS algorithm performs better than the sum method in this simulation.

The theoretical upper bound of the EMSE of the proposed algorithm according to (5.4.6) is also shown in Fig. 5.4(b). It can be seen that over the interval 15,000 to 20,000, where the variance of the noise signal is high, the EMSE of the simulation results is very close to this theoretical upper bound. Thus (5.4.6) can give a good upper bound of the steady-state EMSE for the proposed algorithm, and the conclusion can be made that with a given upper bound of the steady-state EMSE, the parameter P can be properly chosen according to (5.4.7).

Note that all the analysis and simulations are based on a white input signal. When the input signal is correlated, the analysis results obtained from (5.4.3) and (5.4.5) are both incorrect, and smaller than the practical results. In this case, the parameter P should be chosen smaller than the value obtained from (5.4.6). Finally, if both input and noise signals are statistically nonstationary signals, the smoothed gradient vector can not measure the proximity of the adaptive process, and the proposed algorithm has no advantage as compared with the sum method.

Although the proposed algorithm is only compared with the sum method in this section, many experiments which are not included in this section have been performed, and it has been shown that all the other existing VSSLMS algorithms perform poorly with such a nonstationary noise signal, since their EMSE is proportional to the noise variance.

Section 5.5. Conclusion 108

5.5 Conclusion '

In this chapter, an overview of typical existing VSSLMS algorithms and an introduction for a theoretically optimal VSSLMS algorithm has been given. Two new VSSLMS algorithms have also been proposed, which are designed for high level noise conditions. Simulations show that these algorithms can obtain both a fast convergence rate and a small EMSE with robustness to statistically stationary or nonstation­ ary noise signals, and perform better as compared with other existing VSSLMS algorithms. Both methods may be potentially used in many applications of adaptive filtering.

Chapter 6

i

VARIABLE TAP-LENGTH