2mAx2

where V(x) = Z(U(x,t) + G(| ^ |2)). Here, we have assumed that h = 1. For convenience, we represent ^(xj,tn + At) as ^jn+l, ^(xj,tn) as ^jn and 4>(xj — Ax,tn) as ^j-in. With these representations, (9.8) reads jn+1 = j + Atjn n - iAtVj jn

Rewriting this equation in a matrix form one gets

Fn+1 = Fn - iAtH'Fn , (9.10) where the Hamiltonian H' is defined as h2 d2

2m dx2

Subsequently,

Since it is required that the norm of F is F * F =1, U must be an orthonormal operator. Since U in (9.12) does not have such a property, in our simulation we impose the normalization after every step.

The nonlinear equation (9.1) involves four external parameters: h, m, Z and [3. The last parameter ¡3 is necessary to update the synaptic weight vector K(x, t). For simplicity, the parameter h is taken as unity and the other three parameters are tuned accordingly. Looking at the complexity of (9.1), we used a genetic algorithm (GA) based on the concept of the univariate marginal distribution algorithm (UMDA) [5, 29] to select near-optimal parameters. The details of the algorithm and its implementation are as follows:

The univariate marginal distribution algorithm estimates the distribution of gene frequencies using a mean-field approximation. Each string in the population is represented by a binary vector x. The algorithm generates new points according to the following distribution:

The UMDA algorithm is given as follows:

- Step 1: Set t = 1, Generate N(>> 0) binary strings randomly.

- Step 2: Select M < N strings according to a selection method.

- Step 3: Compute the marginal frequencies pf(xi,t) from the selected strings.

- Step 4: Generate N new points according to the distribution p(x,t) = n r=i pf(xi,t) ■

- Set t = t +1. If the termination criteria are not met, go to Step 2.

For infinite populations and proportionate selection, it has been shown [29] that the average fitness never decreases for the maximization problem (increases for the minimization problem).

In general, GA provided the parameter values where m < 1,0 < 1 and Z >> 1. The significance of this finding can be understood in the following manner. Since ¡3 was the learning parameter in the Hebbian learning, it is natural to expect that 3 < 1. The less than unity value for m makes self-excitation larger. Similarly, a large value of Z causes a larger input excitation since it appears as a multiplicand in the Schrodinger equation.

The proposed RQNN has been successfully applied to denoising of various signals like dc signals, sinusoids, shifted sinusoids, amplitude-modulated sine and square waves, speech signals, embedded in high Gaussian or non-Gaussian noise. Some selected results are presented in this section.

Amplitude-modulated and frequency-modulated signals are normally used in coding and transmission of data and appear corrupted at the receiver's end by channel noise [23]. For simulation purpose, we have selected the frequency of the carrier signal to be a sinusoid of frequency 5 Hz, although in reality they are very high frequency signals. The amplitude was modulated by superimposing a triangular variation of frequency 0.5Hz. Thus the expression for the composite signal ya(t) is ya(t) = a(t) • sin(2^5t) ; a(t) = { 1J(2 _ t) 0 11 < 2 , (9.14)

where a(t) is periodic with period 0.5Hz. A similar strategy is applied in generating the amplitude modulated square wave, i. e., the amplitude of a(t) is kept constant over every single period of the carrier sine wave in (9.14).

The expression for the actual signal ya(t) in this case is given below:

(t) = / a(t) 0 < t < 0.1 . (t) J 1.5t 0 < t < 1

ya(t) = \ -a(t) 0.1 < t < 0.2 a(t)=\ 1.5(2- t) 1 < t < 2 ,

where a(t) is periodic with period 0.5Hz.

The amplitude-modulated sinusoid signal ya(t) in (9.14) was immersed in Gaussian noise. The variance of the Gaussian noise was set according to the 20 dB and 6dB SNR measurement. The values selected for the parameters of the Schrodinger wave equation using UMDA are as follows:

The number of neurons along the x-axis are taken as N = 400. The parameters for the finite-difference equation used for integration are selected as

The parameter y is selected as 100. The tracking of the desired signal ya(t) is shown in Fig. 9.3. It can be observed that the tracking is very smooth and accurate. Snapshots of wave packets are shown in the same figure cor-reponding to marker points shown in the left plot. It can be observed that the pdf does not split, sliding along the x-axis back and forth like a particle. Next, the amplitude-modulated square wave ya(t) in (9.15) is immersed in Gaussian noise. The variance of the Gaussian noise was set according to the 20dB and 6dB noise power with the instantaneous period amplitude of ya(t)

Fig. 9.4. (left) Tracking of amplitude-modulated square wave embedded in 20 dB Gaussian noise: "a" represents the actual signal and "b" represents the tracking by the RQNN. (right) Snapshots of wave packets: wave packet "b" at t = 2.1 s, the wave packet "c" at t = 2.5 s, wave packet "d" at t = 2.9 s and the wave packet "e" at t = 3.1 s

Fig. 9.4. (left) Tracking of amplitude-modulated square wave embedded in 20 dB Gaussian noise: "a" represents the actual signal and "b" represents the tracking by the RQNN. (right) Snapshots of wave packets: wave packet "b" at t = 2.1 s, the wave packet "c" at t = 2.5 s, wave packet "d" at t = 2.9 s and the wave packet "e" at t = 3.1 s as reference. The values selected for the parameters of the Schrodinger wave equation using UMDA are as follows:

This shows that the parameter y influences the speed of response of the RQNN filter. The number of neurons along the x-axis is taken as N = 400. The parameters for the finite-difference equation used for integration are selected as

The tracking of the desired signal ya(t) and the movement of the wave packets are shown in Fig. 9.4. It can be observed that the tracking is accurate.

The pdf of the desired signal as estimated by the RQNN clearly exhibits a soliton property Fig. 9.4. The pdf does not split and slides along the x-axis like a particle.

Speech signals are degraded in many ways that limit their effectiveness for communication. One major source of noise in the speech signal is channel noise that is a major concern, especially in speech-recognition systems. Since the RQNN estimates the pdf of the incoming signal at every instant, if the incoming signal is corrupted by zero-mean noise, then the RQNN must be able to filter out that noise. Working on this hypothesis, we added zero-mean Gaussian noise with variance equal to the square of the instantaneous amplitude of the speech signal. The peak amplitude of the speech signal was normalized to 1.0. The original speech signals are recorded spoken digit utterances taken from the Release 1.0 of the Number Corpus. This corpus is distributed by the Center of Spoken Language Understanding of the Oregon Graduate Institute. The speech file names are mentioned in the captions, along with the respective plots for each of the speech signals.

For tracking the speech signals, the number of neurons along the x-axis is taken as N = 400. The parameters for the finite-difference equation used for integration are selected as

The values selected for the parameters of the Schrodinger wave equation using UMDA are as follows:

The parameter y was selected as 800. The tracking for a particular period of the speech signals selected from the database are shown in Figs. 9.5 and 9.6. The snapshots of the wave packet at two time instants are also shown. It is evident from Figs. 9.5 and 9.6 that the RQNN does track the pdf of the input signal at every instant. The wave packet does not split and it maintains an approximate Gaussian nature. It moves slightly along the x -axis like a particle maintaining its soliton property. By estimating the actual signal as the mean of the pdf at every instant, we can filter out the corrupting noise added to the actual signal. In addition, we reconverted the tracked speech signal and the noisy speech signal to the WAV format. On listening to these signals, we

Fig. 9.6. (left) Filtering of speech signal immersed in Gaussian noise: "a" represents the actual speech signal and "b" represents the tracking by the RQNN. The x-axis represents the sample number n of the speech signal. (right) Movement of the wave packet: wave packet "b" at t = 1.35 s and wave packet "d" at t = 1.45 s could verify that the RQNN filtering does improve the quality of the input noisy speech signal if corrupted by zero-mean Gaussian noise.

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