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This example shows how to do adaptive nonlinear noise cancellation using the `anfis`

and `genfis`

commands.

Define a hypothetical information signal, `x`

, sampled at 100Hz over 6 seconds.

time = (0:0.01:6)'; x = sin(40./(time+0.01)); plot(time,x) title('Information Signal x','fontsize',10) xlabel('time','fontsize',10) ylabel('x','fontsize',10)

Assume that `x`

cannot be measured without an interference signal, , which is generated from another noise source, , via a certain unknown nonlinear process.

The plot below shows noise source .

n1 = randn(size(time)); plot(time,n1) title('Noise Source n_1','fontsize',10) xlabel('time','fontsize',10) ylabel('n_1','fontsize',10)

Assume that the interference signal, , that appears in the measured signal is generated via an unknown nonlinear equation:

Plot this nonlinear characteristic as a surface.

domain = linspace(min(n1),max(n1),20); [xx,yy] = meshgrid(domain,domain); zz = 4*sin(xx).*yy./(1+yy.^2); surf(xx,yy,zz); xlabel('n_1(k)','fontsize',10); ylabel('n_1(k-1)','fontsize',10); zlabel('n_2(k)','fontsize',10); title('Unknown Interference Channel Characteristics','fontsize',10);

Compute the interferenace signal, , from the noise source, , and plot both signals.

n1d0 = n1; % n1 with delay 0 n1d1 = [0; n1d0(1:length(n1d0)-1)]; % n1 with delay 1 n2 = 4*sin(n1d0).*n1d1./(1+n1d1.^2); % interference subplot(2,1,1) plot(time,n1); ylabel('noise n_1','fontsize',10); subplot(2,1,2) plot(time,n2); ylabel('interference n_2','fontsize',10);

is related to via the highly nonlinear process shown previously; from the plots, it is hard to see if these two signals are correlated in any way.

The measured signal, `m`

, is the sum of the original information signal, `x`

, and the interference, . However, we do not know . The only signals available to us are the noise signal, , and the measured signal `m`

.

m = x + n2; % measured signal subplot(1,1,1) plot(time, m) title('Measured Signal','fontsize',10) xlabel('time','fontsize',10) ylabel('m','fontsize',10)

You can recover the original information signal, `x`

, using adaptive noise cancellation via ANFIS training.

Use the `anfis`

command to identify the nonlinear relationship between and . While is not directly available, you can assume that `m`

is a "contaminated" version of for training. This assumption treats `x`

as "noise" in this kind of nonlinear fitting.

Assume the order of the nonlinear channel is known (in this case, `2`

), so you can use a 2-input ANFIS structure for training.

Define the training data. The first two columns of `data`

are the inputs to the ANFIS structure, and a delayed version of . The final column of `data`

is the measured signal, `m`

.

delayed_n1 = [0; n1(1:length(n1)-1)]; data = [delayed_n1 n1 m];

Generate the initial FIS structure. By default, the grid partitioning algorithm uses two membership functions for each input variable, which produces four fuzzy rules for learning.

```
genOpt = genfisOptions('GridPartition');
inFIS = genfis(data(:,1:end-1),data(:,end),genOpt);
```

Tune the FIS using the `anfis`

command with an initial training step size of `0.2`

.

trainOpt = anfisOptions('InitialFIS',inFIS,'InitialStepSize',0.2); outFIS = anfis(data,trainOpt);

ANFIS info: Number of nodes: 21 Number of linear parameters: 12 Number of nonlinear parameters: 12 Total number of parameters: 24 Number of training data pairs: 601 Number of checking data pairs: 0 Number of fuzzy rules: 4 Start training ANFIS ... 1 0.761817 2 0.748426 3 0.739315 4 0.733993 5 0.729492 Step size increases to 0.220000 after epoch 5. 6 0.725382 7 0.721269 8 0.717621 9 0.714474 Step size increases to 0.242000 after epoch 9. 10 0.71207 Designated epoch number reached --> ANFIS training completed at epoch 10. Minimal training RMSE = 0.712070

The tuned FIS, `outFIS`

, models the second-order relationship between and .

Calculate the estimated interference signal, `estimated_n2`

, by evaulating the tuned FIS using the original training data.

estimated_n2 = evalfis(data(:,1:2),outFIS);

Plot the and actual signal and the estimated version from the ANFIS output.

subplot(2,1,1) plot(time, n2) ylabel('n_2 (unknown)'); subplot(2,1,2) plot(time, estimated_n2) ylabel('Estimated n_2');

The estimated information signal is equal to the difference between the measured signal, `m`

, and the estimated interference (ANFIS output).

estimated_x = m - estimated_n2;

Compare the original information signal, `x`

, and the estimate, `estimated_x`

.

figure plot(time,estimated_x,'b',time,x,'r') legend('Estimated x','Actual x (unknown)','Location','SouthEast')

Without extensive training, the ANFIS produces a good estimate of the information signal.

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