Learning the Coefficients of Auto Regressive (AR) Model Using Least Mean Squares (LMS) Filter for Signal Prediction

Question

I want to do two things.

1. Estimating Coefficients of AR model using LMS
2. Using Coefficients found in step 1 and predict future samples of a signal using AR equation. I don't have a desired signal so I can't use future samples as well.

The equation of Autoregressive model is $$x(n) = \sum_{i=1}^p a_i x(n-i) + e(n)$$

where $p$ indicates the AR model order ($p=30$ in my case) and $\mathbf{a}$ shows the AR coefficients $\{a_1, a_2, a_3, a_4,\ldots, a_{30}\} $.

And LEAST MEANS SQUARE (LMS) algorithm is as follows:

Order=30;      % AR model order 
M=30;           % No of filter coefficients same as AR model order
mu=0.001;      % Learning rate/ step size
x=signal(1,1:700);  % length of signal is 700 samples, x= input signal
N=length(x); 
Predicted_signal=zeros(1,N);
w=zeros(1,M);               % weights / filter coefficients
for n=M:N
    pp=n-M+1;
    x1=x(n:-1:pp); 
    Predicted_signal(n)=(w*x1'); 
    e(n)=x(n)-Predicted_signal(n);     %e(n)=d(n)-y(n); reference signal - actual signal
    w=w+2*mu*e(n)*x1;
    w1(n-M+1,:)=w(1,:);      % filter coefficients
end 

Coefficients (weights/ w) found in previous code to be used in AR equation.

ts = prediction starting point that is 436th sample

Predicted_signal_ar= orignal_signal(1,1:436);
for i=1:Order
    t=ts-i;
    prediction1(1,i)=(w(1,i)*Predicted_signal_ar(t)); %% putting lms coefficients in AR equation
end

s1(1,a)=sum(prediction1(1,:));
Predicted_signal_ar(1,ts)=s1(1,a); 

There is some error in my code and I am not sure what is it. The above code should give me close to actual Coefficients which then I pass to step 2 for AR prediction. In results of 2nd step there is a sudden jump in prediction of first few points and prediction performance is also poor.

Answer 1

In order to use the LMS to learn an AR Model one should use the predictor variant of the Least Mean Squares (LMS) filter.

Basically we predict the $ x \left[ n \right] $ sample using past samples: $ \left\{ x \left[ n - i \right] \right\}_{i = 1}^{k} $ where $ k $ is the LMS filter order.

This will basically give us the AR Model of any signal we'll drive into the LMS filter.

I don't have you samples so I created a similar signal using a linear combination of 30 sine signals with different amplitudes, frequencies and phases:

I used 75% of the signal for the LMS training (Blue) and the rest is unseen by the LMS.

The prediction is:

Though the input signal isn't generated by an AR Model, using a model of order 101 the results seems to be pretty good.
results might get even better when the model order is tweaked better.

The code is available at my StackExchange Codes Signal Processing Q59325 GitHub Repository (Look at the SignalProcessing\Q59325 folder).