Implementing the noisy AR(1) Process I'm trying to implement noisy AR(1) process and plot it. The observed noisy sequence x(n) = s(n) + w(n) where variance of w(n) = 0.2. 

s(n) is defined as an AR(1) process with s(n) = 0.5 s(n-1) + e(n), for n=1:100, s(0) = 0 and the variance of e(n) = 1. w(n) and e(n) are independent.

I first do the following:

N = 100;                       % number of process samples.
a = [1, 0.5];                  % denominator coefficients, p = 1.
b = 1;                         % numerator coefficient.
s = filter(b,a, randn(1,N));   % generate N sample of AR(1) 

nu = sqrt(0.2) * randn(1,N);
x = s + nu;

figure(1)
plot(s,'r');
hold on;
plot(x,'g');
legend('AR(1) Process','Noisy Process');


Can anybody help me whether I do the correct thing. Any help would be appreciated.
Yes, the method you have used generates an AR-(1) random process by inputting Gaussian white noise of unit variance into the LTI filter defined by the coefficeints $a$ and $b$. 

Then you are adding some uncorrelated noise to it, which means your random process is not anymore a pure AR-(1) but a noisy one.