I'm driving a white noise signal $e(n)$ through an AR random process:
$$ x(n)=e(n)+\sum\limits_{k=1}^N \alpha_k x(n-k) $$
I got five values of $\alpha_k$ and plotted its power spectrum in Matlab for $\omega$ from $0$ to $\pi$.
Then I generated the realizations of $x(n)$ with 10000 samples
Now I'm trying to get the estimate of the spectrum from the expectation data using the periodogram formula
$$ S_{per}(\omega)= \frac{1}{K} \left|\sum\limits_{k=0}^{K-1}x[k]e^{-j\omega k} \right|^2 $$
with $K=256$ the number of first values of $x(n)$. When I run my code what I get is a vector with the same value in every position, so I'm plotting a straight line. I think that just applying the formula I should get the estimate, so what am I doing wrong? :-S
This is the section of the code I'm using to find the estimates
var_e=0.01; % noise variance
b_1=1.4261; % coefficients
b_2=-.7634;
b_3=-.9002;
b_4=1.2548;
b_5=-.5707;
K=256;
wn=rand(1,10000)-0.5; % white noise
samp=1:1:1e4;
T=10000;
x=zeros(1,length(samp));
% here I generate the expectations
for n=6:T+5
x(n)=b_1*x(n-1)+b_2*x(n-2)+b_3*x(n-3)+b_4*x(n-4)+b_5*x(n-5)+wn(n-5);
end
x=x(6:end);
figure
plot(samp,x)
for i=1:length(w) % w is the frequency, has 1000
for k=1:K
per2(i)=sum(x(k)*exp(-j*k*i)); % sum of exponentials
pe2(i)=per2(i)*conj(per2(i)); % squared
end
end
Thanks in advance for your help