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I would like to basically understand what an autoregressive model is used for (so I don't really attempt maths in the answers).

I just started a signal processing course and the model was introduced.

This is what I understood but I don't think I understood it well :

We have $n-1$ samples of a signal and we want to guess what the $n$'th will be.

With Yule-Walker, we have an approximation of the $n$'th sample that is : $\tilde{x}[n]=\sum a_k x[n-k]$

But $\tilde{x}[n]$ is just an approximation of $x[n]$.

So we have : $x[n]=\tilde{x}[n]+\epsilon$ where $\epsilon$ is the "error" I make when I use $\tilde{x}[n]$.

The autoregressive models assumes that $\epsilon$ is a white noise with average zero (and this makes sense because there is no reason in our model to have more often $\tilde{x}[n]>x[n]$ than $\tilde{x}[n]<x[n]$ ).

Did I understand the physical meaning of the model well ?

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Your understanding is right. Just a few points...

  • I think you meant Yule-Walker equations.
  • It is better to distinguish $N$ from $n$. In an AR($N$) model we consider last $N$ terms to approximate the next term.
  • An important interpretation that you missed was the fact that when we consider $$\epsilon=x[n]-\tilde{x}[n]=x[n]-\sum_ka_kx[n-k],$$ or in a more standard way (representing the process by variable $y$ instead of $x$ $$y[n]=\sum_ka_ky[n-k]+\epsilon[n],$$ it can be seen as an input-output relationship of an LTI filter, $\epsilon[n]$ being input and $y[n]$ the output. More specifically,

AR model can be seen as the output of an all-pole IIR filter which is excited by a white noise input.

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  • $\begingroup$ Thank you for your answer. So I assumed that you mainly agreed with what I said you just put further details. But I didn't understand why we can see it as an input-output relation between $y[n]$ and $\epsilon[n]$ only. Indeed we also need all the $y[n-k]$ to compute the ouput. So should't it be a filter with $P+1$ input ($1$ for $\epsilon$ and $P$ for all the $y[n-k]$ needed). $\endgroup$
    – StarBucK
    Commented Nov 7, 2016 at 12:09
  • $\begingroup$ Consider an IIR filter with input signal $x$ and output signal $y$. We just input one sample at each time, and get one sample of output ($x[n]$ and $y[n]$). But the only input is $x[n]$. Although technically the filter considers the outputs from some previous times in the calculation of the next sample, they are not inputs of the filter. $\endgroup$
    – msm
    Commented Nov 7, 2016 at 17:18
  • $\begingroup$ Does it make sense? $\endgroup$
    – msm
    Commented Nov 8, 2016 at 12:42
  • $\begingroup$ Hmm sorry but I don't get it :/. I am not familiar with IIR filter at all (I just saw it on wikipedia now). If at "time" n I need y[n], then I need into my computation $ \epsilon [n]$ AND $y[n-k]$ : I didn't understand why you said that the previous time output are not input of the filter $\endgroup$
    – StarBucK
    Commented Nov 8, 2016 at 19:47
  • $\begingroup$ You don't have another way to explain it ? :s $\endgroup$
    – StarBucK
    Commented Nov 11, 2016 at 11:37

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