# What are Autoregressive Coefficients?

Can anyone explain what are Autoregressive Coefficients? What is their meaning that is.

Consider a method:

public double[] calculateARCoefficients(double[] inputseries, int order)

When this method is called on 256 values, lets say these values represent some kind of signal, I will get an array of n numbers, where n=order. What do these numbers represent?

I mean for example variance tells me how spread out the values of the signal are. The higher variance value is, the more spread out values are. The smaller variance is, the more unified signal is.

Are you able to say, in a similar fashion to above paragraph about variance, what value of AR coefficients says about the signal?

• Autoregressive coefficients represent coefficients of an IIR filter. An autoregressive model can be represented as an IIR filter. Similar tohow filter coefficients work, these coefficients define how thefilter actually works(parameters like pass band of the filter, stop band of the filter and are all dependenton these coefficients). – Naresh Jun 10 '13 at 6:53

## 2 Answers

The coefficients $a_i$ of an autoregressive (AR) model of a signal $x(n)$ enter the model definition in the following way:

$$x(n)=\sum_{i=1}^Na_ix(n-i)+e(n)$$

where $e(n)$ is zero-mean white noise and $N$ is the model order. I.e. the signal is modeled as the output of a linear time-invariant filter with transfer function

$$H(z)=\frac{1}{1-\sum_{i=1}^Na_iz^{-1}}$$

and with input signal $e(n)$. There is no simple interpretation of specific values of the coefficients $a_i$. You have to look at the whole polynomial $1-\sum_{i=1}^Na_iz^{-1}$, e.g. by computing the power spectrum of the AR model of $x(n)$:

$$S_x(z)=\sigma_e^2\left|\frac{1}{1-\sum_{i=1}^Na_iz^{-1}}\right|^2,\quad z=e^{j\theta}$$

where $\sigma_e^2$ is the variance of $e(n)$. So the AR coefficients are the parameters of a specific signal model, and they do not have a simple interpretation comparable to mean or variance of a random process. However, in general they represent a much more detailed description of the signal than mean and variance alone.

BlockquoteCan anyone explain what are Autoregressive Coefficients? What is their meaning that is.

AR coefficients can be thought of as describing the envelope of the spectrum. In your function declaration you need to pass an order argument. The rule of thumb is that the order must be set to two times the expected number of peaks in the spectrum. So if you expect 5 peaks in the spectrum (as in speech) the order should be 10.

In your example where the block size is 256 and you for instance set order to 10 the AR coefficients provides a smooth and compact representation of the signal spectrum. It 'throws away' the details in the spectrum.

@Matt L $z^{-1} \rightarrow z^i$