- #1

Master1022

- 611

- 117

- TL;DR Summary
- Why can an autoregressive AR(p) model provide a spectrum estimate with ##p/2## peaks?

Hi,

I was recently reading about spectral estimation with parametric methods, and specifically auto-regressive models. I came across the statement: "

An AR(p) model predicts the next value in a time series using a linear combination of the ## p ## previous values. This can be written in the following form:

[tex] x_t = - \sum_{k = 1}^{p} a_{k} x_{t - k} + e_t [/tex]

Then, by taking the z-transform, this leads to:

[tex] X(z) \left( 1 + \sum_{i = k}^{p} a_{k} z^{-k} \right) = E(z) \rightarrow G(z) = \frac{X(z)}{E(z)} = 1/\left( 1 + \sum_{i = k}^{p} a_{k} z^{-k} \right) [/tex]

Taking ##z = e^{j\omega T_s} ## where ## \omega ## is the angular frequency and ## T_s ## is the sampling period, we can obtain the frequency response of the discrete system associated with the AR model. Then if we want to find the PSD spectrum of the signal:

[tex] P_{xx}(\omega) = |G(\omega)|^2 P_{ee}(\omega) [/tex]

if the error is assumed to be gaussian with a variance ## \sigma_e ^ 2 ##, then we get:

[tex] P_{xx}(\omega) = \frac{\sigma_e ^ 2}{|1 + \sum_{i = k}^{p} a_{k} e^{-j\omega k T_s}|^2} [/tex]

at which point it just states, without explanation, that it can produce a spectrum with ## p/2 ## peaks. My question is:

From the denominator, it looks as if we have a +1 shifted 'circular-type' shape (with varying coefficients) from the origin. My only guess is that the ## p ## points will be will be symmetric about the real axis, and thus this will equate to ## p/2 ## peaks. Any help or guidance around this would be greatly appreciated.

I was recently reading about spectral estimation with parametric methods, and specifically auto-regressive models. I came across the statement: "

*An AR(p) model can provide spectral estimates with p/2 peaks*" and I was wondering why this was the case. I do apologize if this is the wrong forum - should I be posting signal processing questions in the 'Electrical Engineering' forum, although I think it falls under information engineering?**Here is the context for the statement:**An AR(p) model predicts the next value in a time series using a linear combination of the ## p ## previous values. This can be written in the following form:

[tex] x_t = - \sum_{k = 1}^{p} a_{k} x_{t - k} + e_t [/tex]

Then, by taking the z-transform, this leads to:

[tex] X(z) \left( 1 + \sum_{i = k}^{p} a_{k} z^{-k} \right) = E(z) \rightarrow G(z) = \frac{X(z)}{E(z)} = 1/\left( 1 + \sum_{i = k}^{p} a_{k} z^{-k} \right) [/tex]

Taking ##z = e^{j\omega T_s} ## where ## \omega ## is the angular frequency and ## T_s ## is the sampling period, we can obtain the frequency response of the discrete system associated with the AR model. Then if we want to find the PSD spectrum of the signal:

[tex] P_{xx}(\omega) = |G(\omega)|^2 P_{ee}(\omega) [/tex]

if the error is assumed to be gaussian with a variance ## \sigma_e ^ 2 ##, then we get:

[tex] P_{xx}(\omega) = \frac{\sigma_e ^ 2}{|1 + \sum_{i = k}^{p} a_{k} e^{-j\omega k T_s}|^2} [/tex]

at which point it just states, without explanation, that it can produce a spectrum with ## p/2 ## peaks. My question is:

**why this is the case?**

Attempt to understand:Attempt to understand:

From the denominator, it looks as if we have a +1 shifted 'circular-type' shape (with varying coefficients) from the origin. My only guess is that the ## p ## points will be will be symmetric about the real axis, and thus this will equate to ## p/2 ## peaks. Any help or guidance around this would be greatly appreciated.

Last edited: