- #1
Master1022
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- TL;DR
- 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 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:
x_t = - \sum_{k = 1}^{p} a_{k} x_{t - k} + e_t
Then, by taking the z-transform, this leads to:
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)
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:
P_{xx}(\omega) = |G(\omega)|^2 P_{ee}(\omega)
if the error is assumed to be gaussian with a variance ## \sigma_e ^ 2 ##, then we get:
P_{xx}(\omega) = \frac{\sigma_e ^ 2}{|1 + \sum_{i = k}^{p} a_{k} e^{-j\omega k T_s}|^2}
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:
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:
x_t = - \sum_{k = 1}^{p} a_{k} x_{t - k} + e_t
Then, by taking the z-transform, this leads to:
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)
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:
P_{xx}(\omega) = |G(\omega)|^2 P_{ee}(\omega)
if the error is assumed to be gaussian with a variance ## \sigma_e ^ 2 ##, then we get:
P_{xx}(\omega) = \frac{\sigma_e ^ 2}{|1 + \sum_{i = k}^{p} a_{k} e^{-j\omega k T_s}|^2}
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:
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.
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