Question about Spectral Estimation using AR Models

In summary: Thus, the AR(p) prediction error spectrum will have p/2 peaks, which is why it can provide spectral estimates with p/2 peaks.In summary, the conversation discusses the statement that an AR(p) model can provide spectral estimates with p/2 peaks. The context for this statement is explained, including the use of z-transform and the calculation of the frequency response of the discrete system associated with the AR model. It is then discussed that the AR(p) prediction error spectrum will have p/2 peaks, which is why it can provide spectral estimates with p/2 peaks. The conversation ends with a request for further clarification and agreement on the explanation.
  • #1
Master1022
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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 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:

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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  • #2
I'm not as well-versed in z-tranforms as Fourier transforms, but I'm inclined to agree with your guess.

The FFT produces a two-sided power spectrum, symmetric about a DC value - with half of the energy in either side of this symmetry.
 
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FAQ: Question about Spectral Estimation using AR Models

1. What is spectral estimation?

Spectral estimation is a process used in signal processing and statistics to estimate the spectrum of a signal. It involves determining the frequency content of a signal and can be used to analyze and interpret data in various fields, such as telecommunications, physics, and economics.

2. What is an AR model?

An AR (autoregressive) model is a type of statistical model used to describe the behavior of a time series data. It assumes that the current value of a variable is a linear combination of its past values and a random error term. AR models are commonly used in spectral estimation to analyze signals with a known or suspected underlying structure.

3. How is spectral estimation using AR models different from other methods?

Spectral estimation using AR models differs from other methods in that it takes into account the dependence of a signal on its past values, rather than just looking at the frequency content of the signal. This makes it particularly useful for analyzing signals with a known or suspected underlying structure.

4. What are the advantages of using AR models for spectral estimation?

One advantage of using AR models for spectral estimation is that they can provide more accurate estimates of the frequency content of a signal compared to other methods. They also allow for the identification and characterization of underlying structures in the signal, which can be useful in various applications.

5. What are some common applications of spectral estimation using AR models?

Spectral estimation using AR models has a wide range of applications in various fields. It is commonly used in telecommunications for signal processing and channel estimation, in physics for analyzing the frequency content of physical systems, and in economics for modeling and forecasting financial time series data.

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