Question

Compute the spectral density function of the autoregressive process \(\left\\{X_{n}\right\\}\) satisfying $$ X_{n}=\beta_{1} X_{n-1}+\cdots+\beta_{4} X_{n-4}+\xi_{n} $$ where \(\left\\{\xi_{n}\right\\}\) are uncorrelated zero-mean random variables having unit variance. Assume the \(q\) roots of \(x^{4}-\beta_{1} x^{4-1} \cdots-\beta_{q}=0\) are all less than one in absolute value. Answer: $$ f(\omega)=\left\\{2 \pi \sigma_{x}^{2}\left|1-\sum_{k=1}^{4} \beta_{k} e^{i k \omega}\right|^{2}\right\\}^{-1},-\pi<\omega<\pi $$

Short answer

\[f(\omega) = \left\{2\pi\left|1 - \sum_{k=1}^{4} \beta_{k} e^{i k \omega}\right|^{2}\right\}^{-1} , \ -\pi < \omega < \pi\]

Step-by-step solution

The given autoregressive process, \(X_n = \beta_1 X_{n-1} + \cdots + \beta_4 X_{n-4} + \xi_n\), is of order 4 since the current value of the process depends on the past four values. It is denoted as AR(4). #Step 2: Write the process equation in the z-domain#

Now, we need to find the transfer function of the process in the frequency domain. We can first write the process equation in the z-domain: \[1 - \beta_1z^{-1} - \beta_2z^{-2} - \beta_3z^{-3} - \beta_4z^{-4} = \frac{\xi(z)}{X(z)}\] #Step 3: Solve for X(z)#

By isolating X(z) in this equation, we get: \[X(z) = \frac{\xi(z)}{1 - \beta_1z^{-1} - \beta_2z^{-2} - \beta_3z^{-3} - \beta_4z^{-4}}\] #Step 4: Determine the spectral density function formula in z-domain#

The spectral density function of an autoregressive process in the z-domain can be given as: \[S_X(z) = \sigma_\xi^2 \cdot \frac{1}{\left|(1 - \beta_1z^{-1} - \beta_2z^{-2} - \beta_3z^{-3} - \beta_4z^{-4})\right|^2}\] Here, \(\sigma_\xi^2\) is the variance of the uncorrelated, zero-mean random variables, which is equal to 1. #Step 5: Convert the spectral density function to the frequency domain#

Next, we need to convert the spectral density function to the frequency domain by replacing z with \(e^{i\omega}\): \[S_X(\omega) = \frac{1}{\left|(1 - \beta_1e^{-i\omega} - \beta_2e^{-2i\omega} - \beta_3e^{-3i\omega} - \beta_4e^{-4i\omega})\right|^2}\] #Step 6: Simplify and present the final answer#

To present the spectral density function in a more concise form, we can rewrite the sum in exponential form: \[f(\omega) = \left\{2\pi\left|1 - \sum_{k=1}^{4} \beta_{k} e^{i k \omega}\right|^{2}\right\}^{-1} , \ -\pi < \omega < \pi\]

Key concepts

Autoregressive Process

An autoregressive process (AR) is a type of stochastic process that is widely used in time series analysis. In an autoregressive process, the current value of the series can be predicted as a linear function of its previous values plus a random error term. An AR process is characterized by its order, which tells us how many past values it relies on for the current prediction. For instance, an AR(4) process, like in the exercise, uses the last four values.

The general form of an AR process of order p can be written as:
\[X_{n} = \beta_{1} X_{n-1} + \beta_{2} X_{n-2} + \ ... \ + \beta_{p} X_{n-p} + \/xi_{n}\]
where \(\beta_{i}\) are the parameters of the model and \(\xi_{n}\) represents the white noise or random shocks to the process, assumed to be uncorrelated with zero mean and constant variance. To guarantee the stationarity of an AR process, its roots must lie inside the unit circle when plotted on the complex plane. This requirement ensures that the effects of shocks to the system diminish over time, rather than growing unbounded.

Stochastic Processes

Stochastic processes are mathematical objects that represent systems randomly evolving over time. They are used to model phenomena where there is some inherent randomness, making them fundamental in diverse fields like finance, physics, biology, and engineering. A stochastic process is defined as a collection of random variables indexed by time.

A key feature of stochastic processes is their ability to capture both the random and deterministic components within a system's evolution. Stochastic processes can be classified into discrete-time or continuous-time processes, depending on whether they are indexed by a countable set, like the natural numbers for time series, or a continuous interval. The autoregressive process we've discussed is an example of a discrete-time stochastic process.

Frequency Domain Analysis

Frequency domain analysis is a fundamental concept in signal processing and system analysis. It involves analyzing mathematical functions or signals with respect to frequency, rather than time. This analysis is crucial because it allows us to examine the periodic structure and oscillatory behaviors of signals.

In the context of stochastic processes such as the AR(4) process, the frequency domain analysis provides insight into how different frequency components contribute to the process's behavior. The spectral density function, a key outcome of this analysis, represents the distribution of power into frequency components composing that signal. As shown in the exercise, converting the AR process into the frequency domain helps to understand its internal dynamics in terms of the cyclic patterns it may exhibit.

Z-domain Transformation

The z-domain transformation is a mathematical technique used in the study of discrete-time linear systems, particularly in digital signal processing. It is the application of the Z-transform, which converts a discrete-time signal into a complex frequency domain representation. This transformation takes a sequence of real or complex numbers and maps it into a function of a complex variable Z.

Applying the z-domain transformation to a discrete signal helps us analyze its frequency characteristics and stability. It's especially helpful in the design and analysis of filters and control systems. As in our textbook exercise, we use the z-domain to express the autoregressive process and then find its spectral density function, facilitating the examination of the process in the frequency domain and demonstrating the connection between time-domain representations of stochastic processes and their frequency domain counterparts.