Question

Determine the spectral density funetion corresponding to the covariance. function \(R(v)=\gamma^{|v|}, \quad v=0, \pm 1, \ldots\), where \(|\gamma|<1\) Answer: $$ f(\omega)=\frac{1-\gamma^{2}}{2 \pi\left|1-\gamma e^{i \|}\right|^{2}}, \quad-\pi<\omega<\pi $$

Short answer

\[f(\omega) = \frac{1-\gamma^{2}}{2\pi\left|1 - \gamma e^{i\omega}\right|^2}, \quad -\pi < \omega < \pi\]

Step-by-step solution

Definition of Spectral Density Function

The spectral density function, represented by \(f(\omega)\), is the Fourier transform of the covariance function. To calculate the spectral density function, we need to compute the Fourier transform of \(R(v)\). Mathematically, this is given by: \[f(\omega) = \sum_{v=-\infty}^{\infty} R(v) e^{-iv\omega}\]

Substituting the given covariance function

We have the covariance function as \(R(v) = \gamma^{|v|}\) for \(v=0, \pm1, \ldots\) and \(|\gamma| < 1\). Now, we substitute this into the expression for the spectral density function: \[f(\omega) = \sum_{v=-\infty}^{\infty} \gamma^{|v|} e^{-iv\omega}\]

Splitting the sum into two parts

We can split the above summation into two parts: one for negative values of \(v\) and one for positive values of \(v\): \[f(\omega) = \left[\sum_{v=-\infty}^{-1} \gamma^{-v} e^{iv\omega} \right] + \left[\sum_{v=0}^{\infty} \gamma^{v} e^{-iv\omega} \right]\]

Compute the sum for each part

The left sum is a geometric series with common ratio \(\gamma e^{-i\omega}\). The right sum is also a geometric series with common ratio \( \gamma e^{i\omega}\). We will use the formula for the sum of a geometric series: \[\sum_{k=0}^{n} ar^k = \frac{a(1 - r^{n+1})}{1 - r},\quad \text{for}\; |r| < 1\] For the left sum, we have \(k = -v\) and a common ratio of \(| \gamma e^{-i\omega} | < 1\). Hence, \[\sum_{v=-\infty}^{-1} \gamma^{-v} e^{iv\omega} = \frac{\gamma e^{i\omega}}{1 - \gamma e^{i\omega}}\] For the right sum, we have a common ratio of \(| \gamma e^{-i\omega}| < 1\). Thus, we find: \[\sum_{v=0}^{\infty} \gamma^{v} e^{-iv\omega} = \frac{1}{1 - \gamma e^{-i\omega}}\]

Combine the two sums

We can now combine the two sums to get the spectral density function: \[f(\omega) = \frac{\gamma e^{i\omega}}{1 - \gamma e^{i\omega}} + \frac{1}{1 - \gamma e^{-i\omega}}\]

Simplify the expression

Now, we simplify the expression for the spectral density function: \[\begin{aligned} f(\omega) &= \frac{(1 - \gamma e^{-i\omega})(\gamma e^{i\omega}) + (1 - \gamma e^{i\omega})}{(1 - \gamma e^{-i\omega})(1 - \gamma e^{i\omega})} \\ &= \frac{1 - \gamma^2}{(1 - \gamma e^{-i\omega})(1 - \gamma e^{i\omega})} \end{aligned}\]

Final expression for the spectral density function

Finally, we write the expression for \(f(\omega)\) in terms of the magnitude of the denominator: \[f(\omega) = \frac{1-\gamma^{2}}{2\pi\left|1 - \gamma e^{i\omega}\right|^2}, \quad -\pi < \omega < \pi\]

Key concepts

Covariance Function

Understanding the covariance function is fundamental when dealing with time series or stochastic processes. This mathematical concept quantifies the degree to which two random variables change together. In simpler terms, if we have two data points measured at different times, the covariance function helps us understand how one's movement might affect the other's.

For instance, in the context of the spectral density function exercise, the covariance function given by \(R(v) = \texttt{gamma}^{|v|}\), where \(v\) represents the time lag, is a measure of memory in the process. The smaller the \(\gamma\), the weaker the correlation between different time points. This decay of correlation with time is typical in many natural and economic phenomena, which makes it very relevant for modeling such processes.

Fourier Transform

The Fourier transform is a powerful mathematical tool that transforms a function of time into a function of frequency. In other words, it allows us to analyze what frequencies are present in a signal and how strong those frequencies are.

In our exercise, we use the Fourier transform to convert the covariance function into a spectral density function, thus moving from time domain to frequency domain analysis. This conversion is critical in signal processing and communications, as it highlights the periodic characteristics of a stochastic process and aids in the identification and filtering of the frequency components.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This concept is particularly useful when the process has a recursive or iterative nature, creating patterns that expand or shrink systematically.

In the context of the given exercise, the geometric series arises when we split the Fourier transform summation based on the sign of the index \(v\), resulting in two geometric series that need to be summed. Recognizing this leads to a simplified calculation using the geometric series formula, illustrating the interconnectedness between algebraic structures and their applications in analyses such as spectral density functions.

Stochastic Processes

Stochastic processes describe systems that evolve over time with an inherent randomness – think weather patterns, stock markets, or even the noise in electronic circuits. They are characterized by a set of random variables indexed by time.

In our exercise, we're dealing with a stochastic process through its covariance function. We tackle the randomness not by looking into one particular outcome, but by analyzing the average behavior over time. The spectral density function we compute provides a deeper understanding of the underlying process, revealing how its frequency components contribute to the overall variability. This insight is critical for diverse applications such as forecasting, engineering, economics, and environmental science.