Covariance and power spectral density of a signal? I am reading an article on signal processing and I'm not sure of a few things. It defines the covariance function of a signal $y = y(t)$ as
$$\text{cov}(y) = E[y(t)y^\ast(t-k)].$$
And then it says the power spectral density is given by the Fourier transform of this covariance function.


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*Say we have the covariance function $E[XY]$ for two random variables $X$ and $Y$. A positive covariance in this case means that as $X$ increases $Y$ increases. What does it mean in the context of $E[y(t)y^\ast(t-k)]$ as in this case we only have one random variable $y$ which is taken at two different times? What does a positive covariance mean here and what is the significance of variable $k$?

*If we take the Fourier transform of a signal and look at the resulting plot, a strong peak on this plot at a point $k$ signifies that one of the components of the original signal is a sinusoidal wave corresponding to that $k$. If the power spectral density is the Fourier transform of the covariance what does a strong peak in the power spectral density plot signify?

*Why do we take the complex conjugate of $y^\ast(t-k)$? Why isn't the covariance $\text{cov}(y) = E[y(t)y(t-k)]$?

 A: *

*In the case in which $y(t)$ is real (to keep  it simple) it means that values above the mean of $y(t-k)$ tend to be followed by values above the mean of $Y(t)$; the interpretation is the same as for your $X$ and $Y$, just replace $y(t)$ and $y(t-k)$ in place of $X$ and $Y$.

*The Fourier transform of a signal will in general be complex valued; the square modulus of the Fourier transform would show a peak as you said. It turns our that the Fourier transform of the covariance sequence is the square modulus of the Fourier transform of the signal (up to a constant, depending on how you define things). This is not hard to see: the covariance sequence is the convolution of the (centered) series with itself, and the Fourier transform of the convolution of two sequences is the product of the respective Fourier transforms.

*Sometimes the series are complex valued and that is the way of defining the covariance in such cases.
I think looking at books such as Brillinger's Time Series: Data Analysis and Theory or Jenkins-Watts Spectral analysis and its Applications, just to name two off the top of my head will help you with issues such as these.
