Solved – Covariance and power spectral density of a signal

autocorrelationcorrelationcovariancefourier transformspectral analysis

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.

  1. 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 $k$?
  2. 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?
  3. Why do we take the complex conjugate of $y^\ast(t-k)$? Why isn't the covariance $\text{cov}(y) = E[y(t)y(t-k)]$?

Best Answer

  1. 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 $Y$.

  2. 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.

  3. 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.

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