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Consider a Gaussian random vector x with mean and covariance and P_x, respectively. As mentioned previously, for Gaussian densities, these two parameters are sufficient to completely describe the full pdf of the random vector.

Next consider a linear transformation from x to y which is governed by the transformation matrix H. The resulting equation for y is given by

(36.27)

In this case, the transformed random vector, y, can be shown to be a Gaussian random vector with mean and covariance

(36.28)

(36.29)

This preservation of Gaussian nature when transformed via linear operations is an important property of Gaussian densities that makes the linear Kalman filter relatively simple to implement.

Now consider a generalized nonlinear transformation

(36.30)

In this case, the density of y can become difficult to calculate exactly. While we will address this issue in more detail later in the chapter, generally speaking, the resulting density function is clearly non‐Gaussian, thus limiting the performance of the linear Kalman filter algorithm. Nonlinear estimators attempt to maintain a higher‐fidelity estimate of the overall density function as it transforms over time.

In the next section, we present our first class of estimators designed to support systems with non‐Gaussian pdfs.