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The following notation is used in this chapter:

 State vector: The state vector at time k is represented by the vector xk.

 State estimate: An estimated quantity is represented using the hat operator. For example, the estimated state vector at time k is .

 A priori/a posteriori estimates: A priori and a posteriori estimates are represented using the + and – superscript notation. For example, the a priori state estimate at time k is , and the a posteriori state estimate at time k is .

 State error covariance estimates: The state error covariance matrix is represented using the matrix P with superscripts and subscripts as required. For example, the a priori state error covariance matrix at time k is given by .

 State transition matrix: The state transition matrix from time k – 1 to k is given by . Note that the time indices may be omitted when they are explained contextually.

 Process noise vector and covariance: The process noise vector at time k is wk. The process noise covariance matrix at time k is Qk.

 Observation vector: The observation vector at time k is given by zk.

 Observation influence matrix: The observation influence matrix at time k is given by Hk. Note that the time index may be omitted when contextually unnecessary.

 Measurement noise vector and covariance: The measurement noise vector at time k is represented by vk. The measurement noise covariance is represented by Rk.

 Probability density function: Probability density functions are expressed as p(·).