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lect10a

# lect10a - Kevin Buckley 2007 1 ECE 8770 Topics in Digital...

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Kevin Buckley - 2007 1 ECE 8770 Topics in Digital Communications - Sp. 2007 Lecture 10a 4 Channel Equalization 4.4 Adaptive Equalization 4.5 Alternative Adaptation Schemes (continued) The Kalman Filter as an Adaptive Equalizer: In this Subsection we develop the Kalman filtering algorithm for adaptation of the coefficient vector of a linear equalizer or DFE. We start with the general Kalman fil- tering problem formulation and solution, and we then discuss its application to channel equalization. 1. Kalman Filtering: The Kalman filter is popular as an effective estimator of the state of a random process because it is the minimum mean-squared state estimator . It is also the conditional mean estimator of the state . We will see that it can also be used as an effective adaptive equalizer. The State-Space Model or a Random Process: Consider the following general discrete-time linear state-space model w k +1 = A k w k + G k v k (1) z k = H k w k + η k (2) where z k is the L × 1 dimensional observation (i.e. data) vector at time k , w k is the M × 1 dimensional state vector, H k is the L × L dimensional transition matrix from the state to the output at time k , η k is the L × 1 measurement noise vector, A k is the M × M dimensional state transition matrix at time k , v k is the P × 1 dimensional process noise vector, and G k is the M × P dimensional transition matrix at time k from the state transition noise to the next state. We assume that η k and v k are zero-mean vector sequences, uncorrelated across time, with covariance matrices S k and Q k at time k , respectively. η k and v n are assumed uncorrelated with one another for all k and n .

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Kevin Buckley - 2007 2 Figure 1 illustrates this general discrete-time linear state-space model. This can
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lect10a - Kevin Buckley 2007 1 ECE 8770 Topics in Digital...

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