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The formula used to calculate the a posteriori error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our

Kalman gain formula on the right by SkKkT, it follows that

Referring back to our expanded formula for the a posteriori error covariance,

we find the last two terms cancel out, giving

This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with numerical stability, or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the a posteriori error covariance formula as derived above must be used.

An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorithms, most adaptive filters are digital filters. Adaptive filters are required for some applications because some parameters of the desired processing operation (for instance, the locations of reflective surfaces in areverberant space) are not known in advance or are changing. The closed loop adaptive filter uses feedback in the form of an error signal to refine its transfer function.

Generally speaking, the closed loop adaptive process involves the use of a cost function, which is a criterion for optimum performance of the filter, to feed an algorithm, which

determines how to modify filter transfer function to minimize the cost on the next iteration. The most common cost function is the mean square of the error signal.

As the power of digital signal processors has increased, adaptive filters have become much more common and are now routinely used in devices such as mobile phones and other communication devices, camcorders and digital cameras, and medical monitoring equipment.

Assuming the hospital is monitoring a patient's heart beating, namely, ECG, the signal is 50 Hz (frequency is used by many countries supply) noise

Notch filter method to eliminate noise of this is the use of 50Hz (en:notch filter) of the signal filtering. However, because of the power supply in hospital. There will be a little fluctuation, sowe assume that the power supply real may fluctuate in the 47Hz to 53Hz. In order to eliminate47 to static filters will greatly reduce the frequency of 53Hz between the ECG quality, this isbecause in the stopband within might well have a frequency component of beating heart.

In order to avoid the possible loss of information, you can use the adaptive filter. The adaptive filter will supply signal and power of patients directly as the input signal, dynamicallytracking noise fluctuation frequency. Adaptive filter this usually stopband width is smaller,which means in this case an output signal for medical diagnosis is more accurate.

Hybrid Kalman filter[edit]

Most physical systems are represented as continuous-time models while discrete-time measurements are frequently taken for state estimation via a digital processor. Therefore,

the system model and measurement model are given by

where

.

Initialize

Predict

The prediction equations are derived from those of continuous-time Kalman filter without update from measurements,

i.e., . The predicted state and covariance are calculated respectively by solving a set of differential equations with the initial value equal to the estimate at the previous step.

Update

The update equations are identical to those of the discrete-time Kalman filter.

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