Discrete Stochastic Processes and Optimal Filtering

6.3. Data representation

The general shape of an adaptive filter might be the following.

Figure 6.7. Theoretical schema with multiple inputs

The input signal can be simultaneously the result of sensors (case of adaptive antennae, for example) or as well they can represent the different samples, taken at different instants of a single signal.

We will take as notation:

– multiple input: ;

– single input: .

In the case of a single input that we will consider next, we will have the following configuration.

Figure 6.8. Schema of predictor principle

X_K, Y_K, Z_K, D_K and representing the signal to be predicted, the filter input, the filter output, the desired output and the signal error respectively.

Let us write the output Z_K:

By calling λ_K the weight vector or coefficient vector at instant “K”, also written in the form , we can use a single vectorial notation:

Our system not being perfect we ...