The prediction-error filter results directly from the prediction filter. We define the prediction-error series as the difference between the true value
and the estimated or predicted value
. The prediction error at time instant
is thus
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(10)
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If
is replaced by n in the above equation, the result is the expression for the prediction error, at the present time n, given by
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(11)
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The prediction error
is a signal that represents the nonpredictable part of
. The above equation shows that the filter is
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(12)
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This filter is called the prediction-error filter or the prediction-error operator. There are
zeros in the prediction-error operator that lie between the leading coefficient, namely 1, and the negative prediction-operator coefficients. These
zeros constitute the gap. Let
represent the zero-delay unit spike (where the 1 occurs at time instant 0). The Z-transform of the zero-delay unit spike is 1. Let
represent the unit spike for delay
(where there are
zeros before the 1). The Z-transform of the unit spike for delay
is
. The prediction-error operator is now the difference between the zero-delay unit spike and the prediction operator delayed by the prediction distance; that is,
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(13)
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A matrix equation can be derived for the prediction-error operator. Normal equations 9 for the prediction filter can be augmented in such a way that the prediction operator is converted into the prediction-error operator. First, in equations 9, we subtract the left-hand side from the right-hand side. The result is
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(14)
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which is
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(15)
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and which also can be written as
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(16)
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Next, we define the quantities
with the equation
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(17)
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Then, we combine equations 16 and 17 to yield
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(18)
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The column vector on the left-hand side is the prediction-error operator
. Thus, matrix equation 17 is a representation for the prediction-error operator with prediction distance
Such an operator performs what is known as gap deconvolution.
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Vínculos externos
find literature about The prediction-error filter/es
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