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Estimating Spectra and Frequency Functions
This section describes methods that estimate the frequency functions and spectra (3-11) directly. The cross-covariance function between
and
is defined as
analogously to (3-7). Its Fourier transform, the cross spectrum,
is defined analogously to (3-6). Provided that the input
is independent of
, the relationship (3-1) implies the following relationships between the spectra.
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(3-32) |
By estimating the various spectra involved, the frequency function and the disturbance spectrum can be estimated as follows.
Form estimates of the covariance functions (as defined in (3-7)) ,
, and
, using
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(3-33) |
and analog expressions for the others. Then, form estimates of the corresponding spectra
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(3-34) |
and analogously for and
. Here
is the so-called lag window and M is the width of the lag window. The estimates are then formed as
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(3-35) |
This procedure is known as spectral analysis. (See Chapter 6 in Ljung (1999).)
![]() | Estimating Impulse Responses | Estimating Parametric Models | ![]() |