Estimate frequency response and spectrum using spectral analysis with frequency-dependent resolution
g = spafdr(data)
g = spafdr(data,Resol,w)
g = spafdr(data) estimates the input-to-output frequency response G(ω)
and noise spectrum Φυ of the general linear
model
where Φυ(ω)
is the spectrum of υ(t). data contains
the output-input data as an iddata object. The
data can be complex valued, and either time or frequency domain. It
can also be an idfrd object containing frequency-response
data. g is an idfrd object
with the estimate of at the frequencies ω specified
by row vector w. g also includes
information about the spectrum estimate of Φυ(ω)
at the same frequencies. Both results are returned with estimated
covariances, included in g. The normalization of
the spectrum is the same as described in spa.
Information about the estimation results and options used is
stored in the model's Report property. Report has
the following fields:
Status — Summary of the
model status, which indicates whether the model was created by construction
or obtained by estimation.
Method — Estimation command
used.
WindowSize — Frequency resolution.
DataUsed — Attributes of
the data used for estimation. Structure with the following fields:
Name — Name of the data
set.
Type — Data type.
Length — Number of data
samples.
Ts — Sample time.
InterSample — Input intersample
behavior.
InputOffset — Offset removed
from time-domain input data during estimation.
OutputOffset — Offset removed
from time-domain output data during estimation.
g = spafdr(data,Resol,w) specifies frequencies
and frequency resolution.
The frequency variable w is either specified as a row vector of
frequencies in rad/TimeUnit, where TimeUnit
refers to the TimeUnit property of data, or as a cell array
{wmin,wmax}. In the latter case the covered frequencies will
be 50 logarithmically spaced points from wmin to
wmax. You can change the number of points to
NP by entering {wmin,wmax,NP}.
Omitting w or entering it as an empty matrix gives the default value, which
is 100 logarithmically spaced frequencies between the smallest and largest frequency
in data. For time-domain data, the default range goes from to , where Ts is the sample time of data and
N is the number of data points.
The argument Resol defines the frequency resolution of the estimates. The
resolution (measured in rad/TimeUnit) is the size of the smallest
detail in the frequency function and the spectrum that is resolved by the estimate.
The resolution is a tradeoff between obtaining estimates with fine, reliable
details, and suffering from spurious, random effects: The finer the resolution, the
higher the variance in the estimate. Resol can be entered as a
scalar (measured in rad/TimeUnit), which defines the resolution
over the whole frequency interval. It can also be entered as a row vector of the
same length as w. Then Resol(k) is the local,
frequency-dependent resolution around frequency w(k).
The default value of Resol, obtained by omitting
it or entering it as the empty matrix, is Resol(k) = 2(w(k+1)-w(k)),
adjusted upwards, so that a reasonable estimate is guaranteed. In
all cases, the resolution is returned in the variable g.Report.WindowSize.
If the data is given in the time domain, it is first converted
to the frequency domain. Then averages of Y(w)Conj(U(w)) and U(w)Conj(U(w)) are
formed over the frequency ranges w, corresponding
to the desired resolution around the frequency in question. The ratio
of these averages is then formed for the frequency-function estimate,
and corresponding expressions define the noise spectrum estimate.