System Identification Toolbox™

spa

Estimate frequency response and spectrum using spectral analysis returning idfrd object

Syntax

g = spa(data)
g = spa(data,M,w,maxsize)
[g,phi,spe] = spa(data)

Description

spa estimates the transfer function g and the noise spectrum of the general linear model

where is the spectrum of .

data contains the output-input data as an iddata object. The data can be complex valued. data can be both time domain and frequency domain. data can also be an idfrd object.

g is returned as an idfrd object (see idfrd) with the estimate of at the frequencies specified by row vector w. The default value of w is

w = [1:128]/128*pi/Ts

Here Ts is the sampling interval of data.

g also includes information about the spectrum estimate of at the same frequencies. Both outputs are returned with estimated covariances, included in g. See idfrd.

M is the length of the lag window used in the calculations. The default value is

M = min(30,length(data)/10)

Changing the value of M controls the frequency resolution of the estimate. The resolution corresponding to M is approximately /M rad/sampling interval. The value of M exchanges bias for variance in the spectral estimate. As M is increased, the estimated functions show more detail, but are more corrupted by noise. The sharper peaks a true frequency function has, the higher M it needs. See etfe as an alternative for narrowband signals and systems. The function spafdr allows the frequency resolution to depend on the frequency. See also Identifying Frequency-Response Models.

maxsize controls the memory-speed tradeoff (see Algorithm Properties).

For time series, where data contains no input channels, g is returned with the estimated output spectrum and its estimated standard deviation.

When spa is called with two or three output arguments,

• g is returned as an idfrd model with just the estimated frequency response from input to output and its uncertainty.

• phi is returned as an idfrd model, containing just the spectrum data for the output spectrum and its uncertainty.

• spe is returned as an idfrd model containing spectrum data for all output-input channels in data. That is, if z = [data.OutputData, data.InputData], spe contains as spectrum data the matrix-valued power spectrum of z.

  

Here win(m) is weight at lag m of an M-size Hamming window and W is the frequency value i rad/s. Note that ' denotes complex-conjugate transpose.

The normalization of the spectrum differs from the normalization used by the Signal Processing Toolbox spectrum function. SeeUnderstanding Spectrum Normalization for a more precise definition.

Algorithm

The covariance function estimates are computed using covf. These are multiplied by a Hamming window of lag size M and then transformed using a Fourier transform. The relevant ratios and differences are then formed. For the default frequencies, this is done using a fast Fourier transform, which is more efficient than for user-defined frequencies. For multivariable systems, a straightforward for loop is used.

Note that M = is in Table 6.1 of Ljung (1999). The standard deviations are computed as on pages 184 and 188 in Ljung (1999).

Examples

With default frequencies,

g = spa(z);
bode(g)

With logarithmically spaced frequencies,

w = logspace(-2,pi,128);
g= spa(z,[],w); % (empty matrix gives default)
bode(g,'sd',3)
bode(g('noise'),'sd',3) % noise spectrum with confidence
interval of 3 standard deviations.

See Also

size

spafdr