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When you estimate the noise model of your linear system, you can plot the spectrum of the estimated noise model. Noise-spectrum plots are available for all linear parametric models and spectral analysis (nonparametric) models.

For nonlinear models and correlation analysis models, noise-spectrum plots are not available. For time-series models, you can only generate noise-spectrum plots for parametric and spectral-analysis models.

The general equation of a linear dynamic system is given by:

$$y(t)=G(z)u(t)+v(t)$$

In this equation, *G* is an operator that
takes the input to the output and captures the system dynamics, and *v* is
the additive noise term. The toolbox treats the noise term as filtered
white noise, as follows:

$$v(t)=H(z)e(t)$$

where *e*(*t*) is a white-noise
source with variance λ.

The toolbox computes both *H* and $$\lambda $$ during the estimation of the
noise model and stores these quantities as model properties. The *H(z)* operator
represents the noise model.

Whereas the frequency-response plot shows the response of *G*,
the noise-spectrum plot shows the frequency-response of the noise
model *H*.

For input-output models, the noise spectrum is given by the following equation:

$${\Phi}_{v}(\omega )=\lambda {\left|H\left({e}^{i\omega}\right)\right|}^{2}$$

For time-series models (no input), the vertical axis of the noise-spectrum plot is the same as the dynamic model spectrum. These axes are the same because there is no input for time series and $$y=He$$.

You can avoid estimating the noise model by selecting the Output-Error
model structure or by setting the `DisturbanceModel`

property
value to `'None'`

for a state space model. If you
choose to not estimate a noise model for your system, then *H* and
the noise spectrum amplitude are equal to 1 at all frequencies.

In addition to the noise-spectrum curve, you can display a confidence interval on the plot. To learn how to show or hide confidence interval, see the description of the plot settings in Plot the Noise Spectrum Using the System Identification App.

The *confidence interval* corresponds to
the range of power-spectrum values with a specific probability of
being the actual noise spectrum of the system. The toolbox uses the
estimated uncertainty in the model parameters to calculate confidence
intervals and assumes the estimates have a Gaussian distribution.

For example, for a 95% confidence interval, the region around the nominal curve represents the range where there is a 95% chance that the true response belongs.. You can specify the confidence interval as a probability (between 0 and 1) or as the number of standard deviations of a Gaussian distribution. For example, a probability of 0.99 (99%) corresponds to 2.58 standard deviations.

The calculation of the confidence interval assumes that the model sufficiently describes the system dynamics and the model residuals pass independence tests.

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