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## Noise Spectrum Plots

### Supported Models

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.

### Note

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.

### What Does a Noise Spectrum Plot Show?

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

`$y\left(t\right)=G\left(z\right)u\left(t\right)+v\left(t\right)$`

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:

`$v\left(t\right)=H\left(z\right)e\left(t\right)$`

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:

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

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 $y=He$.

### Note

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.

### Displaying the Confidence Interval

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.

### Note

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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