Learn more about System Identification Toolbox

Noise Spectrum Plots

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Supported Models What Does a Noise Spectrum Plot Show? Displaying the Confidence Interval How to Plot the Noise Spectrum Using the GUI How to Plot the Noise Spectrum at the Command Line

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.

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What Does a Noise Spectrum Plot Show?

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

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:

The toolbox computes both H and during the estimation of the noise model and stores these quantities as model properties. The H(z) operator represents the noise model. e(t) is a white-noise source with variance .

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:

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 .

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.

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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 How to Plot the Noise Spectrum Using the GUI.

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 of values that have a 95% probability of being the true system noise spectrum. 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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How to Plot the Noise Spectrum Using the GUI

To create a noise spectrum plot for parametric linear models in the GUI, select the Noise spectrum check box in the Model Views area. For general information about creating and working with plots, see Working with Plots in the System Identification Tool GUI.

To include or exclude a model on the plot, click the corresponding model icon in the System Identification Tool GUI. Active models display a thick line inside the Model Board icon.

The following figure shows a sample Noise Spectrum plot.

The following table summarizes the Noise Spectrum plot settings.

Noise Spectrum Plot Settings

Action	Command
Display the confidence interval.	To display the dashed lines on either side of the nominal model curve, select Options > Show confidence intervals. Select this option again to hide the confidence intervals. To change the confidence value, select Options > Set % confidence level, and choose a value from the list. To enter your own confidence level, select Options > Set confidence level > Other. Enter the value as a probability (between 0 and 1) or as the number of standard deviations of a Gaussian distribution.
Change the frequency values for computing the noise spectrum. The default frequency vector is 128 linearly distributed values, greater than zero and less than or equal to the Nyquist frequency.	Select Options > Frequency range and specify a new frequency vector in units of radians per second. Enter the frequency vector using any one of following methods: MATLAB expression, such as [1:100]*pi/100 or logspace(-3,-1,200). Cannot contain variables in the MATLAB workspace. Row vector of values, such as [1:.1:100] Tip   To restore the default frequency vector, enter [].
Change frequency units between hertz and radians per second.	Select Style > Frequency (Hz) or Style > Frequency (rad/s).
Change frequency scale between linear and logarithmic.	Select Style > Linear frequency scale or Style > Log frequency scale.
Change amplitude scale between linear and logarithmic.	Select Style > Linear amplitude scale or Style > Log amplitude scale.
(Multiple-output system only) Select an input-output pair to view the noise spectrum corresponding to those channels. Note   You cannot view cross spectra between different outputs.	Select the output by name in the Channel menu.

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How to Plot the Noise Spectrum at the Command Line

You can plot the frequency-response of the noise model.

First, select the portion of the model object that corresponds to the noise model H. For example, to select the noise model in the model object m, type the following command:

m_noise=m('noise')

Tip   You can abbreviate the command to m_noise=m('n').

To plot the frequency-response of the noise model, use the bode command:

bode(m_noise)

To determine if your estimated noise model is good enough, you can compare the frequency-response of the estimated noise-model H to the estimated frequency response of v(t). To compute v(t), which represents the actual noise term in the system, use the following commands:

ysimulated = sim(m,data);
v = ymeasured-ysimulated;

ymeasured is data.y. v is the noise term v(t), as described in What Does a Noise Spectrum Plot Show? and corresponds to the difference between the simulated response ysimulated and the actual response ymeasured.

To compute the frequency-response model of the actual noise, use spa:

V = spa(v);

The toolbox uses the following equation to compute the noise spectrum of the actual noise:

The covariance function is given in terms of E, which denotes the mathematical expectation, as follows:

To compare the parametric noise-model H to the (nonparametric) frequency-response estimate of the actual noise v(t), use bode:

bode(V,m('noise'))

If the parametric and the nonparametric estimates of the noise spectra are different, then you might need a higher-order noise model.

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Frequency Response Plots

Pole and Zero Plots

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