## Treating Noise Channels as Measured Inputs

A linear models is given by:

$$y=Gu+He$$

Where *G* is
an operator that takes the measured inputs *u* to
the outputs and captures the system dynamics. *H* is
an operator that describes the properties of the additive output disturbance
and takes the hypothetical (unmeasured) noise source inputs to the
outputs. *H* represents the noise model. When you
specify to estimate a noise model, the resulting model include one
noise channel *e* at the input for each output
in your system.

To study noise contributions in more detail,
it might be useful to convert the noise channels to measured channels
using `noisecnv`

:

m_GH = noisecnv(m)

This operation creates a model `m_GH`

that
represents both measured inputs `u`

and noise inputs *e*,
treating both sources as measured signals. `m_GH`

is
a model from `u`

and *e* to `y`

,
describing the transfer functions *G* and *H*.

Converting noise channels to measured inputs loses
information about the variance of the innovations `e`

.
For example, step response due to the noise channels does not take
into consideration the magnitude of the noise contributions. To include
this variance information, normalize *e* such that *v* becomes
white noise with an identity covariance matrix, where

$$e=Lv$$

To
normalize *e*, use the following command:

m_GH = noisecnv(m,'Norm')

This
command creates a model where *u* and *v* are
treated as measured signals, as follows:

$$y(t)=Gu(t)+HLv=\left[\begin{array}{cc}G& HL\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$$

For example, the scaling by *L* causes
the step responses from *v* to *y* to
reflect the size of the disturbance influence.

The converted noise sources are named in a way
that relates the noise channel to the corresponding output. Unnormalized
noise sources *e* are assigned names such as `'e@y1'`

, `'e@y2'`

,
..., `'e@yn'`

, where `'e@yn'`

refers
to the noise input associated with the output `yn`

.
Similarly, normalized noise sources `v`

, are named `'v@y1'`

, `'v@y2'`

,
..., `'v@yn'`

.

If you want to create
a model that has only the noise channels of an identified model as
its measured inputs, use the `noise2meas`

command.
It results in a model with *y*(*t*)
= *H**e* or *y*(*t*)
= *H**L**v*, where *e* or *v* is
treated as a measured input.

**Note**

When you plot models in the app that include noise sources, you can select to view the response of the noise model corresponding to specific outputs. For more information, see Selecting Measured and Noise Channels in Plots.