# noise2meas

Noise component of linear identified model

## Description

returns the noise component of a linear identified model, `noiseModel`

= noise2meas(`sys`

)`sys`

.
Use `noise2meas`

to convert a time-series model (no inputs) to an
input/output model. You can use the converted model for linear analysis, including
viewing pole/zero maps and plotting the step response.

specifies the noise variance normalization method.`noiseModel`

= noise2meas(`sys`

,`noise`

)

## Examples

### Convert Noise Component of Linear Identified Model into Input/Output Model

Convert a time-series model to an input/output model that may be used by linear analysis tools.

Identify a time-series model.

load iddata9 z9 sys = ar(z9,4,'ls');

`sys`

is an `idpoly`

model with no inputs.

Convert `sys`

to a measured model.

noise_model = noise2meas(sys);

`noise_model`

is an `idpoly`

model with one input.

You can use `noise_model`

for linear analysis functions such as `step`

, `iopzmap`

, etc.

### Normalizing Noise Variance

Convert an identified linear model to an input/output model, and normalize its noise variance.

Identify a linear model using data.

```
load twotankdata;
z = iddata(y,u,0.2);
sys = ssest(z,4);
```

`sys`

is an `idss`

model, with a noise variance of 6.6211e-06. The value of $$L$$ is `sqrt(sys.NoiseVariance)`

, which is 0.0026.

View the disturbance matrix.

sys.K

`ans = `*4×1*
0.2719
1.6570
0.6318
0.2877

Obtain a model that absorbs the noise variance of `sys`

.

`noise_model_normalize = noise2meas(sys,'normalize');`

`noise_model_normalize`

is an `idpoly`

model.

View the $$B$$ matrix for `noise_model_normalize`

.

noise_model_normalize.B

`ans = `*4×1*
0.0007
0.0043
0.0016
0.0007

As expected, `noise_model_normalize.B`

is equal to `L*sys.K`

.

Compare the bode response with a model that ignores the noise variance of `sys`

.

noise_model_innovation = noise2meas(sys,'innovations'); bodemag(noise_model_normalize,noise_model_innovation); legend('Normalized noise variance','Ignored noise variance');

The difference between the bode magnitudes of the `noise_model_innovation`

and `noise_model_normalized`

is approximately 51 dB. As expected, the magnitude difference is approximately equal to `20*log10(L)`

.

## Input Arguments

`sys`

— Identified linear model

`idss`

| `idtf`

| `idproc`

| `idpoly`

| `idfrd`

| `idgrey`

Identified linear model, specified as one of the following model objects.

`sys`

represents the system:

$$y(t)=Gu(t)+He(t)$$

*G* is the transfer function between the measured input,
*u*(*t*), and the output,
*y*(*t*). *H* is the
noise model and describes the effect of the disturbance,
*e*(*t*), on the model response.

An equivalent state-space representation of `sys`

is

$$\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+Ke(t)\\ y(t)=Cx(t)+Du(t)+e(t)\\ e(t)=Lv(t)\end{array}$$

*v*(*t*) is white noise with independent
channels and unit variances. The white-noise signal
*e*(*t*) represents the model
innovations and has variance
*LL ^{T}*. The noise-variance
data is stored using the

`NoiseVariance`

property of
`sys`

.`noise`

— Noise variance normalization method

`'innovations'`

(default) | `'normalize'`

Noise variance normalization method, specified as one of the following values.

`'innovations'`

— Noise sources are not normalized and remain as the innovations process.`'normalize'`

— Noise sources are normalized to be independent with unit variance.

## Output Arguments

`noiseModel`

— Noise component of identified model

`idss`

| `idtf`

| `idpoly`

| `idfrd`

Noise component of identified model, returned as an `idss`

,
`idtf`

, `idpoly`

, or
`idfrd`

object.

The model type of `noiseModel`

depends on the model
type of `sys`

.

`noiseModel`

is an`idtf`

model if`sys`

is an`idproc`

model.`noiseModel`

is an`idss`

model if`sys`

is an`idgrey`

model.`noiseModel`

is the same type of model as`sys`

for all other model types.

To obtain the model coefficients of `noiseModel`

in
state-space form, use `ssdata`

. Similarly, to obtain
the model coefficients in transfer-function form, use `tfdata`

.

#### Noise Sources Not Normalized

If `noise`

is `'innovations'`

,
then `noise2meas`

returns *H* and
`noiseModel`

represents the system

$$y(t)=He(t)$$

An equivalent state-space representation of
`noiseModel`

is

$$\begin{array}{l}\dot{x}(t)=Ax(t)+Ke(t)\\ y(t)=Cx(t)+e(t)\end{array}$$

`noise2meas`

returns the noise channels of
`sys`

as the input channels of
`noiseModel`

. The input channels are named using
the format `'e@yk'`

, where `yk`

corresponds to the `OutputName`

property of an output.
The measured input channels of `sys`

are discarded
and the noise variance is set to zero.

#### Noise Sources Normalized

If `noise`

is `'normalize'`

, then
`noise2meas`

first normalizes

$$e(t)=Lv(t)$$

`noiseModel`

represents the system

$$y(t)=HLv(t)$$

or, equivalently, in state-space representation

$$\begin{array}{l}\dot{x}(t)=Ax(t)+KLv(t)\\ y(t)=Cx(t)+Lv(t)\end{array}$$

The input channels are named using the format
`'v@yk'`

, where `yk`

corresponds
to the `OutputName`

property of an output.

## Version History

**Introduced in R2012a**

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