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# Documentation

## Representing Frequency-Response Data Using idfrd Objects

### idfrd Constructor

The idfrd represents complex frequency-response data. Before you can create an idfrd object, you must import your data as described in Frequency-Response Data Representation.

 Note:   The idfrd object can only encapsulate one frequency-response data set. It does not support the iddata equivalent of multiexperiment data.

Use the following syntax to create the data object fr_data:

```fr_data = idfrd(response,f,Ts)
```

Suppose that ny is the number of output channels, nu is the number of input channels, and nf is a vector of frequency values. response is an ny-by-nu-by-nf 3-D array. f is the frequency vector that contains the frequencies of the response.Ts is the sampling time, which is used when measuring or computing the frequency response. If you are working with a continuous-time system, set Ts to 0.

response(ky,ku,kf), where ky, ku, and kf reference the kth output, input, and frequency value, respectively, is interpreted as the complex-valued frequency response from input ku to output ky at frequency f(kf).

 Note:   When you work at the command line, you can only create idfrd objects from complex values of G(eiw). For a SISO system, response can be a vector.

You can specify object properties when you create the idfrd object using the constructor syntax:

```fr_data = idfrd(response,f,Ts,
'Property1',Value1,...,'PropertyN',ValueN)```

### idfrd Properties

To view the properties of the idfrd object, you can use the get command. The following example shows how to create an idfrd object that contains 100 frequency-response values with a sampling time interval of 0.1 s and get its properties:

```f = logspace(-1,1,100);
[mag, phase] = bode(idtf([1 .2],[1 2 1 1]),f);
response = mag.*exp(1j*phase*pi/180);
fr_data = idfrd(response,f,0.1);
get(fr_data)
```
```      FrequencyUnit: 'rad/TimeUnit'
Report: [1x1 idresults.frdest]
SpectrumData: []
CovarianceData: []
NoiseCovariance: []
InterSample: {'zoh'}
ResponseData: [1x1x100 double]
ioDelay: 0
InputDelay: 0
OutputDelay: 0
Ts: 0.1000
TimeUnit: 'seconds'
InputName: {''}
InputUnit: {''}
InputGroup: [1x1 struct]
OutputName: {''}
OutputUnit: {''}
OutputGroup: [1x1 struct]
Name: ''
Notes: {}
UserData: []
SamplingGrid: [1x1 struct]
Frequency: [100x1 double]

```

For a complete description of all idfrd object properties, see the idfrd reference page.

To change property values for an existing idfrd object, use the set command or dot notation. For example, to change the name of the idfrd object, type the following command sequence at the prompt:

```fr_data.Name = 'DC_Converter';
```

### Select I/O Channels and Data in idfrd Objects

You can reference specific data values in the idfrd object using the following syntax:

`fr_data(outputchannels,inputchannels)`

Reference specific channels by name or by channel index.

 Tip   Use a colon (:) to specify all channels, and use the empty matrix ([]) to specify no channels.

For example, the following command references frequency-response data from input channel 3 to output channel 2:

`fr_data(2,3) `

You can also access the data in specific channels using channel names. To list multiple channel names, use a cell array. For example, to retrieve the power output, and the voltage and speed inputs, use the following syntax:

`fr_data('power',{'voltage','speed'})`

To retrieve only the responses corresponding to frequency values between 200 and 300, use the following command:

`fr_data_sub = fselect(fr_data,[200:300])`

You can also use logical expressions to subreference data. For example, to retrieve all frequency-response values between frequencies 1.27 and 9.3 in the idfrd object fr_data, use the following syntax:

`fr_data_sub = fselect(fr_data,fr_data.f>1.27&fr_data.f<9.3)`
 Tip   Use end to reference the last sample number in the data. For example, data(77:end).
 Note:   You do not need to type the entire property name. In this example, f in fr_data.f uniquely identifies the Frequency property of the idfrd object.

### Adding Input or Output Channels in idfrd Objects

The horizontal and vertical concatenation of idfrd objects combine information in the ResponseData properties of these objects. ResponseData is an ny-by-nu-by-nf array that stores the response of the system, where ny is the number of output channels, nu is the number of input channels, and nf is a vector of frequency values (see Properties).

#### Horizontal Concatenation of idfrd Objects

The following syntax creates a new idfrd object data that contains the horizontal concatenation of data1,data2,...,dataN:

`data = [data1,data2,...,dataN]`

data contains the frequency responses from all of the inputs in data1,data2,...,dataN to the same outputs. The following diagram is a graphical representation of horizontal concatenation of frequency-response data. The (j,i,:) vector of the resulting response data represents the frequency response from the ith input to the jth output at all frequencies.

 Note:   Horizontal concatenation of idfrd objects requires that they have the same outputs and frequency vectors. If the output channel names are different and their dimensions are the same, the concatenation operation resets the output names to their default values.

#### Vertical Concatenation of idfrd Objects

The following syntax creates a new idfrd object data that contains the vertical concatenation of data1,data2,...,dataN:

`data = [data1;data2;... ;dataN]`

The resulting idfrd object data contains the frequency responses from the same inputs in data1,data2,...,dataN to all the outputs. The following diagram is a graphical representation of vertical concatenation of frequency-response data. The (j,i,:) vector of the resulting response data represents the frequency response from the ith input to the jth output at all frequencies.

 Note:   Vertical concatenation of idfrd objects requires that they have the same inputs and frequency vectors. If the input channel names are different and their dimensions are the same, the concatenation operation resets the input names to their default values.

#### Concatenating Noise Spectrum Data of idfrd Objects

When the SpectrumData property of individual idfrd objects is not empty, horizontal and vertical concatenation handle SpectrumData, as follows.

In case of horizontal concatenation, there is no meaningful way to combine the SpectrumData of individual idfrd objects and the resulting SpectrumData property is empty. An empty property results because each idfrd object has its own set of noise channels, where the number of noise channels equals the number of outputs. When the resulting idfrd object contains the same output channels as each of the individual idfrd objects, it cannot accommodate the noise data from all the idfrd objects.

In case of vertical concatenation, the toolbox concatenates individual noise models diagonally. The following shows that data.SpectrumData is a block diagonal matrix of the power spectra and cross spectra of the output noise in the system:

$data.s=\left(\begin{array}{ccc}data1.s& & 0\\ & \ddots & \\ 0& & dataN.s\end{array}\right)$

s in data.s is the abbreviation for the SpectrumData property name.

### Operations That Create idfrd Objects

The following operations create idfrd objects:

• Constructing idfrd objects.

• Estimating nonparametric models using etfe, spa, and spafdr. For more information, see Identifying Frequency-Response Models.

• Converting the Control System Toolbox™ frd object. For more information, see Using Identified Models for Control Design Applications.

• Converting any linear dynamic system using the idfrd command.

For example:

```sys_idpoly = idpoly([1 2 1],[0 2],'Ts',1);
G = idfrd(sys_idpoly,linspace(0,pi,128))
```
```G =
IDFRD model.
Contains Frequency Response Data for 1 output(s) and 1 input(s), and the spectra for disturbances at the outputs.
Response data and disturbance spectra are available at 128 frequency points, ranging from 0 rad/s to 3.142 rad/s.

Sample time: 1 seconds
Status:
Created by direct construction or transformation. Not estimated.
```