## What is a Frequency-Response Model?

A *frequency-response model* is the frequency
response of a linear system evaluated over a range of frequency values.
The model is represented by an `idfrd`

model
object that stores the frequency response, sample time, and input-output
channel information.

The frequency-response function describes the steady-state response of a system to sinusoidal inputs. For a linear system, a sinusoidal input of a specific frequency results in an output that is also a sinusoid with the same frequency, but with a different amplitude and phase. The frequency-response function describes the amplitude change and phase shift as a function of frequency.

You can estimate frequency-response models and visualize the responses on a Bode plot, which shows the amplitude change and the phase shift as a function of the sinusoid frequency.

For a discrete-time system sampled with a time interval *T*,
the transfer function *G(z)* relates the Z-transforms
of the input *U(z)* and output *Y(z)*:

$$Y(z)=G(z)U(z)+H(z)E(z)$$

The frequency-response is the value of the transfer function, *G(z)*,
evaluated on the unit circle (*z * = exp^{iwT})
for a vector of frequencies, *w*. *H(z)* represents
the noise transfer function, and *E(z)* is the Z-transform
of the additive disturbance *e(t)* with variance *λ*.
The values of *G* are stored in the `ResponseData`

property
of the `idfrd`

object. The noise spectrum is stored
in the `SpectrumData property`

.

Where, the noise spectrum is defined as:

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

A MIMO frequency-response model contains frequency-responses corresponding to each input-output pair in the system. For example, for a two-input, two-output model:

$$\begin{array}{l}{Y}_{1}(z)={G}_{11}(z){U}_{1}(z)+{G}_{12}(z){U}_{2}(z)+{H}_{1}(z){E}_{1}(z)\\ {Y}_{2}(z)={G}_{21}(z){U}_{1}(z)+{G}_{22}(z){U}_{2}(z)+{H}_{2}(z){E}_{2}(z)\end{array}$$

Where, *G*_{ij} is the
transfer function between the *i*^{th} output
and the *j*^{th} input. *H _{1}(z)* and

*H*represent the noise transfer functions for the two outputs.

_{2}(z)*E*and

_{1}(z)*E*are the Z-transforms of the additive disturbances,

_{2}(z)*e*and

_{1}(t)*e*, at the two model outputs, respectively.

_{2}(t)Similar expressions apply for continuous-time frequency response.
The equations are represented in Laplace domain. For more details,
see the `idfrd`

reference page.