## 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):

`$Y\left(z\right)=G\left(z\right)U\left(z\right)+H\left(z\right)E\left(z\right)$`

The frequency-response is the value of the transfer function, G(z), evaluated on the unit circle (z = expiwT) 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:

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

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:

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

Where, Gij is the transfer function between the ith output and the jth input. H1(z) and H2(z) represent the noise transfer functions for the two outputs. E1(z) and E2(z) are the Z-transforms of the additive disturbances, e1(t) and e2(t), at the two model outputs, respectively.

Similar expressions apply for continuous-time frequency response. The equations are represented in Laplace domain. For more details, see the `idfrd` reference page.