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:

$Φ_{v} (ω) = λ T {| H (e^{i ω T}) |}^{2}$

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. 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.

Data Supported by Frequency-Response Models

You can estimate spectral analysis models from data with the following characteristics:

Complex or real data.
Time- or frequency-domain iddata or idfrd data object. To learn more about estimating time-series models, see Time Series Analysis.
Single- or multiple-output data.

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Data Supported by Frequency-Response Models