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
object that stores the frequency response, sample time, and input-output
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):
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
idfrd object. The noise spectrum is stored
Where, the noise spectrum is defined as:
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:
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,
idfrd reference page.