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## 5.8  Non-parametric transfer function

Response data structures are the classical format to store non-parametric transfer functions. Multi-dim curve can also be used.

For a linear system at a given frequency ω, the response vector {y} at NS sensor locations to a vector {u} of NA inputs is described by the NS by NA rectangular matrix of Frequency Responses (FRF)

{
 y1(ω) ⋮ yNS(ω)
} = [H] {u} = [
 H11(ω) H12(ω) … H21(ω) H22(ω) ⋮ ⋱
]NS× NA {
 u1(ω) ⋮ uNA(ω)
}     (5.29)

The SDT stores frequencies at which the FRF are evaluated as a column vector w

w = {
 ω1 ⋮ ωNW
}NW× 1     (5.30)

and SISO FRFs Hij are stored as columns of the matrix xf where each row corresponds to a different frequency (indicated in w). By default, it is assumed that the correspondence between the columns of xf and the sensors and actuator numbers is as follows. The NS transfer functions from actuator 1 to the NS sensors are stored as the first NS columns of xf, then the NS transfer functions of actuator 2, etc.

xf = [
 H11(ω1) H21(ω1) … H12(ω1) H22(ω1) … H11(ω2) H21(ω2) … H12(ω2) H22(ω2) … ⋮ ⋱ ⋮ ⋱
]NW× (NS × NA)     (5.31)

Further characterization of the properties of a given set of FRFs is given by a structure detailed under Response data section.

Frequency response functions corresponding to parametric models can be generated in the xf format using qbode (transformation from ss and tf formats), nor2xf, or res2xf. These functions use robustness/speed trade-offs that are different from algorithms implemented in other MATLAB toolboxes and are more appropriate for applications in structural dynamics.