What Is the Difference between Transfer Function (TF) and Frequency Response Function (FRF)? : MALMIJAL-M Blog

In system analysis, two terms are often used

They look very similar and are often used interchangeably. But they are not exactly the same.

What Is the Difference between Transfer Function (TF) and Frequency Response Function (FRF)?

transfer function

Derived from equations (differential for continuous or difference equation for discrete-time system)
Ideal (no noise)
Exact mathematical description
e.g. Digital filter form

z-Transform TF

FRF (Frequency Response Function)

Computed from real data
Includes noise and measurement effects
Practical approximation of system behavior
Sxy: Cross Power Spectral Density, Sxx: Auto Power Spectral Density, H1 estimator

Type	How Obtained
Transfer Function (TF)	Derived from physics / equations
Frequency Response Function (FRF)	Measured using input/output signals

FRF property window

FRF of noisy input and output

FRF of random input and output

In ideal conditions, FRF ≈ Transfer Function

But in reality, FRF is an estimate

In practice, FRF is computed using

H1 estimator: minimize output noise → output noise dominant, impact hammer used
H2 estimator: minimize input noise → input noise dominant, random excitation used
Hv estimator: total least squares form → compromise between H1 and H2, used when both input and output noise are present

Example

H1 estimator of FRF

FRF property window

This reduces output noise effects

Systems, Filters & ModelingWhat Is the Difference between Transfer Function (TF) and Frequency Response Function (FRF)?