Transfer Function / FRF

The Frequency Response Function (FRF) describes the input-output relationship of a linear system as a function of frequency. It is the fundamental measurement in experimental modal analysis, structural dynamics, and acoustics, providing both magnitude and phase information about the system response.

Definition

The FRF H(f) relates the output spectrum Y(f) to the input spectrum X(f) of a linear time-invariant system:

Y(f) = H(f) · X(f)

In practice, measurement noise on either the input or the output signal requires the use of spectral averaging and specific estimator formulations to obtain a reliable estimate of the true FRF.

FRF Estimators

H1 Estimator

The H1 estimator minimizes the effect of noise on the output signal:

H₁(f) = S_xy(f) / S_xx(f)

H1 is the preferred estimator when the input (excitation) signal has a good signal-to-noise ratio and noise is primarily present on the output (response). This is the most common scenario in impact testing and shaker excitation.

H2 Estimator

The H2 estimator minimizes the effect of noise on the input signal:

H₂(f) = S_yy(f) / S_yx(f)

H2 is appropriate when the input signal is noisy, for example when measuring in an operating environment where additional unmeasured forces act on the structure.

Hv Estimator

The Hv (or Hc) estimator provides an unbiased estimate when noise is present on both channels. It is computed by solving an eigenvalue problem that accounts for noise on both input and output:

Hv = solution that minimizes total least-squares error on both channels

The Hv estimator always produces a magnitude between H1 and H2. When coherence is high (γ² > 0.9), all three estimators converge to the same value.

The relationship between the estimators and coherence is: γ² = H₁ / H₂. This identity provides a useful cross-check of your measurement data.

FRF Types in Structural Dynamics

Depending on the physical quantities of input and output, different FRF types are obtained:

FRF Type	Input	Output	Units
Receptance (Compliance)	Force	Displacement	m/N
Mobility	Force	Velocity	(m/s)/N
Accelerance (Inertance)	Force	Acceleration	(m/s²)/N
Dynamic Stiffness	Displacement	Force	N/m
Mechanical Impedance	Velocity	Force	N·s/m
Apparent Mass	Acceleration	Force	N/(m/s²) = kg

Impulse Response Function

The inverse FFT of the FRF yields the Impulse Response Function (IRF) in the time domain. The IRF represents the system response to a unit impulse (Dirac delta) and contains the same information as the FRF:

h(t) = IFFT{ H(f) }

The IRF is useful for verifying measurement quality (it should decay to zero within the FFT block length) and for time-domain modal parameter extraction methods such as the Ibrahim Time Domain (ITD) method.

Display Options

Magnitude and Phase

The most common display format shows FRF magnitude (in dB or linear scale) and phase (in degrees) as functions of frequency. At resonance, the magnitude peaks and the phase undergoes a 180-degree shift for a lightly damped single-degree-of-freedom system.

Real and Imaginary Parts

Displaying the real and imaginary parts separately is useful for modal parameter estimation. For proportionally damped systems, the imaginary part of the mobility FRF peaks at resonance and the real part passes through zero.

Nyquist Plot

The Nyquist plot displays the FRF as a curve in the complex plane (real vs. imaginary). Each resonance appears as a circle, and the diameter of the circle is related to the modal damping ratio.

Parameters

Parameter	Options	Description
Input Channel	Force / reference signal	The excitation channel (denominator of FRF).
Output Channel(s)	Response signal(s)	One or more response channels (numerator of FRF).
Estimator	H1 / H2 / Hv	FRF estimator type. Use H1 for most applications.
Spectrum Size	256 – 65536	FFT block size.
Overlap	0% – 95%	Block overlap.
Window Type	Hanning, Force, Exponential	Window function. Impact testing typically uses Force + Exponential windows.
Display Format	Magnitude+Phase / Real+Imaginary / Nyquist	Visualization format.

Use in Modal Testing

The FRF matrix is the primary input to experimental modal analysis algorithms. A complete FRF matrix consists of measurements at all combinations of excitation and response points (or a sufficient subset). Key considerations include:

Driving-point FRFs (excitation and response at the same point) must show alternating resonance/anti-resonance behavior and positive imaginary parts (for mobility).
Reciprocity should be verified: H_ij ≈ H_ji for linear structures.
Coherence should exceed 0.9 across the frequency range of interest. See Coherence Function for troubleshooting.

Practical Tips

For impact testing, use a force window on the input channel and an exponential window on the output channel to minimize leakage.
Ensure that the response signal decays to near zero within the FFT block. If it does not, increase the block size or apply a heavier exponential window.
Monitor the coherence function in real time during measurement. Reject individual hits with poor coherence before averaging.
When the FRF is imported from an external source, verify that the frequency resolution and format are compatible with the modal analysis module. See EMA Overview.

Double hits during impact testing cause a notch in the force spectrum, leading to poor coherence and unreliable FRF values at the affected frequencies. Always verify the force time signal and auto-spectrum before accepting a measurement.