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Measuring linear response functions with multifrequency measurement

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Dr. Daniel Forchhheimer

Dr. Daniel Forchhheimer

Intermodulation Products, Sweden

TutorialProduct: MLA-3
Figure 1

A schematic showing experimental data on an AFM cantilever. The resonance curve, both amplitude and phase, is extracted by simple division of response comb with the excitation comb. No frequency sweep is needed. All frequencies are measured at the 'same time', or in the same measurement time window.

Tutorial: Measuring linear response functions with multifrequency measurement

Goal: Characterize a linear device-under-test (DUT) rapidly by measuring its transfer function across multiple frequencies simultaneously.

Theory of Operation:
The MLA-3 synthesizes a multifrequency signal consisting of a precise linear superposition of many tones. In time domain, a frequency comb of closely spaced tones manifests as a sinc pulse. When this signal drives a linear system, the system responds to each tone independently of all others.

The MLA measures the response at every individual tone with zero spectral leakage between them using the concept of tuning. Because the system is linear, the Linear Response Function is determined by a simple complex division at each frequency point:
H(ω)=Drive(ω)Response(ω)​H(ω)=\frac{Drive(ω)}{Response(ω)}​

Note: This simple division is valid only for linear systems. In nonlinear systems, the tones would intermodulate, requiring more complex analysis.

What is the Linear Response Function? The Linear Response Function (often called the Transfer Function) is the mathematical "fingerprint" of a linear system. It quantifies exactly how the system will respond to any signal, defining both the change in amplitude (gain/attenuation) and the shift in phase at any given frequency.

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