What an impulse response is
If an object behaves as a linear, time-invariant system, its response to any input is fixed by one function: its impulse response, the output you would see after an ideal instantaneous tap. Its Fourier transform is the transfer function. Peaks in that transfer function are the resonances.
Why a sweep beats a click
A real tap is short, so it carries little energy and is limited by the hammer or speaker. A long sweep puts far more energy into each frequency while keeping the signal below the actuator's limits, which raises the signal-to-noise ratio. The exponential sine sweep method has a second advantage: harmonic distortion from a slightly nonlinear speaker or amplifier lands as separate lines before the linear response after deconvolution, so a time window can remove it. Angelo Farina described the method in detail for audio measurements.
x(t) = sin[ (w1*T / ln(w2/w1)) * ( exp((t/T) * ln(w2/w1)) - 1 ) ] w1, w2: start and stop angular frequency T: sweep duration
How the deconvolution works
- Play the sweep through the actuator and record the response with the sensor, using one shared clock.
- Build the inverse filter: the sweep reversed in time, with an amplitude envelope that offsets the sweep's pink spectrum.
- Convolve the recording with the inverse filter. The sweep compresses into an impulse, delayed by the sweep length.
- Window the result. The linear impulse response sits at the main peak; harmonic-distortion products arrive earlier and can be discarded or analysed separately.
Practical points: sweep beyond the band of interest at both ends, leave time after the sweep for the response to decay, keep levels below clipping, and measure at more than one level to check linearity.
What you can do with the impulse response
| Use | How | Limit |
|---|---|---|
| Convolution | Convolve any input with the measured response to predict the output | Passive and linear. It reproduces the measured behaviour, no more. |
| Additive modal synthesis | Sum one damped sinusoid per mode | Needs accurate frequencies, damping and phases for every mode that matters. |
| Parametric fit | Fit frequencies and damping ratios, then compare repeats | Mode count and model order must be chosen with care. |
| Inverse-filtered drive | Design an actuator signal that makes a sensor follow a target waveform | Needs regularisation; exact inversion of deep antiresonances is unstable. |
| Time reversal | Reverse a recorded response and re-emit it to refocus energy at a point | Focuses vibration; it does not retune the object. |
The additive form is s(t) = Σ An e−t/τn sin(2π fn t + φn), where τn = 1/(2π ζn fn).
What none of this does: retune the object
Convolution and modal synthesis recreate the response that was measured. They do not alter the object that produced it. Changing a frequency in a synthesis model describes a different object, not a changed copy of the real one. Replaying a recorded response through an actuator drives the object at the frequencies in the recording, and the object rings down at its own natural frequencies when the drive stops. Time reversal on plates has been demonstrated to refocus a chirp-derived response at a chosen point, enough to make salt grains jump there, which is energy focusing rather than retuning.
Whether any treatment changed the object has to be settled by an independent low-amplitude measurement afterwards. See can sound change an object's resonant frequency?
In this project
The record-and-replay proposal uses these tools to test, not assume, whether replay changes anything. The full article covers the replay chain, the confounds and the proposed controls. At the project's proposed band above 230 kHz, a sweep-based measurement needs megasample-per-second acquisition and a shared reference clock. No measurement has been made.
Frequently asked questions
What is an exponential sine sweep?
A sine tone whose frequency rises exponentially with time, so each octave takes the same time. Recorded through a system and deconvolved, it yields the impulse response and separates harmonic distortion from the linear response.
Can an impulse response change an object's resonance?
No. An impulse response describes how the object behaves. Convolving with it, or synthesising from it, recreates that behaviour but does not alter the object that produced it.
Sources
- Advancements in impulse response measurements by sine sweeps, Angelo Farina, 122nd Audio Engineering Society Convention, Vienna (2007).
- Time reversal focusing of elastic waves in plates for an educational demonstration, Heaton, Anderson and Young, J. Acoust. Soc. Am. 141(2) (2017).