# What type of test signal is appropriate for frequency response analysis of string instruments?

I want to test the frequency response of stringed musical instruments . My test set-up includes a piezo film transducer which is just taped onto the instrument and fed a signal which then vibrates the instrument. The resulting sound is picked up by a calibrated microphone and sent to a spectrum analyzer. The strings are damped with memory foam so their vibrations are damped.

I would like to know if it matters what kind of test signal I send to piezo element. Should I use single frequencies, a sweep or "chirp" signal , or a broadband noise signal such as pink or white noise.

• Depends how good your analysis tools are. I would recommend a series of single-freq. sine waves so you can see how much the instrument builds up energy at resonant frequencies. A fast chirp might "miss" some responses. OTOH, a square-wave is often used when analyzing speaker response, as the output will show the relative response at all wavelengths comprising the square wave. – Carl Witthoft Dec 30 '15 at 21:21
• What about white noise, there is nothing to "miss" because the noise provides a continuous stimulus for any and all resonances ? – William Hird Jan 2 '16 at 19:22
• @willliam herd white noise is also defined as being fully random (in frequency and amplitude), so you don't know what you're putting into the system. A clean square wave, by comparison, has a well-defined Fourier decomposition. – Carl Witthoft Jan 2 '16 at 19:57
• @CarlWitthoft, Then why is white noise used to measure the frequency response of loudspeakers if , as you say, "you don't know what your putting into the system? – William Hird Jan 3 '16 at 0:48

First, can you really actuate white noise? If yes, White noise would probably show many natural frequencies and harmonics. However, make sure the actuation bandwidth is sufficient for your frequency range.

If the piezo can be adjusted to your frequencies of interest, a more deterministic approach would be to do a frequency sweep. For that, start from the lowest frequency and increase the frequency towards the highest frequency that the piezo permits. This would give the complete frequency response for the instrument (its strings, structure, etc).

You would be able to capture the frequencies of the strings (all of them), the instrument structure, and maybe even the objects in the room, so a non-echo room would be an advantage.

• I should have mentioned that the strings are damped so they don't vibrate, and the piezo actuator cannot be tuned, its frequency resonance is way above audio frequencies so it doesn't come into play when taking actual measurements. Your response unfortunately doesn't really answer my question though. What are the intricacies of using various stimuli , single frequency, swept frequency, chirp, or noise , in testing stringed instruments? – William Hird Dec 29 '15 at 19:53
• @WilliamHird, your explanations deserve being in the question. a "stringed musical instrument" directly made me jump to the frequency response of the strings and the instrument (structure). The tuning I've mentioned is a frequency sweep in fact. and by "tuning the piezo" I meant to adjust the frequency of actuation.. I'll clarify that part. – Gürkan Çetin Dec 29 '15 at 20:28

It is helpful to be clear about what you are trying to measure. The frequency response (aka transfer function) of the string will be given by $$T(f)=\frac{O(f)}{I(f)},$$ where $T(f)$ is the frequency response, $I(f)$ is the input signal, and $O(f)$ is the output signal and all three are a function of frequency. In order to have a meaningful frequency response function, it is important that $I(f)$ and $O(f)$ have meaningful units, which is achieved through proper calibration of your drive and sensor.

One difficulty in your situation is that you don't have a good measurement of $I(f)$, so you have to make some assumptions about it's responsivity to your electrical drive. Since audio frequencies are much lower than typical piezo resonances, you can probably assume that the frequency response of the piezo element has a flat amplitude, but the phase may start to roll off at higher audio frequencies. If your goal is simply to measure the amplitude of the $T(f)$ and ignore the phase, then you should be safe simply driving the piezo with white noise and taking the $O(f)$ to be the amplitude of the frequency response function.

In a perfect world where both your sensor and transducer have no noise and no saturation, then a quick white noise measurement is enough to fully measure the frequency response of any system. In the real world, you have to be careful of the following (I'm sure I'm missing some):

• Saturation of the peizo
• Saturation of the microphone
• Saturation of the electronics driving the piezo
• Saturation of the electronics reading the microphone
• Producing a sound which is below the background noise
• Producing a sound which is below the noise level of the microphone.
• Nonlinearities in the instrument causing response at frequencies other than the driving frequency.

Some example reasons that you might want to use a signal other than white noise:

• Measuring at discrete frequencies (swept sine) allows you to make higher SNR measurements at the specific frequencies which you are interested in because you can integrate for longer.
• If the response is very low at some of the frequencies you are interested in, then your sensor may not pick up the sound. In this case, you may want to drive harder in certain frequency bands (pink noise).
• On the other hand, if the response is very strong at certain frequencies, it may drive your sensor into the nonlinear regime which will destroy the measurement. In this case you may want to drive less in certain frequency bands (pink noise).
• Wouldn't it help if you just place the piezo next to the microphone and measure its frf, which can then be used to eliminate its effects on the measured frf of total system with the music instrument, assuming that there everything acts approximately linear. Such that you can find the dynamics of only the instrument. – fibonatic Dec 30 '15 at 4:56
• @fibonatic That might work. I'm not sure if the coupling of the bare piezo to the air is strong enough to produce a measurable sound though, especially at low frequencies. – Chris Mueller Dec 30 '15 at 12:40
• @fibonatic - You need to determine the output of the transducer regardless of the method you choose because, as Chris describes, you need to know the input to get a meaningful output. You should notice also that where you place the transducer matters - if you put it on a resonant node then it won't excite that particular frequency. You are liable to get a new spectrum at each transducer mounting position because of the complexity of the shape of the instrument. – Chuck Dec 30 '15 at 13:14
• @Chuck: Yes good point, that's why I put the transducer right under the left foot of the bridge, all the string vibrations have to go thru this point anyway so we are simulating the playing of the instrument as close as possible. – William Hird Dec 30 '15 at 20:11