10.4.2 Frequency response measurement: hints and tips

A: Philosophy

Making decent measurements is a skill of a similar kind to making a decent instrument. Doing it by recipe is like making a violin from instructions in a book — it will work, but it probably won’t be great. Some people have more natural talent than others, but all beginners at instrument making or acoustical measurement benefit from occasional hands-on guidance from someone with more experience. In the case of measurements the issue is not so much “which buttons to press when”, but how to recognise quickly that something looks wrong, then concentrate on fixing that rather than wasting time collecting a lot of misleading data.

As was emphasised in section 10.4B, you should always ask yourself the key questions “What do I need to know?” and “How well do I need to know it?” before starting any test project. Asking these questions can greatly simplify the selection of a test method. For example, a test which only requires tracking of resonant frequency changes or relative level changes is much simpler than one that involves absolute admittance levels, which requires a fully calibrated force and velocity measurement system.

B: Digital choices

You are no doubt using some kind of FFT package, which can sample your signals and turn them into frequency spectra. There are a couple of choices you will need to make before you start measuring: the sampling rate, and the length of time to sample for. Both have important consequences.

The sampling rate governs the frequency range of your tests: your top frequency will be half the sampling rate, known as the Nyquist frequency. If the signal you are measuring contains significant energy at frequencies above the Nyquist frequency, you may get seriously misleading results because of a phenomenon called aliasing. Figure 1 shows an example. The black curve shows a sine wave with a frequency of 4.5 kHz. The red stars show what happens if you sample this sine wave with a sampling rate of 5 kHz. The Nyquist frequency is then 2.5 kHz, lower than the frequency of our signal. You can see that something remarkable has happened: the red stars do indeed dot out a sine wave, but it is at the wrong frequency, a much lower frequency than it should be.

Figure 1. An illustration of aliasing. A sine wave at frequency 4.5 kHz is shown in black. The red stars are the result of sampling this sine wave at a frequency of 5 kHz. They dot out a sine wave with the much lower frequency 500 Hz, which is 5000-4500 Hz: the sampling frequency minus the sine wave frequency.

Specifically, the frequency marked by the red stars is 500 Hz, which is the difference between the sampling rate and the true frequency of the sine wave. This aliasing effect is not just a mathematical artefact: you can hear it. Sound 1 is a sine wave which sweeps up in frequency from zero to 7 kHz. Sound 2 is the result of sampling this same sine wave sweep at a sampling rate of 5 kHz. The note rises exactly as before until it reaches the Nyquist frequency, 2.5 kHz. It then turns round and goes back down again! Eventually this aliased sound reaches zero frequency (when the true frequency reaches the sampling rate). It then turns round again and starts rising. The case plotted in Fig. 1 occurs shortly before this second reversal, when the sweep reaches 5 kHz and you hear a tone at 500 Hz.

Sound 1. A linear frequency sweep from zero to 7 kHz, sampled at 20 kHz
Sound 2. The same linear frequency sweep as in Sound 1, but sampled at 5 kHz so that the sound is aliased when it goes above the Nyquist frequency 2.5 kHz

You absolutely do not want aliasing to be going on in your measurement! Frequency peaks will appear in entirely the wrong places. So you must choose a sampling rate high enough that all the frequencies making up your actual signal lie below the Nyquist frequency. But how do you know, before you have started making the measurement? There is a simple trick. Make a guess, and do a measurement at that sampling rate. Save it. Now increase the sampling rate — for example, you might double it. Repeat the same measurement, and compare the spectrum with the one you saved. Do they look different? Does the new spectrum show any significant output in the frequency range which is above the Nyquist frequency of the first measurement? If either answer is “yes”, the original sampling rate was too low. Now double the rate again, and repeat. By exploring in this way, you can quite rapidly find an acceptable sampling rate.

Next, you have to decide how long to collect data for, or equivalently how many samples to collect. This sampling time governs the frequency resolution of your results, in a very straightforward way: the step between frequency points will be the inverse of your sampling time. So if you collect data for 1 s your resolution will be 1 Hz, with 2 s it will be 0.5 Hz, and so on. Figure 2 shows a typical hammer-tap response of an instrument body (a banjo in this particular example). You can see that the response has decayed to low levels by 0.5 s, and if this is the complete data sample, the resolution will be 2 Hz.

Figure 2. The response from an accelerometer attached to the bridge of a banjo, following a hammer tap.

Now you have to consider a trade-off. Higher resolution seems desirable because it leads to smoother plots, but it requires you to store a larger amount of data, and more insidiously there are implications for the amount of noise included in your signal. Figure 2 shows why. If you collected data for 2 s, in order to increase your resolution of 0.5 Hz, you would be including an extra 1.5 s in which the real signal was essentially zero. But the actual measured signal will never be zero, because there will always be some level of noise — both physical noise being picked up by your sensor, and electrical noise from your instrumentation. So the price of the increased resolution will be extra noise.

A standard fix for this is called “zero padding”: you set a threshold level below which you think the signal will be dominated by noise, and whenever the measured signal falls below that threshold, you replace the measured values by zero. This trick is particularly important for the force signal from an impulse hammer. The real force is only non-zero for the short time when the hammer is in contact with the instrument. Outside that time the force must be zero, so it makes sense to detect that contact time via a threshold, and then zero out all the rest of the signal before you do your FFT to deduce the spectrum of the hammer force.

Returning to Fig. 2, it is easy to see that there is also a minimum time for which you should collect the data. For this particular example, suppose you only collected 0.1 s of the signal. You would miss quite a significant part of the decay. This lost data will distort the frequency spectrum, as well as giving you a very low resolution of 10 Hz. You need to collect enough data to catch “most” of the decay, although the exact meaning of “most” will depend on the actual noise level in your measurement. But as a rule of thumb, the amount plotted in Fig. 2 feels about right for the minimum sampling time.

C: What could possibly go wrong?

C.1: Support fixtures

There are other things you need to think about when designing your experiment. You probably need to support the instrument you are testing in some way. All support fixtures are likely to change the behaviour to a greater or lesser extent, and you should obviously seek to minimise that influence. The most realistic, of course, is to do a measurement while a player holds the instrument in the normal manner. But that is impractical for some measurements, and in any case it will introduce a source of variability of you want to compare results for many instruments, collected over a long period of time.

So usually people design some kind of supporting fixture. You can try to make something that mimics a player’s hold: perhaps a foam-lined support around the neck where a hand might hold it, perhaps some kind of soft grip on the chinrest of a violin… The main thing is to be alert to the issue — ideally you should try to play the instrument, at least a little, while it is in the rig and with any attached instrumentation in place. Does it sound normal, or is it muted or muffled? Is something buzzing or rattling? An example of a suitable holding arrangement is shown in Fig. 3. This treble viol is resting on soft foam on its end block, and is supported by a foam-lined “hand” on the neck.

Figure 3. A treble viol being tested. Note the holding arrangement and the card for string damping

There is one further issue about support structures if you want to use the wire-break method of testing. You don’t want the instrument to fall over when you pull the wire to break it! So for that kind of testing you need a rather firm support. If an instrument is being held by a player in the usual way, that generally gives a sufficiently firm support. But artificial support frames need to be a bit more robust than the one seen in Fig. 3. For example, the lower bouts of this viol could be held between foam-lined side supports, mirroring the way a player of this instrument would grip the body with their legs.

C.2: String damping

If you are testing a stringed instrument, you need to think carefully about whether you want the strings to be damped or not. It all depends on what you want to do with your measured frequency response. There is, of course, an argument in favour of leaving strings undamped, since that is what happens to the non-played strings of an instrument during normal performance. So if your aim if to characterise the sound of the instrument, in some sense, perhaps you want to include the contribution from the ringing strings.

However, for other purposes you definitely do not want undamped strings: they are at best a nuisance and a distraction, and at worst they invalidate what you are trying to do. Think back to the synthesised guitar and banjo sounds presented in earlier chapters. Many of those used measured body response, but the strings were added in the computer program. This allowed variations in string choice and playing details to be simulated, while keeping the body response constant. In other cases, variations were made in the body response while keeping the strings and playing details the same. None of that would have been possible if the body measurement already included the effects of coupling to undamped strings.

More commonly, you do not want undamped strings in a frequency response measurement because they produce extra narrow peaks and dips in the response, determined by the resonances of the strings. Figure 4 shows an example of the bridge admittance of a violin, with and without string damping. If you want to identify peaks as body modes, and perhaps to do modal analysis (which we will come to in section 10.5), these extra peaks are a distraction: notice how the string modes have had a particularly strong effect on the low-frequency “signature modes”. Furthermore, if you want to compare your measurements with ones done by other people, the comparison is cleaner and simpler without string effects: the other instruments may have been fitted with different strings, they may not have been tuned the same, and so on.

Figure 4. Measured bridge admittance of a violin, with strings damped (blue) and undamped (red)

However, it is important that the strings are in place and under normal tension during a measurement. The violin gives a particularly clear example here. The bridge is held in place by the strings, and the static stresses produced by the string tension also influence other aspects of the behaviour. The tightness of the soundpost would be different without string tension, for example. Also, the geometry of the violin means that the longitudinal stiffness of the strings is a contributing factor to the vibration modes of the body. We met some consequences of this kind of influence of the strings when we discussed the banjo in section 5.5 — the banjo shares with the violin the arrangement of a “floating” bridge and a tailpiece to anchor the strings.

If you damp the strings, you need to do it carefully. If you were to do it by putting foam between the strings and the fingerboard, for example, you would be adding damping to the body modes as well as damping the strings. A method that often works very effectively is to weave a piece of paper or light card (like a business card) over and under the strings. This can be done without making contact with the fingerboard or body, and it can produce very effective damping via frictional rubbing of the strings against the card. An example can be seen in Fig. 3. A useful check is to strum the strings with the card in place — if you can still hear a pitch associated with a string frequency, the damping is not enough.

C.3: Checks

There are some things you can do to check that a measured frequency response looks reliable. The first is something we mentioned back in section 5.1.1. If you are doing a hammer test and your software package allows it, you should repeat each test several times, and calculate an average response. This will reduce the effects of noise, and it also gives you the coherence function, which is an indication of how well your measurements are behaving. Coherence close to 1 (or 0 dB if you are looking at plots with a decibel scale) indicates that your measurement passes at least some of the tests for reliability.

The coherence plot often gives a very direct indication of the bandwidth over which your frequency response can be trusted. At high frequencies your hammer taps will not be putting very much energy into the instrument, and this will lead to an increase in noise and a reduction in coherence. Figure 5 shows an example. In this case, the coherence suggests that the measured bridge admittance is reliable up to about 6 kHz.

Figure 5. The bridge admittance of a violin (blue) with the associated coherence function (red)

There are two other checks that can be made if your measurement is of a structural response, such as admittance. If you are doing a driving-point measurement (tapping essentially at the same position as the sensor) then the frequency response function has an important property which can be checked if your software package allows you to view the phase response, or else complex version of the function. Figure 6 illustrates this property for an example of the driving-point admittance at the bridge of a violin. It shows two alternative views of the same response: on the left the real and imaginary parts of the complex admittance are plotted, while on the right the phase angle is plotted.

Notice that the real part of the admittance (blue curve on the left) is always positive, while the phase angle always stays within the range between $-90^\circ$ and $+90^\circ$. These both follow from the requirement that if you apply a force to the violin at the measurement point, energy must always flow into the violin, never out of it. Any measurement of a driving-point admittance should show this behaviour, and it is a useful thing to check. One thing you may find is that the real part of your admittance is always negative, rather than always positive. This is simply an indication that when you come to calibrate your measuring system (see subsection D) you will need a negative calibration factor to correct the error.

If your measurement is of accelerance rather than admittance (i.e. you are measuring acceleration rather than velocity in response to a hammer tap), there is an equivalent property but the details are slightly different. It is now the imaginary part, not the real part, which must always be positive. The equivalent condition on the phase angle is that it must lie in the range between $0^\circ$ and $180^\circ$.

On the other hand, if your tapping position is not the same as your sensor position then you can do a different check. There is a very general reciprocal theorem, which tells us that if you swap the positions of tapping and measurement, the frequency response should be exactly the same. So do the measurement you planned, then move your sensor to where you were tapping, and tap with your hammer at the original sensor position. Compare the results of the two measurements: they should be identical (within the accuracy limits of the measurement you are doing).

D: Reproducibility

All measurements have noise and accuracy limits. To avoid over-interpreting your measurements, and as a way of getting to know your particular measurement rig and its quirks, it is really important to go through something like the checklist given below. The important thing is the idea and the mindset, rather than the exact numbers of how many of each kind of test you do. But don’t shortcut this learning and checking stage, the time it takes will be amply repaid in mistakes avoided later.

D.1: Repeat tests

Set your rig up for the measurement you intend. But don’t just do the test once, repeat the measurement several times. Display the results on top of each other — on the screen if your software allows it, but if necessary print the plots out, stack them and hold them up to a window. Look carefully at the detailed comparison. How different are they? Close enough for what you wanted to know, or would you like the reproducibility to be better? Is the difference more or less uniform? Perhaps it is different in different frequency ranges, or the peaks are more reproducible than the dips, or there is some other pattern. Patterns are always trying to tell you something….so think about it. Can you guess what kind of thing might cause this pattern? Is it something you could tighten up in your procedure? Try it!

You don’t have to go through this rigmarole every time you make a test, but you should do it occasionally — more often when you are learning, less often as you gain experience.

In this repeat-testing process, think about the limits on the resolution of the measurement. If the frequency step is 1.5 Hz then looking for changes smaller than about 6 Hz will be problematic. Set the FFT analyzer so that it can clearly resolve the differences that are being measured, to the best of the analyzer’s capabilities. This is no different than selecting a scale for weighing things: if the scale only reads out in tenths of a gram, weighing items that are on the order of a half of gram or less will have poor resolution.

D.2: Deliberate variations

Now do a deliberate experiment about variability. Keep the same test instrument (or wood sample, or whatever you are trying to measure), and repeat the test with “irrelevant” things deliberately varied. The first thing to do is to simply take the instrument (or whatever) out of the rig, put everything away, and then take it out again as if it was another day’s testing, and try to put it all back together exactly as you had it before. The next thing to do is to move the rig around in the room, or to a different room. Move the furniture around a bit. Choose days with different temperature and humidity. Anything else you can think of. For all these tests, ask the same kind of questions as before. How repeatable are the results? Are there patterns to the variation?

Now you will have gained some experience about how far you can trust your measurement, and in the course of doing that you will also have got better at controlling what you are doing. Just keep thinking of the analogy with instrument making, and remember your first instrument: you were proud of it at the time, but many established makers would like to get that first instrument back and destroy it (or at least take their label out). Your first measurements may be similar. There is no shame in that, everyone is the same.

The caution can be extended: when you make your second instrument the “it really shouldn’t take this long” voice is heard in the back of your mind. As it in many things in life, and for measurements in particular, ‘haste make waste’.

A good idea for a final step in checking out your test rig and data analysis processes before you get down to serious measurement is to try to reproduce the results of a known test. This is best done as a non-destructive test such as adding lumps of mass to the instrument. Look around for a simple experiment someone else has done and see if you can get comparable results. The violin in your rig is different from the one tested by the previous author, so what you want to see is a similar trend, and broadly similar numbers.

D.3: Real tests

Now get back to what you were trying to do. Probably you wanted to measure differences between instruments, or changes after some procedure. Are the differences you measure bigger than the ones you saw in these reproducibility tests? If they are, you (probably) have meaningful data. If not, you may still have meaningful data, but it isn’t obvious. Look for the patterns again — does the deliberate variation produce a different pattern from the ones you saw before, or does it look suspiciously similar to the effect of the weather changing or moving the furniture?

Hypothesis testing is the best way to explore the data. A hypothesis is based on your current understanding and expectations. A hypothesis is a guess, hopefully an educated guess. It maybe the null hypothesis “If I do this nothing will change” or based on conjecture “If I block one f-hole A0 will change by a fifth”. The hypothesis may be right or wrong. If the hypothesis is correct you most likely understand how the system is working (great!). If the hypothesis turns out to be incorrect, what you learn from the experiment will help you pose a new hypothesis. After you have checked the hypothesis, look around — did something else change in a pattern? For example, in most cases changing something that moves a low frequency mode will also change the system at high frequencies.

E. Calibration

You have now learned to make measurements that you understand and trust, at least to an extent. Now you may want to compare with someone else’s measurements of what is supposed to be the same thing. Maybe something published, or maybe you are exchanging results with a friend with a similar test rig. The first thing you need is some kind of calibration procedure, to fix the absolute level of your plots so that it makes sense to compare. The essence of any calibration procedure is to apply your measurement procedure exactly as you were doing it, with no changes whatever to measure something that you already know the answer to. You compare what you get with what you expect. If all is well, the pattern should look right, but you need to apply a scaling factor to your measurements to convert to the right answer. This is the calibration factor for your rig, write it down and apply exactly the same factor to all measurements made with the same rig and the same settings.

If you change a setting by twiddling a knob or switching a range switch, you should repeat the calibration experiment — don’t trust the manufacturer to have marked the dial accurately enough, check it out yourself. Any instrumentation rig that has ‘knobs’ which vary the gain continuously (as opposed to switching between preset ranges) will be a constant source of problems with maintaining calibration. One approach is to arrange to work with the knob right at the top of the range, which is something you can reliably repeat every time. Unfortunately top of the range settings can have other consequences related to signal limiting or signal distortion. The best advice to work around the ‘knob’ problem is find a good safe setting, then tape over the knob securely, do your calibration measurement in that condition, and never touch the knob again!

E.1: Absolute and relative calibration

There are two levels of calibration you might do. The “gold standard” is an absolute calibration, which turns your results into internationally agreed units: for example admittance (or mobility) in N s/m (or s/kg, which is the same thing although it looks different). If you can easily do such a calibration, as you can for admittance, then it is obviously the best thing. For many measurements, especially for sound pressure measured by a microphone, it is not so easy to do an absolute calibration using reasonably cheap equipment. Then you have to think what you are really trying to do. Do you want to express your radiated sound power in absolute terms (in milliwatts, for instance)? Then you need to bite the bullet and find a way to get an absolute calibration. But perhaps you just want to be able to compare measured sound levels with someone else’s results from a similar rig, being sure that differences are correctly captured. Then all you really need is the relative calibration factor which would convert your friend’s results onto the same scale as yours.

E.2: How do you do it?

There is a choice of two approaches: calibrate each part of the system separately, or do an overall calibration. The second approach is usually much easier. With enough effort, you can calibrate each separate piece of equipment — for example accelerometer, hammer, conditioning amplifiers, computer card, and software FFT algorithm — then combine all these factors to give the overall factor for your frequency response function measurement. Some of these factors may be supplied by manufacturers, but the others may take some effort to determine. The chain is no stronger than its weakest link, so this approach is quite error-prone if you miss a step out or get one thing wrong.

Usually, it is much simpler to do a single calibration for your entire frequency response function in one go. For an absolute calibration, you need to find a system for which the particular property you are measuring is already known — either because of some fundamental law of nature, or because someone else has already calibrated it using some standard test. For admittance, you can use Newton’s law of motion “force equals mass times acceleration”. If you measure a system which is essentially just a mass moving in a straight line, then the acceleration per unit force is simply the inverse of the mass, which you can determine with a weighing scale (but remember that the standard unit of mass is the kilogram, not the gram, so express the mass in kilograms!). In practice your mass might be suspended on strings as a pendulum, or supported on soft foam. You have to be careful that your hammer tap really does just make it move in a more-or-less straight line, without any significant rotation or cross-direction motion. You must also remember to choose a mass which is convenient to measure using the same settings of gain etc. that you use for your actual measurements. But then with care you can obtain an absolute calibration and compare your results with, for example, published ones by Erik Jansson or George Bissinger. (To compare with Jansson measurements you need to read the small print — his plots are not given in calibrated units, but he gives the required calibration correction in the text.)

For a relative calibration of a hammer-to-microphone measurement, for example, you need to choose some kind of stable and easily reproducible system which makes a noise when tapped. By agreeing to tap in a given position and place your microphone at certain place, you should get the same answer every time (but check it out by the approach of section 2). Once that is verified, then the same structure or a careful copy of it can be tested by your friend. You should obtain matching frequency response plots, except for a scale factor (i.e. a shift on a dB scale). That scale factor is what you need to convert your measurements ready to be compared.

Pay careful attention in two-channel measurements where one spectrum is being divided by another — it is easy to get the input and output cables swapped. Mobility (velocity over force, admittance) is quite different from impedance (force over velocity). Whereas mobility peaks at resonances, impedance peaks at anti-resonances. If the signature modes have suddenly jumped when you do a measurement on your standard test article, check the channel assignments: another reason to add that standard test article to your test rig kit bag.

A final note: with all these calibration factors, of any kind, it is fatally easy to multiply by the factor when you meant to divide, or vice versa. You need to be awake when you think about how to do this! It always sounds obvious when someone explains, but this is a very common mistake.