# Advice on calibrating complex stimuli that vary in amplitude and also decay over time

So I am finding it difficult to calibrate for a particular signal that changes in amplitude over time, and is randomly presented to a listener over a time span. I have attached a picture of the waveform for reference. Basically it sounds like wind chimes.

There is about a 15 dB difference between the largest peak of the most intense presentations and the largest peak of a less intense stimulus presentation. Additionally, the stimuli decay about 2-3 dB over the duration of each signal presentation (ranging from 1-2s).

Using our Larson Davis 831 SLM or our Fluke 45 volt meter for a stimuli like this is not accurately capturing one value that represents the signal amplitude very well.

I am presenting everything else for the study (pure tones, narrow bands of noise, broad bands of noise) at 55 dB SPL at a fixed frequency, and those stimuli are easy to calibrate, but this signal is proving very difficult to capture and assign an output voltage, as the signals change in amplitude during each presentation, and over time. Not to mention the frequency composition of each sound is different.

Basically, though the RMS of a pure tone and the complex stimulus are close to equal, the loudness of the chimes is much greater than for the pure tone.

Any suggestions on how to calibrate this complex stimulus?

• Can you calibrate using people? Play two signals and ask which is louder. Adjust the amplitudes so they are closer together. Play them again in random order. Or play two other randomly chosen signals. There must be descriptions of this process somewhere (such as for studies of loudness curves). Aug 24 '17 at 17:42
• I have to ask... calibrate to what? Calibration implies some pre-determined standard, which you haven't mentioned. 55dB SPL isn't a deterministic factor if what you're aiming for is 'equal [or unsurprising] apparent loudness', especially with those transient peaks at the start of your sounds. I'm with Dale... get some ears involved. Aug 24 '17 at 18:31
• Have you considered something like integrated LUFS? I might be misunderstanding the question, but it's much better than simple RMS for human loudness purposes. Aug 25 '17 at 15:42

I don't know if there's an instrument that can measure this but you can calculate the RMS value of complex waveforms in many ways. It would include some maths so I would personally recommend using something like Scilab for the calculations

From the wikipedia article on RMS (root mean square):

...the root mean square (abbreviated RMS or rms) is defined as the square root of mean square (the arithmetic mean of the squares of a set of numbers).3 The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle.

So what you need to do in your case (in the discrete or digital time domain)is to square each sample value, add them together, divide the sum by the number of samples and work out its square root. In the frequency domain, (from the same source):

Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).

If you already know the power of each frequency component from FFT, you can square it, add them together and find the square root of the result. In the continuous time domain, first you square the waveform so that the negative values don't cancel out the positive ones, the you integrate the result (find the area under the curve) and then divide by the amount of time passed (a second). And finally you work out the square root of the integration. Please note that although I've studied this, I haven't used it in practice for over a decade so please excuse any inaccuracies or omissions. If there's any, let me know in the comments and I will edit my answer.

P.S. I'm not sure I understand what you mean by calibrating. If you mean to set an output level so you don't exceed the maximum of a DAC, then you only need to look at maximum values.

• Thanks so much! Good thought on the power of each frequency component. I actually did go and did compute the RMS, and scaled the complex tone to that of the RMS of a pure tone. Thanks for your help! Aug 25 '17 at 13:40