Calibration Math¶

This page collects every equation behind the calibrations cftscal runs. You don't need it to run a calibration, but it's the place to look when you want to check cftscal's numbers against your own, or work out why a measured value came out where it did.

Conventions¶

Two conventions matter before any of the equations make sense, because getting either one backwards flips a sign or inverts a ratio.

Sensitivity is expressed in V/Pa¶

Throughout these pages, sensitivity is expressed in \(\frac{V}{Pa}\) — volts per Pascal — which is the convention used throughout the technical literature and on most manufacturer datasheets.

This is the inverse of the EPL CFTS convention

The legacy EPL cochlear function test suite expresses sensitivity as \(\frac{Pa}{V}\). If you are comparing cftscal's numbers against EPL's, or reading older lab notes, one of the two is the reciprocal of the other (or, in dB, the negative). See Reading cftscal's reported numbers below for what each field in the GUI actually holds.

The word "sensitivity" also means two different physical things depending on which end of the chain you are at:

Device	Sensitivity means	Units
Microphone	The voltage the microphone generates in response to a given sound pressure.	\(\frac{V}{Pa}\)
Speaker	The sound pressure produced for a given drive voltage — or, equivalently, the drive voltage needed per Pascal of output.	\(\frac{Pa}{V}\) or \(\frac{V}{Pa}\)

Do the arithmetic in dB¶

Calibration measurements can involve very small numbers. A 20 µPa reference pressure, a microphone output of a few hundred microvolts, and a power spectral density value are all small enough that chaining several multiplications and divisions together in linear units can run into the limits of floating-point arithmetic.

What the floating-point problem actually is

A computer stores a number as a fixed number of significant digits plus an exponent — roughly 16 significant decimal digits for the double-precision floats NumPy uses by default. The exponent handles the scale fine, so a value like \(20 \times 10^{-6}\) is not itself a problem. What degrades is everything beyond those 16 digits, in three ways that calibration arithmetic runs into:

Rounding error compounds. Each multiplication or division rounds its result to the nearest representable value. Chain enough of them together — microphone voltage, divided by sensitivity, squared into a power, divided by a reference — and those tiny individual errors accumulate.
Subtracting two nearly-equal numbers destroys precision. If two quantities agree in their first 14 digits, their difference retains only the 2 digits where they differed, and the rest is noise. This is called catastrophic cancellation, and it is the most damaging of the three.
Summing values of wildly different magnitudes loses the small ones. Adding a very small number to a very large one can change nothing at all, because the result rounds straight back to the large value. Summing power across thousands of spectral bins is exactly this pattern.

Working in dB helps because it keeps every quantity in a narrow, well- conditioned range (tens to low hundreds) and replaces the long chains of multiplication and division with addition and subtraction, so there is far less opportunity for error to compound in the first place. It is a substantial improvement rather than a guarantee — dB arithmetic is still floating-point arithmetic, and a subtraction of two nearly-equal dB values can still cancel.

This is why almost every equation below is given in both a linear and a dB form, and why cftscal and psiaudio store sensitivity internally in dB. Prefer the dB form when you implement anything yourself.

The flat-response assumption¶

A measurement microphone is assumed to have the same sensitivity at every frequency, so a single number describes it. That assumption holds well for a precision microphone — if you spend enough money on the microphone, it is genuinely flat across its rated range.

It does not hold for a cheap microphone. Sometimes you only need to record audio during an experiment and a precision microphone is overkill; an inexpensive microphone's sensitivity will vary substantially as a function of frequency, so it has to be described by a curve rather than a single number. That is exactly the difference between the two microphone workspaces in cftscal:

Measurement Microphone Calibration produces a single flat sensitivity value (psiaudio's FlatCalibration).
Generic Microphone Calibration produces sensitivity as a function of frequency (psiaudio's InterpCalibration).

Symbols¶

Symbol	Meaning	Units
\(f\)	Frequency	Hz
\(O(f)\)	Acoustic output (sound pressure)	Pa
\(O_{dBSPL}\)	The same output, in dB SPL	dB re 20 µPa
\(S_{cal}\)	Sensitivity of the calibration (measurement) microphone	\(\frac{V}{Pa}\)
\(V_{cal}(f)\)	Voltage measured at the calibration microphone	\(V_{rms}\)
\(S_{exp}(f)\)	Sensitivity of the experiment microphone being calibrated	\(\frac{V}{Pa}\)
\(V_{exp}(f)\)	Voltage measured at the experiment microphone	\(V_{rms}\)
\(S_{PT}(f)\)	Sensitivity of a probe-tube (starship) microphone	\(\frac{V}{Pa}\)
\(V_{PT}(f)\)	Voltage measured at the probe-tube microphone	\(V_{rms}\)
\(S_{s}(f)\)	Speaker sensitivity, as drive voltage required per Pascal	\(\frac{V}{Pa}\)
\(V_{DAC}(f)\)	Voltage generated by the digital-to-analog converter	\(V_{rms}\)

A subscript \(dB\) on any sensitivity (e.g. \(S_{PT_{dB}}\)) means \(20 \times log_{10}\) of that value.

Reading \(f\)

A quantity written as a function of frequency — \(S_{s}(f)\), \(V_{cal}(f)\) — is one that takes a different value at every frequency, as opposed to a single number like \(S_{cal}\). So read \(S_{s}(f)\) as simply "the speaker's sensitivity at this frequency".

\(f\) is ordinary frequency, in hertz, which is what you handle everywhere in practice: cftscal's plot axes, the frequency column of a saved chirp_sens.csv or golay_sens.csv, and psiaudio's frequency arguments are all in Hz.

Much of the acoustics literature states these same relationships in terms of angular frequency \(\omega\), in radians per second, where \(\omega = 2 \pi f\). The two notations describe identical quantities; only the label on the axis differs. Nothing on this page needs the \(2 \pi\), because frequency appears here purely as the argument identifying which frequency a quantity is evaluated at. The factor does matter once you compute a phase from a frequency — see Chirps and frequency ramps.

Step 1: Measurement microphone sensitivity¶

The chain starts with a physical standard: a pistonphone, which generates a precise, stable sound pressure at a known frequency and level (e.g. 114 dB SPL at 1 kHz). Record the microphone's voltage while the pistonphone is running, and its sensitivity follows directly from the known pressure:

\[ S_{cal} = \frac{V_{cal}}{O_{pistonphone}} \]

where the pistonphone's rated level in dB SPL is converted to Pascals using the dB SPL conversions below. This is what Measurement Microphone Calibration computes.

Use a flattop window for this one measurement

Because the pistonphone is a single tone, this is the one calibration in cftscal where a flattop window is the right choice rather than a Hamming window. See Choosing a window.

Step 2: Speaker output¶

Once you have a microphone with a known sensitivity, you can measure what a speaker actually produces. Play a signal with a known RMS voltage, \(V_{DAC}(f)\), and measure the voltage at the calibration microphone, \(V_{cal}(f)\). The speaker's output in Pascals is:

\[ O(f) = \frac{V_{cal}(f)}{S_{cal}} \]

Or in dB:

\[ O_{dB}(f) = 20 \times log_{10}\left(\frac{V_{cal}(f)}{S_{cal}}\right) \]

\[ O_{dB}(f) = 20 \times log_{10}(V_{cal}(f)) - 20 \times log_{10}(S_{cal}) \]

Note how the division became a subtraction — that is the reason for working in dB.

Step 3: Experiment microphone sensitivity¶

To calibrate an experiment microphone (one that is not a precision measurement microphone), record its voltage \(V_{exp}(f)\) at the same time as you measure the speaker's output in step 2. Two microphones, co-located, recording the same stimulus. Since you know what the speaker actually produced, the experiment microphone's sensitivity follows:

\[ S_{exp}(f) = \frac{V_{exp}(f)}{O(f)} \]

Substituting \(O(f)\) from step 2:

\[ S_{exp}(f) = \frac{V_{exp}(f)}{\frac{V_{cal}(f)}{S_{cal}}} = \frac{V_{exp}(f) \times S_{cal}}{V_{cal}(f)} \]

The result is in \(\frac{V}{Pa}\). In dB, that is sensitivity as dB re 1 V/Pa:

\[ S_{exp_{dB}}(f) = 20 \times log_{10}(V_{exp}(f)) + 20 \times log_{10}(S_{cal}) - 20 \times log_{10}(V_{cal}(f)) \]

This is what Generic Microphone Calibration computes, and it is why that workspace requires the two microphones to be genuinely co-located — any difference in distance or angle from the speaker shows up in \(S_{exp}\) as if it were a property of the microphone.

Step 4: In-ear speaker calibration¶

The acoustics of the system change as soon as the experiment microphone is inserted into an ear: the ear canal presents a different acoustic load from a coupler — largely a compliance, set by the enclosed volume — which shifts the resonances of the system. This means the calibration has to be redone every time you reposition the experiment microphone while it is in an animal's ear. It is not a one-time measurement.

The quantity you need is the speaker transfer function \(S_{s}(f)\), in \(\frac{V_{rms}}{Pa}\), which tells you the voltage required to drive the speaker to a given level. Generate a stimulus through the DAC with known frequency content \(V_{DAC}(f)\), and measure the result with the probe-tube microphone whose sensitivity \(S_{PT}(f)\) you already know:

\[ O(f) = \frac{V_{PT}(f)}{S_{PT}(f)} \]

\[ S_{s}(f) = \frac{V_{DAC}(f)}{O(f)} = \frac{V_{DAC}(f) \times S_{PT}(f)}{V_{PT}(f)} \]

In dB:

\[ S_{s_{dB}}(f) = 20 \times log_{10}(V_{DAC}(f)) + 20 \times log_{10}(S_{PT}(f)) - 20 \times log_{10}(V_{PT}(f)) \]

\[ S_{s_{dB}}(f) = 20 \times log_{10}(V_{DAC}(f)) + S_{PT_{dB}}(f) - 20 \times log_{10}(V_{PT}(f)) \]

This is the calibration that Starship Calibration establishes against a coupler, and that Starship Check re-verifies in the ear.

dB SPL and Pascals¶

dB SPL is referenced to 20 µPa, the nominal threshold of human hearing:

\[ O_{dBSPL} = 20 \times log_{10}\left(\frac{O}{20 \times 10^{-6}}\right) \]

Solving for pressure in Pascals:

\[ O = 10^{\frac{O_{dBSPL}}{20}} \times 20 \times 10^{-6} \]

The constant \(20 \times log_{10}(20 \times 10^{-6}) \approx -93.98\) dB appears throughout the equations below; it is just this reference pressure expressed in dB.

Generating a tone at a specific level¶

Given the speaker sensitivity \(S_{s}(f)\), the voltage needed at the DAC to produce an output of \(O\) Pascals is simply:

\[ V_{DAC}(f) = S_{s}(f) \times O \]

In practice you want to specify the level in dB SPL rather than Pascals. Substituting the pressure conversion from above:

\[ V_{DAC}(f) = S_{s}(f) \times 10^{\frac{O_{dBSPL}}{20}} \times 20 \times 10^{-6} \]

Expressed in dB, the multiplications collapse into additions:

\[ V_{DAC_{dB}}(f) = 20 \times log_{10}(S_{s}(f)) + 20 \times log_{10}\left(10^{\frac{O_{dBSPL}}{20}}\right) + 20 \times log_{10}(20 \times 10^{-6}) \]

\[ V_{DAC_{dB}}(f) = S_{s_{dB}}(f) + O_{dBSPL} + 20 \times log_{10}(20 \times 10^{-6}) \]

That last form is the useful one, because it is written entirely in terms of quantities the calibration already produced. To actually drive the DAC, convert back to a linear voltage:

\[ V_{DAC}(f) = 10^{\frac{S_{s_{dB}}(f) + O_{dBSPL} + 20 \times log_{10}(20 \times 10^{-6})}{20}} \]

Estimating the output at a given voltage¶

Solving the same relationship for level instead of voltage answers the reverse question — "what will I get out if I drive the speaker at this voltage?":

\[ O_{dBSPL}(f) = 20 \times log_{10}(V_{DAC}) - S_{s_{dB}}(f) - 20 \times log_{10}(20 \times 10^{-6}) \]

Or, if you want the answer in Pascals:

\[ O(f) = \frac{V_{DAC}}{S_{s}(f)} \]

Quick reference¶

These are the two calculations you will use most often, both written in terms of the dB-domain sensitivities that cftscal stores.

Voltage required at the DAC for a given dB SPL:

\[ V_{DAC}(f) = 10^{\frac{S_{s_{dB}}(f) + O_{dBSPL} + 20 \times log_{10}(20 \times 10^{-6})}{20}} \]

Microphone voltage converted to dB SPL:

\[ O_{dBSPL} = 20 \times log_{10}(V_{PT}(f)) - S_{PT_{dB}}(f) - 20 \times log_{10}(20 \times 10^{-6}) \]

Sensitivity from a level measured at 1 V_rms. If you know the speaker produces \(O_{dBSPL}(f)\) when driven at 1 V_rms:

\[ S(f) = \left(10^{\frac{O_{dBSPL}(f)}{20}} \times 20 \times 10^{-6}\right)^{-1} \]

\[ S_{dB}(f) = -\left[O_{dBSPL}(f) + 20 \times log_{10}(20 \times 10^{-6})\right] \]

Referring a level measured at some other voltage back to 1 V_rms. If you measured the speaker's output as \(O_{dBSPL,x}(f)\) while driving it at \(x\) V_rms (say 10 V_rms) rather than 1 V_rms, the level referred to a 1 V_rms drive is:

\[ O_{dBSPL,1V}(f) = O_{dBSPL,x}(f) - 20 \times log_{10}(x) \]

So a measurement of 120 dB SPL at 10 V_rms corresponds to 100 dB SPL at 1 V_rms. This is exactly the correction psiaudio's calibration constructors apply through their vrms= argument (sensitivity = level - db(vrms)), and the same one used by the attenuation calculation in the legacy neurogen package.

This rescaling applies to the level, not to a V/Pa sensitivity

It is easy to state this as "converting a sensitivity measured at \(x\) V_rms to one at 1 V_rms", but that phrasing is wrong for the \(\frac{V}{Pa}\) sensitivity used elsewhere on this page. For a linear system, \(S_{s} = x / O\) comes out the same no matter what \(x\) you drove at — doubling the drive doubles the output, so the ratio is unchanged, and subtracting \(20 \times log_{10}(x)\) from it would be an error.

The quantity that genuinely needs the correction is the one psiaudio stores: dB SPL produced at 1 V_rms, which is defined relative to a specific drive voltage and so has to be referred back to 1 V if you measured at anything else.

Reading cftscal's reported numbers¶

The equations above use the \(\frac{V}{Pa}\) convention. cftscal and psiaudio report things in the units that are most convenient per device type, so here is the mapping.

Where	What is actually stored/shown	Convention
Measurement microphone list, Sens column	`mV/Pa`	\(\frac{V}{Pa}\), scaled to mV
Measurement microphone list, Sens (dB) column	`dB(mV/Pa)`	\(20 \times log_{10}\) of the mV/Pa value
Input Recording, Nominal sensor's mV/Pa field	`mV/Pa`	same as above; typed in by hand from a datasheet
Speaker / Starship / Generic Mic. sensitivity plots	The dB SPL produced at a 1 V_rms drive, per frequency	dB SPL per V_rms
psiaudio `FlatCalibration` / `InterpCalibration` internals	dB SPL at 1 V_rms	dB SPL per V_rms

The plotted curves and psiaudio's internal sensitivity are therefore in the \(\frac{Pa}{V}\) sense (how much output you get per volt), not the \(\frac{V}{Pa}\) sense used in the equations above. The two are related by the sensitivity-at-1-V_rms equation in the previous section:

\[ O_{dBSPL}(f) = -S_{dB}(f) - 20 \times log_{10}(20 \times 10^{-6}) \]

Worked example. A speaker plot reads 100 dB SPL at 1 V_rms for some frequency. In Pascals that is \(10^{100/20} \times 20 \times 10^{-6} = 2\) Pa, so \(S_{s} = 1\,V / 2\,Pa = 0.5\ \frac{V}{Pa}\), i.e. \(S_{s_{dB}} = -6.02\) dB re 1 V/Pa. Checking against the formula: \(-[100 + (-93.98)] = -6.02\). To then generate 80 dB SPL at that frequency, \(V_{DAC} = 0.5 \times 10^{80/20} \times 20 \times 10^{-6} = 0.1\) V_rms.

Gain is removed before any of this arithmetic runs

None of the equations above include preamp gain, because it has already been divided out by the time the analysis sees the recording. The Gain you enter in a Settings panel is passed through to the acquisition engine as the input channel's gain (in dB), and the engine scales the incoming samples down by it — in the sound-card engine, literally data / dbi(total_gain). Every voltage in the equations above is therefore referred to the microphone's own output, before the preamp.

That is why a gain value which does not match the physical hardware shifts a whole curve by exactly that many dB: the samples get divided by the wrong constant, and a constant scale error in volts is a constant offset in dB. A mismatch of 20 dB moves the reported sensitivity by 20 dB, no more and no less, at every frequency equally.