I have a measurement performed using GRAS 46AE microphone calibrated at 410mVrms at 114dB (250Hz).
I have a .wav file recorded and performed stft and obtained the magnitude. Now I would like to scale my signal and obtain my power spectra in dB SPL (the level I measured). How do I incorporate the scaling in the dB calculation?
Power = 10*log10((magnitude^2)/?)
Because you are interested in recording a .wav file and calibrating that frequency spectrum back to the SPL, there is a chain of gain factors you would need to deal with:
Your choices are either to chase each of these down one by one, or to find a way to calibrate from a known input to a measured output, the Fourier spectrum of your .wav file. I think you are better off trying to calibrate.
Calibration
You have a great start by knowing the microphone sensitivity of 410 mV RMS for 114 dB SPL. This spares you from having to create a sound of known SPL -- you can just measure voltage and make the final connection to SPL with this calibration factor. But what you need now is to calibrate the entire chain that includes all those other gain factors, so that you know that some number on your Fourier Transform spectrum means some SPL contribution at that frequency.
If you play a sine wave tone out a speaker, you can then pick it up with the mic and measure the RMS on your oscilloscope (or take the amplitude and divide by the square root of 2). Probably the easiest way to generate this is to find an online tone generator and play it out of your phone, tablet, or computer. Since you know the sensitivity of your microphone, 410 mV for 114 dB SPL, you can use that to make the final conversion from mV to SPL. But first you need to know how many mV corresponds to a particular value in your calculated Fourier spectrum of your .wav file.
If you acquire this sine wave signal, record the .wav file, and run it through your Fourier transform, you should see a single spike at one frequency. Knowing the RMS voltage that created this (and hence the SPL), you can then come up with the calibration factor you are looking for. The amplitude of that spike corresponds to the SPL that your microphone picked up, which you know because you measured the RMS voltage on an oscilloscope and used your known microphone sensitivity.
Now, due to details of how one properly takes the Fourier transform of a signal over finite time, you might actually see a main peak and a little bit of signal in the neighboring frequency bin of your digital Fourier Transform. This is due to the inherently finite frequency resolution of the signal spectrum measurement, since you are not sampling for infinite time. If this happens, simply square those amplitudes, sum them, and take the square root, and use that as your amplitude. This is effectively just adding up the total power in those bins and attributing it to your original sine wave that created it.
If you take your Fourier spectrum \$F(\nu)\$ and calculate \$20log(|F(\nu)|)\$, then calculate what you need to add to this to get your sine wave peak to have the numerical value equal to the SPL that your microphone/oscilloscope measurement is telling you, you will have calculated the necessary calibration factor.
One final note: the only reason you can get away with this single-frequency measurement is because your microphone has such a flat spectral response:
You have the known sensitivity of the microphone: 410 mv for a SPL of 114 dB. You also need to know 2 other parameters: 1) the gain between your microphone and the measuring device and 2) the measured voltage. As an example, assume the gain is X10 and the measured voltage is 100 mv. That means that the voltage out of the microphone is 100/10 = 10 mv. Next calculate 10/410 = 0.0244 and convert to dB: 20log(0.0244) = -32 Db. Finally take 114 + (-32) = 82 dB SPL. You should perform the same calculations with your actual data.
The SPL reference of 0dB is the threshold of hearing.
The 0dB reference for voltage depends on the reference either Peak or RMS. One cannot assume either without checking, but every application I have used, uses peak. ( although some simple meters measure peak or average and scale to RMS)
Z- or ZERO frequency-weighting was introduced in the International Standard IEC 61672 in 2003 and was intended to replace the "Flat" or "Linear" frequency weighting often fitted by manufacturers
The definition of dB SPL is the 20 log of the ratio between the measured sound pressure level and the reference point is defined as 0.000002 Newtons per square meter, the threshold of hearing.
However, the threshold of hearing (and sensitivity to level) changes by frequency and for soft and loud sounds, as discovered by Fletcher and Munson in 1933, so different SPL reference filters are used for industrial uses, unlike flat spectrum SPL for recording high quality audio.
For industrial uses there are A,B,C,D frequency response curves for different purposes other than here.
The Z- or ZERO frequency-weighting was introduced in the International Standard IEC 61672 in 2003 and was intended to replace the "Flat" or "Linear" frequency weighting often fitted by manufacturers.
The definition of dB in voltage is also 20 log or ratio to any reference, then the units indicate the reference.
E.g. 1V = 0 dBV = 60 dBmV = 120 dBuV
In Spectrum Analyzers , S.A. and Audacity, the reference for 1V is peak sine. Others might use RMS but peak amplitude is more often used for VU meters and SA’s.
If 114 dBA @250Hz = 410 mV(RMS)= -7.74 dBV = 52.26 dBmV = 112 dBuV
If you wanted to use the peak reference for 1V then you add 3.01 dB.
