
The average power in a sound during a certain time range and in a certain frequency range, expressed in Pa^{2}/Hz.
The complex spectrum of a sound x(t) in the time range (t_{1}, t_{2}) is
X(f) ≡ ∫_{t1}^{t2} x(t) e^{2πift} dt 
for any frequency f in the twosided frequency domain (F, +F). If x(t) is expressed in units of Pascal, X(f) is expressed in units of Pa/Hz. In Praat, this complex spectrum is the quantity stored in a Spectrum.
From the complex spectrum we can compute the onesided power spectral density in Pa^{2}/Hz as
PSD(f) ≡ 2X(f)^{2} / (t_{2}  t_{1}) 
where the factor 2 is due to adding the contributions from positive and negative frequencies. In Praat, this power spectral density is the quantity stored in a Spectrogram.
The PSD divides up the total power of the sound. To see this, we integrate it over its entire onesided frequency domain (0, F):
∫_{0}^{F} PSD(f) df = ∫_{0}^{F} 2X(f)^{2}/(t_{2}t_{1}) df = 
= 1/(t_{2}t_{1}) ∫_{F}^{+F} X(f)^{2} df = 1/(t_{2}t_{1}) ∫_{t1}^{t2} x(t)^{2} dt 
where the last step uses Parceval's theorem. The result is precisely the average power of the sound in the time range (t_{1}, t_{2}).
It is often useful to express the power spectral density in dB relative to P_{ref} = 2·10^{5} Pa:
PSD_{dB}(f) = 10 log_{10} { PSD(f) / P_{ref}^{2} } 
Since the argument of the logarithm is in units of Hz^{1}, this spectral measure can loosely be said to be in units of `dB/Hz'. In Praat, this logarithmic power spectral density is the quantity stored in an Ltas; it is also the quantity shown in pictures of a Spectrum and a Spectrogram.
© ppgb, October 26, 2010