power spectral density

The average power in a sound during a certain time range and in a certain frequency range, expressed in Pa2/Hz.

### Mathematical definition

The complex spectrum of a sound x(t) in the time range (t1, t2) is

 X(f) ≡ ∫t1t2 x(t) e-2πift dt

for any frequency f in the two-sided 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 one-sided power spectral density in Pa2/Hz as

 PSD(f) ≡ 2|X(f)|2 / (t2 - t1)

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 one-sided frequency domain (0, F):

 ∫0F PSD(f) df = ∫0F 2|X(f)|2/(t2-t1) df =
 = 1/(t2-t1) ∫-F+F |X(f)|2 df = 1/(t2-t1) ∫t1t2 |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 (t1, t2).

### The logarithmic power spectral density

It is often useful to express the power spectral density in dB relative to Pref = 2·10-5 Pa:

 PSDdB(f) = 10 log10 { PSD(f) / Pref2 }

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.