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-2&#π;ift 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.

© ppgb, October 26, 2010