The log-spectral distance (LSD), also referred to as log-spectral distortion or root mean square log-spectral distance, is a
distance measure
Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some ''observable'' quantity (such as the luminosity of a distant quasar, the red ...
between two
spectra
Spectra may refer to:
* The plural of spectrum, conditions or values that vary over a continuum, especially the colours of visible light
* ''Spectra'' (journal), of the Museum Computer Network (MCN)
* The plural of spectrum (topology), an object ...
. The log-spectral distance between spectra
and
is defined as
p-norm
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...
:
:
where
and
are
power spectra.
Unlike the
Itakura–Saito distance, the log-spectral distance is symmetric.
[
]
In speech coding, log spectral distortion for a given frame is defined as the root mean square difference between the original LPC log power spectrum and the quantized or interpolated LPC log power spectrum. Usually the average of spectral distortion over a large number of frames is calculated and that is used as the measure of performance of
quantization or
interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has ...
.
Meaning
When measuring the distortion between signals, the scale or temporality/spatiality of the signals can have different levels of significance to the distortion measures. To incorporate the proper level of significance, the signals can be transformed into a different domain.
When the signals are transformed into the spectral domain with transformation methods such as
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
and
DCT, the spectral distance is the measure to compare the transformed signals. LSD incorporates the logarithmic characteristics of the power spectra, and it becomes effective when the processing task of the power spectrum also has logarithmic characteristics, ''e.g.'' human listening to the sound signal with different levels of loudness.
Moreover, LSD is equal to the cepstral distance which is the distance between the signals'
cepstrum
In Fourier analysis, the cepstrum (; plural ''cepstra'', adjective ''cepstral'') is the result of computing the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. The method is a tool for investigating periodic st ...
when the p-numbers are the same by
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originate ...
.
Other Representations
As LSD is in the form of p-norm, it can be represented with different p-numbers and log scales.
For instance, when it is expressed in dB with L2 norm, it is defined as:
.
When it is represented in the discrete space, it is defined as:
where
and
are power spectra in discrete space.
See also
*
Itakura–Saito distance
References
Signal processing
{{Signal-processing-stub