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Line spectral pairs (LSP) or line spectral frequencies (LSF) are used to represent linear prediction coefficients (LPC) for transmission over a channel. LSPs have several properties (e.g. smaller sensitivity to quantization noise) that make them superior to direct quantization of LPCs. For this reason, LSPs are very useful in
speech coding Speech coding is an application of data compression of digital audio signals containing speech. Speech coding uses speech-specific parameter estimation using audio signal processing techniques to model the speech signal, combined with generic da ...
. LSP representation was developed by
Fumitada Itakura is a Japanese scientist. He did pioneering work in statistical signal processing, and its application to speech analysis, synthesis and coding, including the development of the linear predictive coding (LPC) and line spectral pairs (LSP) methods. ...
, at
Nippon Telegraph and Telephone , commonly known as NTT, is a Japanese telecommunications company headquartered in Tokyo, Japan. Ranked 55th in Fortune Global 500, ''Fortune'' Global 500, NTT is the fourth largest telecommunications company in the world in terms of revenue, as w ...
(NTT) in 1975. From 1975 to 1981, he studied problems in speech analysis and synthesis based on the LSP method. In 1980, his team developed an LSP-based
speech synthesizer Speech synthesis is the artificial production of human speech. A computer system used for this purpose is called a speech synthesizer, and can be implemented in software or Computer hardware, hardware products. A text-to-speech (TTS) system conve ...
chip. LSP is an important technology for speech synthesis and coding, and in the 1990s was adopted by almost all international speech coding standards as an essential component, contributing to the enhancement of digital speech communication over mobile channels and the internet worldwide. LSPs are used in the
code-excited linear prediction Code-excited linear prediction (CELP) is a linear predictive speech coding algorithm originally proposed by Manfred R. Schroeder and Bishnu S. Atal in 1985. At the time, it provided significantly better quality than existing low bit-rate algori ...
(CELP) algorithm, developed by
Bishnu S. Atal Bishnu S. Atal (born 1933) is an Indian physicist and engineer. He is a noted researcher in acoustics, and is best known for developments in speech coding. He advanced linear predictive coding (LPC) during the late 1960s to 1970s, and develope ...
and
Manfred R. Schroeder Manfred Robert Schroeder (12 July 1926 – 28 December 2009) was a German physicist, most known for his contributions to acoustics and computer graphics. He wrote three books and published over 150 articles in his field. Born in Ahlen, he stud ...
in 1985.


Mathematical foundation

The LP
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
A(z) = 1- \sum_^p a_k z^ can be expressed as A(z) = 0.5 (z) + Q(z)/math>, where: * P(z) = A(z) + z^A(z^) * Q(z) = A(z) - z^A(z^) By construction, ''P'' is a
palindromic polynomial In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n = ...
and ''Q'' an antipalindromic polynomial; physically ''P''(''z'') corresponds to the vocal tract with the
glottis The glottis is the opening between the vocal folds (the rima glottidis). The glottis is crucial in producing vowels and voiced consonants. Etymology From Ancient Greek ''γλωττίς'' (glōttís), derived from ''γλῶττα'' (glôtta), va ...
closed and ''Q''(''z'') with the
glottis The glottis is the opening between the vocal folds (the rima glottidis). The glottis is crucial in producing vowels and voiced consonants. Etymology From Ancient Greek ''γλωττίς'' (glōttís), derived from ''γλῶττα'' (glôtta), va ...
open. It can be shown that: * The
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of ''P'' and ''Q'' lie on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
in the complex plane. * The roots of ''P'' alternate with those of ''Q'' as we travel around the circle. * As the coefficients of ''P'' and ''Q'' are real, the roots occur in conjugate pairs The Line Spectral Pair representation of the LP polynomial consists simply of the location of the roots of ''P'' and ''Q'' (i.e. \omega such that z = e^, P(z) = 0). As they occur in pairs, only half of the actual roots (conventionally between 0 and \pi) need be transmitted. The total number of coefficients for both ''P'' and ''Q'' is therefore equal to ''p'', the number of original LP coefficients (not counting a_0=1). A common algorithm for finding thesee.g. lsf.c in http://www.ietf.org/rfc/rfc3951.txt is to evaluate the polynomial at a sequence of closely spaced points around the unit circle, observing when the result changes sign; when it does a root must lie between the points tested. Because the roots of ''P'' are interspersed with those of ''Q'' a single pass is sufficient to find the roots of both polynomials. To convert back to LPCs, we need to evaluate A(z) = 0.5 (z)+ Q(z)/math> by "clocking" an impulse through it ''N'' times (order of the filter), yielding the original filter, ''A''(''z'').


Properties

Line spectral pairs have several interesting and useful properties. When the roots of ''P''(''z'') and ''Q''(''z'') are interleaved, stability of the filter is ensured if and only if the roots are monotonically increasing. Moreover, the closer two roots are, the more resonant the filter is at the corresponding frequency. Because LSPs are not overly sensitive to quantization noise and stability is easily ensured, LSP are widely used for quantizing LPC filters. Line spectral frequencies can be interpolated.


See also

* Log area ratios


Sources


Speex manual
and source code (lsp.c)
"The Computation of Line Spectral Frequencies Using Chebyshev Polynomials"
P. Kabal and R. P. Ramachandran. IEEE Trans. Acoustics, Speech, Signal Processing, vol. 34, no. 6, pp. 1419–1426, Dec. 1986. Includes an overview in relation to LPC.
"Line Spectral Pairs" chapter
as an online excerpt (pdf) / "Digital Signal Processing - A Computer Science Perspective" () Jonathan Stein.


References

{{Compression Methods Lossy compression algorithms Digital signal processing