Line spectral pairs (LSP) orline spectral frequencies (LSF) are used to representlinear prediction coefficients (LPC) for transmission over a channel.^[1] 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 inspeech coding.

LSP representation was developed byFumitada Itakura,^[2] atNippon Telegraph and Telephone (NTT) in 1975.^[3] From 1975 to 1981, he studied problems in speech analysis and synthesis based on the LSP method.^[4] In 1980, his team developed an LSP-basedspeech synthesizer 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.^[3] LSPs are used in thecode-excited linear prediction (CELP) algorithm, developed byBishnu S. Atal andManfred R. Schroeder in 1985.

The LPpolynomial${\displaystyle A(z)=1-\sum _{k=1}^p{a}_k{z}^{-k}}$ can be expressed as${\displaystyle A(z)=0.5[P(z)+Q(z)]}$, where:

• ${\displaystyle P(z)=A(z)+z^{-(p+1)}A(z^{-1})}$
• ${\displaystyle Q(z)=A(z)-z^{-(p+1)}A(z^{-1})}$

By construction,P is apalindromic polynomial andQ anantipalindromic polynomial; physicallyP(z) corresponds to the vocal tract with theglottis closed andQ(z) with theglottis open.^[5] It can be shown that:

• Theroots ofP andQ lie on theunit circle in the complex plane.
• The roots ofP alternate with those ofQ as we travel around the circle.
• As the coefficients ofP andQ are real, the roots occur inconjugate pairs

The Line Spectral Pair representation of the LP polynomial consists simply of the location of the roots ofP andQ (i.e.${\displaystyle \omega }$ such that${\displaystyle z=e^{i\omega },P(z)=0}$). As they occur in pairs, only half of the actual roots (conventionally between 0 and${\displaystyle \pi }$) need be transmitted. The total number of coefficients for bothP andQ is therefore equal top, the number of original LP coefficients (not counting${\displaystyle a_0=1}$).

A common algorithm for finding these^[6] 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 ofP are interspersed with those ofQ a single pass is sufficient to find the roots of both polynomials.

To convert back to LPCs, we need to evaluate${\displaystyle A(z)=0.5[P(z)+Q(z)]}$by "clocking" an impulse through itN times (order of the filter), yielding the original filter, A(z).

Line spectral pairs have several interesting and useful properties. When the roots ofP(z) andQ(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.

• Log area ratios

• 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" (ISBN 0-471-29546-9)Jonathan Stein.

• Lossy compression algorithms
• Digital signal processing
• Data compression

• Articles with short description
• Short description matches Wikidata