Line spectral pairs

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. LSP representation was developed by Fumitada Itakura in the 1970s.

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

By construction, P is a palindromic polynomial and Q an antipalindromic polynomial; physically P(z) corresponds to the vocal tract with the glottis closed and Q(z) with the glottis open. It can be shown that:

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