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Line Spectrum Pair (LSP) was first introduced by Itakura [1,2] as an alternative LPC spectral representations. It was found that this new representation has such interesting properties as (1) all zeros of LSP polynomials are on the unit circle, (2) the corresponding zeros of the symmetric and anti-symmetric LSP polynomials are interlaced, and (3) the reconstructed LPC all-pole filter preserves its minimum phase property if (1) and (2) are kept intact through a quantization procedure. In this paper we prove all these properties via a "phase function." The statistical characteristics of LSP frequencies are investigated by analyzing a speech data base. In addition, we derive an expression for spectral sensitivity with respect to single LSP frequency deviation such that some insight on their quantization effects can be obtained. Results on multi-pulse LPC using LSP for spectral information compression are finally presented.