Line spectrum pair (LSP) and speech data compression

Line Spectrum Pair (LSP) was first introduced by Itakura [1,21 as an alternative LPC spectral representations. It was found that this new representation has such interesting properties as (I) 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 (I) 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. LINE SPECTRUM PAIR (LSP) For a given order m, LPC analysis results in an inverse filter A,,, (z) I + a,z' + a2z2 + + a,,,z" which minimizes the residual energy [3]. In speech compression, the LPC coefficients {a1,a2, , a,,,) are known to be inappropriate for quantization because of their relatively large dynamic range and possible filter instability problems. Different set of parameters representing the same spectral information, such as reflection coefficients and log area ratios, etc., were thus proposed [4,5] for quantization in order to alleviate the above-mentioned problems. LSP is one such kind of representation of spectral information. LSP parameters have both well-behaved dynamic range and filter stability preservation property, and can be used to encode LPC spectral information even more efficiently than many other parameters. The LSP representation is rather artificial. For a given mlh order inverse filter as in (1), we can extend the order to (tn+l) without introducing any new information by letting the (m+1)'° reflection coefficients, k,,,+1, be 1 or —I. This is equivalent to setting the corresponding acoustic tube model completely closed or completely open at the (ni+l)°' stage. We thus have, for k,,,+ = ± I respectively, A,,,+t(z) = A,,,(z) ± For convenience, we shall call these two polynomials P(z) (for k,,,+1 1) and Q(z) (for k,,,+1 —1), respectively. It is obvious that P(z) is a symmetric polynomial and Q (z) is an anti-symmetric polynomial and A,,,(z) = 1⁄2 [P(z) + Q(z)] Three important properties of P(z) and Q (z) are listed as follows: (1) All zeros of P(z) and Q(z) are on the unit circle; (2) Zeros of P (z) and Q (z) are interlaced with each other; and (3) Minimum phase property of A,,,(z) is easily preserved after quantization of the zeros of P(z) and Q(z). Since zeros of P(z) and Q(z) are on the unit circle, they can be expressed as e'" and w's are then called the LSP frequencies. The first two properties are useful for finding the zeros of P (z) and Q (z). The third property ensures the stability of the synthesis filter. LSP PROPERTIES We shall prove LSP properties in this section. Rewriting equation (2) in a product form we have P(z) = A(z) 1 ÷ A (z) A(z) Q(z) = A (z) I — A(z) A(z) 11(z) 4 -"'' A(z') A (z)