Linear prediction coefficients are used for speech coding to compactly characterise the short-time power spectrum. Unfortunately, LPCs are not very suitable for quantisation due to their large dynamic range, also small quantisation errors in individual LPC coefficients can produce relatively large spectral errors and can also result in instability of the filter. To get around this, it is necessary to transform the LPCs into other representations which ensure the stability of the LPC filter after quantisation. The available representations include: reflection coefficients (RCs), log area ratios (LARs), inverse sine coefficients (ISs) (or arcsine reflection coefficients), and line spectral frequencies (LSFs).

Reflection coefficients are computed from LPCs using the levinson durbin recursion, and have 2 main advantages over LPCs: 1) they are less sensitive to quantisation than LPCs, and 2) the stability of the resulting filter can easily be ensured by making all reflections coefficients stay between -1 and 1.

Though RCs are less sensitive to quantisation than LPCs, they have the drawback that their spectral sensitivity is U-shaped, having large values whenever the magnitude of these coefficients is close to unity. This means the coefficients are very sensitive to quantisation distortion when they represent narrow bandwidth poles. However, this can be overcome by using a non-linear transformation of the RCs. Two such transformations are LARs and ISs.

More information on this topic can be sourced from Speech Coding and Synthesis.

## Log Area Ratios

Log area ratios (LAR) are computed from reflection coefficients for transmission over a channel. They are not as good as line spectral pairs (LSFs), but they are much simpler to compute. If \(k\) is a a reflection coefficent, LARs can be computed using the following formula:

\[ \text{LAR} = \log \left( \dfrac{1+k}{1-k} \right) \]The inverse operation, computing reflection coefficents from the LARs is:

\[ k = \dfrac{e^{\text{LAR}}-1}{e^{\text{LAR}}+1} \]As an example, if you have the following reflection coefficents: ` -0.9870 0.9124 -0.2188 0.0547 -0.0383 -0.0366 -0.0638 -0.0876 -0.1178 -0.1541`, then the corresponding LARs are ` -5.0304 3.0836 -0.4447 0.1095 -0.0766 -0.0732 -0.1279 -0.1757 -0.2367 -0.3106`.

## Inverse Sine Coefficients

IS parameters are another transformation of reflection coefficients, their formula is as follows:

\[ \text{IS} = \dfrac{2}{\pi} \sin^{-1} (k) \]The inverse operation, computing reflection coefficents from the ISs is:

\[ k = \sin \left( \dfrac{\pi \text{IS}}{2} \right) \]Using the same reflection coefficients as previously, the IS parameters are ` -0.8973 0.7316 -0.1404 0.0348 -0.0244 -0.0233 -0.0407 -0.0559 -0.0752 -0.0985`

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