Bandwidth expansion

Bandwidth expansion is a technique for widening the bandwidth or the resonances in an LPC filter. This is done by moving all the poles towards the origin by a constant factor $gamma$. The bandwidth-expanded filter $A\text{'}(z)$ can be easily derived from the original filter $A(z)$ by:

:$A\text{'}(z)\; =\; A(z/gamma)$

Let $A(z)$ be expressed as:

:$A(z)\; =\; sum\_\{k=0\}^\{N\}a\_kz^\{-k\}$

The bandwidth-expanded filter can be expressed as:

:$A\text{'}(z)\; =\; sum\_\{k=0\}^\{N\}a\_kgamma^kz^\{-k\}$

In other words, each coefficient $a\_k$ in the original filter is simply multiplied by $gamma^k$ in the bandwidth-expanded filter. The simplicity of this transformation makes it attractive, especially in CELP coding of speech, where it is often used for the perceptual noise weighting and/or to stabilize the LPC analysis. However, when it comes to stabilizing the LPC analysis, lag windowing is often preferred to bandwidth expansion.

References

P. Kabal, "Ill-Conditioning and Bandwidth Expansion in Linear Prediction of Speech", "Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing", pp. I-824-I-827, 2003.