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 ${\displaystyle \gamma }$. The bandwidth-expanded filter ${\displaystyle A'(z)}$ can be easily derived from the original filter ${\displaystyle A(z)}$ by:

${\displaystyle A'(z)=A(z/\gamma )}$

Let ${\displaystyle A(z)}$ be expressed as:

${\displaystyle A(z)=\sum _{k=0}^{N}a_{k}z^{-k}}$

The bandwidth-expanded filter can be expressed as:

${\displaystyle A'(z)=\sum _{k=0}^{N}a_{k}\gamma ^{k}z^{-k}}$

In other words, each coefficient ${\displaystyle a_{k}}$ in the original filter is simply multiplied by ${\displaystyle \gamma ^{k}}$ 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.