Loop theorem

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 $A^{\prime }(z)$ can be easily derived from the original filter $A(z)$ by:

$A^{\prime }(z)=A(z/\gamma )$

Let $A(z)$ be expressed as:

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

The bandwidth-expanded filter can be expressed as:

$A^{\prime }(z)=\sum _{k=0}^N{a}_k{\gamma }^k{z}^{-k}$

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.