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

[math]\displaystyle{ A'(z) = A(z/\gamma) }[/math]

Let [math]\displaystyle{ A(z) }[/math] be expressed as:

[math]\displaystyle{ A(z) = \sum_{k=0}^{N}a_kz^{-k} }[/math]

The bandwidth-expanded filter can be expressed as:

[math]\displaystyle{ A'(z) = \sum_{k=0}^{N}a_k\gamma^kz^{-k} }[/math]

In other words, each coefficient [math]\displaystyle{ a_k }[/math] in the original filter is simply multiplied by [math]\displaystyle{ \gamma^k }[/math] 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.