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 $$A'(z)$$ can be easily derived from the original filter $$A(z)$$ by:


 * $$A'(z) = A(z/\gamma)$$

Let $$A(z)$$ be expressed as:


 * $$A(z) = \sum_{k=0}^{N}a_kz^{-k}$$

The bandwidth-expanded filter can be expressed as:


 * $$A'(z) = \sum_{k=0}^{N}a_k\gamma^kz^{-k}$$

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.