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Bandwidth expansion

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

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

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

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

The bandwidth-expanded filter can be expressed as:

A ′ ( z ) = ∑ k = 0 N a k γ k z − k {\displaystyle A'(z)=\sum _{k=0}^{N}a_{k}\gamma ^{k}z^{-k}}

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

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.

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