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
. The bandwidth-expanded filter
A
′
(
z
)
can be easily derived from the original filter
A
(
z
)
by:
A
′
(
z
)
=
A
(
z
/
γ
)
Let
A
(
z
)
be expressed as:
A
(
z
)
=
∑
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
In other words, each coefficient
a
k
in the original filter is simply multiplied by
γ
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.
