Bandwidth expansion is a technique for widening the

bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...

or the

resonances Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...

in an

LPC LPC may refer to: Science and technology * Linear predictive coding, a method used in audio signal processing and speech processing * Leaf protein concentrate, a concentrated form of the proteins found in the leaves of plants * Long period comet, ...

filter. This is done by moving all the

poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...

towards the origin by a constant factor

\gamma

. The bandwidth-expanded filter

A'(z)

can be easily derived from the original

filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...

A(z)

by: :

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

Let

A(z)

be expressed as: :

A(z) = \sum_^a_kz^

The bandwidth-expanded filter can be expressed as: :

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

In other words, each

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

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 Code-excited linear prediction (CELP) is a linear predictive speech coding algorithm originally proposed by Manfred R. Schroeder and Bishnu S. Atal in 1985. At the time, it provided significantly better quality than existing low bit-rate algori ...

coding of speech, where it is often used for the perceptual noise

weighting The process of weighting involves emphasizing the contribution of particular aspects of a phenomenon (or of a set of data) over others to an outcome or result; thereby highlighting those aspects in comparison to others in the analysis. That i ...

and/or to stabilize the LPC analysis. However, when it comes to stabilizing the LPC analysis,

lag windowing Lag windowing is a technique that consists of windowing the autocorrelation coefficients prior to estimating linear prediction coefficients (LPC). The windowing in the autocorrelation domain has the same effect as a convolution (smoothing) in the ...

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. Signal processing