Bandwidth expansion

Last updated March 13, 2019

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:

Bandwidth (signal processing) difference between the upper and lower frequencies in a continuous set of frequencies

Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

Linear predictive coding (LPC) is a tool used mostly in audio signal processing and speech processing for representing the spectral envelope of a digital signal of speech in compressed form, using the information of a linear predictive model. It is one of the most powerful speech analysis techniques, and one of the most useful methods for encoding good quality speech at a low bit rate and provides extremely accurate estimates of speech parameters.

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

Let $A(z)$ be expressed as:

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

The bandwidth-expanded filter can be expressed as:

A'(z)=\sum _{k=0}^{N}a_{k}\gamma ^{k}z^{-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.

Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples.

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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 power spectral domain and helps in stabilizing the result of the Levinson-Durbin algorithm. The window function is typically a Gaussian function.

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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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