4.2 LPC Spectral Envelope

LPC (linear predictive coding , see [MG80,Opp78,Rob98]) is an early method of digital signal processing, developed originally for speech transmission and compression. By the special properties of the method, it can also be used for spectral envelope estimation.

The idea behind LPC analysis is to represent each sample of a
signal *s*(*n*) in the time-domain by a linear combination of the *p*preceding values
*s*(*n* - *p* - 1) through *s*(*n* - 1). *p* is called the
**order** of the LPC. The approximated value
is computed from the preceding values and *p***predictor-coefficients** (also called **LPC-coefficients** )
*a*_{i} as follows:

Now, for each time-frame, the coefficients

Transmitter and receiver can also be regarded as a linear system with
an adaptive filter, as shown in figure 3.1. What happens
when the residual signal *e*(*n*) is minimized, is that the **analysis
filter** with a transfer function given by

tries to suppress the frequencies in the input signal

As can be seen, the synthesis filter 1/*A*(*z*) is an **all-pole
filter** , since its transfer function is defined by a rational function
with no zero points in the numerator, but with *p* zero points in the
denominator *A*(*z*). Because these zero points come in
compex-conjugate pairs, the absolute value (the magnitude) of the
transfer function of the resulting filter shows *p*/2 **poles** , or peaks.

As the analysis filter tries to flatten the spectrum, it will adapt to it in a way that its inverse filter will describe the spectral envelope of the signal. As the order decreases (i.e. fewer poles are available), the approximation of the spectral envelope will become coarser, but the envelope will nevertheless reflect the rough distribution of energy in the spectrum. This can be seen in figure 3.2.

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For the actual evaluation of the predictor-coefficients to minimize
the prediction error, two classes of methods exist: the
**autocovariance method** and the **autocorrelation method** . Both
have their advantages and disadvantages [Opp78], however, the
autocorrelation method is more widely used, since it can be
efficiently implemented using the **Durbin-Levinson recursion** . I
won't elaborate on the methods here, since they are amply described in
the literature.

In the course of evaluation of the predictor-coefficients, an
intermediate set of parameters, the **reflection coefficients**
*k*_{i} are obtained, which, in fact, correspond to the reflection of
acoustic waves at the boundaries between successive sections of an
acoustic tube, as presented in section 2.4. These
coefficients have advantages for synthesis, and can be interpolated
without problems for the validity (stability) of the resulting
synthesis filter.

Various other parameter sets exist [MG80,Rob98]: the
roots of the analysis filter *A*(*z*), log area ratios
(LAR ), the logarithm of the ratios of the areas of the sections
of the acoustic tube model given by
,
the line spectral pairs , and others. Since it
is possible to convert between them, they don't need to be considered
separately for representation.

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A disadvantage of the LPC spectral envelope in analysing harmonic sounds (sounds with a prevalent partial structure) is that it will tend to envelope the spectrum as tightly as possible, and will under certain conditions descend down to the level of residual noise in the gap between two harmonic partials. This will happen whenever the space between partials is large, as in high pitched sounds, and when the order is high enough, i.e. there are enough poles to come to lay on every partial peak. See figure 3.3 for an example of this effect.