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LPC (Linear predictive coding)

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Linear Predictive Coding (LPC) is a widely used technique for speech analysis and compression. It is a mathematical tool that can be used to model the spectral envelope of speech signals. The technique is based on the assumption that the speech signal can be approximated as a linear combination of its past samples, with the coefficients of this combination being determined by a mathematical optimization process. In this article, we will discuss the principles of LPC, its applications, and some of its variants.

Principles of LPC:

The basic idea behind LPC is to model the spectral envelope of a speech signal by using a linear combination of its past samples. The spectral envelope of a speech signal represents the shape of the speech spectrum, which contains information about the different formants or resonances in the vocal tract. These formants are responsible for the perception of different vowels and consonants in speech.

The LPC model assumes that the speech signal can be represented as a linear combination of its past samples:

x(n) = a1x(n-1) + a2x(n-2) + ... + ap*x(n-p)

where x(n) is the current sample of the speech signal, x(n-1) is the previous sample, x(n-2) is the sample before that, and so on. The coefficients a1, a2, ..., ap represent the weighting factors for each past sample, and p is the order of the LPC model.

The goal of the LPC model is to find the coefficients a1, a2, ..., ap that minimize the mean squared error between the original speech signal and its LPC approximation. This can be done by solving a set of linear equations known as the Yule-Walker equations:

R(1) + a1R(2) + a2R(3) + ... + apR(p+1) = r(1) R(2) + a1R(1) + a2R(2) + ... + apR(p) = r(2) ... R(p+1) + a1R(p) + a2R(p-1) + ... + ap*R(1) = r(p+1)

where R(k) is the autocorrelation function of the speech signal at lag k, and r(k) is the cross-correlation function between the speech signal and its delayed version at lag k. The Yule-Walker equations can be solved using various numerical methods, such as Levinson-Durbin recursion, Cholesky decomposition, or QR decomposition.

Once the LPC coefficients are obtained, the speech signal can be synthesized by using an inverse filter that mimics the spectral envelope of the LPC model:

y(n) = x(n) - a1y(n-1) - a2y(n-2) - ... - ap*y(n-p)

where y(n) is the output of the inverse filter, which represents the synthesized speech signal. This equation can be interpreted as subtracting the LPC approximation of the speech signal from the original signal to obtain the residual signal, which contains the fine details of the speech waveform that are not captured by the LPC model.

Applications of LPC:

LPC has been widely used in speech analysis, speech coding, and speech synthesis. Some of its main applications are:

  1. Speech analysis: LPC can be used to extract various features of the speech signal, such as the formant frequencies, the pitch period, and the glottal waveform. These features can be used for various purposes, such as speaker identification, emotion recognition, and speech quality assessment.
  2. Speech coding: LPC can be used to compress the speech signal by reducing its bit rate while maintaining a high level of perceptual quality. This is achieved by transmitting only the LPC coefficients and the residual signal, instead of the raw speech waveform. The LPC-based speech codecs are widely used in various communication systems, such as mobile phones, VoIP, and video conferencing.
  3. Speech synthesis: LPC can be used to generate synthetic speech by using the LPC coefficients to mimic the spectral envelope of the original speech signal. This technique is used in various applications, such as text-to-speech synthesis, voice conversion, and singing voice synthesis.

Variants of LPC:

There are several variants of the basic LPC model that have been proposed over the years to improve its performance or to extend its applicability to different types of signals. Some of these variants are:

  1. Cepstral analysis: Cepstral analysis is a technique that involves taking the inverse Fourier transform of the log magnitude spectrum of a signal. This technique can be used to separate the spectral envelope of the signal from its fine spectral details, which can be modeled separately using LPC. The resulting technique is known as cepstral LPC, or LPC cepstrum.
  2. Warped LPC: Warped LPC is a variant of LPC that uses a frequency warping function to transform the frequency axis of the speech spectrum to a more uniform scale. This can improve the accuracy of the LPC model by reducing the effect of the non-uniform distribution of formants in the speech spectrum.
  3. Robust LPC: Robust LPC is a variant of LPC that uses a weighted objective function to reduce the impact of outliers or noise in the speech signal. This technique can improve the robustness of the LPC model to various types of distortions or artifacts in the speech signal.

Conclusion:

In summary, LPC is a powerful technique for speech analysis and compression that has been widely used in various applications. The basic LPC model assumes that the speech signal can be represented as a linear combination of its past samples, and that the coefficients of this combination can be obtained by solving a set of linear equations. The LPC model can be used to model the spectral envelope of the speech signal, which contains information about the formants and resonances in the vocal tract. The LPC model has several variants that can improve its performance or adapt it to different types of signals.