Computation of Linear Prediction Coefficients

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

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A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

Minimum-phase polynomials have all their zeros inside the unit circle of the complex plane. — Click for https://ccrma.stanford.edu/~jos/filters/Minimum_Phase_Polynomials.html

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Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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Linear Prediction is an important signal modeling tool which captures correlation information in the form of prediction coefficients. In digital audio signal processing, linear prediction computes a superior spectral envelope that emphasizes spectral peaks. — Click for http://en.wikipedia.org/wiki/Linear_prediction

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Linear Prediction Order Selection

Computation of Linear Prediction Coefficients

In the autocorrelation method of linear prediction, the linear prediction coefficients $\{a_i\}_{i=1}^M$ are computed from the Bartlett-window-biased autocorrelation function (Chapter 6):

$\displaystyle r_{y_m}(l) \isdefs \sum_{n=-\infty}^\infty y_m(n)y_m(n+l) \eqsp \hbox{\sc DFT}^{-1}\left\vert Y_m\right\vert^2 \protect$

(11.11)

where denotes the th data frame from the signal . To obtain the th-order linear predictor coefficients $\{a_1,\ldots,a_M\}$ , we solve the following $M\times M$ system of linear normal equations (also called Yule-Walker or Wiener-Hopf equations):

$\displaystyle \sum_{i=1}^M a_i r_{y_m}(\vert i-j\vert) \eqsp -r_{y_m}(j), \qquad j=1,2,\ldots,M \protect$

(11.12)

In matlab syntax, the solution is given by `` $\verb+a=R\p+$ '', where $\verb+p(j)+ = r_{y_m}(j)$ , and $\verb+R(i,j)+=r_{y_m}(\vert i-j\vert)$ . Since the covariance matrix is symmetric and Toeplitz by construction,^11.4 an solution exists using the Durbin recursion.^11.5

If the rank of the $M\times M$ autocorrelation matrix $R[i,j]=r_{y_n}(\vert i-j\vert)$ is , then the solution to (10.12) is unique, and this solution is always minimum phase [162] (i.e., all roots of are inside the unit circle in the plane [263], so that is always a stable all-pole filter). In practice, the rank of is (with probability 1) whenever includes a noise component. In the noiseless case, if is a sum of sinusoids, each (real) sinusoid at distinct frequency $0<\omega_i T < \pi$ adds 2 to the rank. A dc component, or a component at half the sampling rate, adds 1 to the rank of .

The choice of time window for forming a short-time sample autocorrelation and its weighting also affect the rank of . Equation (10.11) applied to a finite-duration frame yields what is called the autocorrelation method of linear prediction [162]. Dividing out the Bartlett-window bias in such a sample autocorrelation yields a result closer to the covariance method of LP. A matlab example is given in §10.3.3 below.

The classic covariance method computes an unbiased sample covariance matrix by limiting the summation in (10.11) to a range over which stays within the frame--a so-called ``unwindowed'' method. The autocorrelation method sums over the whole frame and replaces by zero when points outside the frame--a so-called ``windowed'' method (windowed by the rectangular window).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Computation of Linear Prediction Coefficients

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University