Linear prediction

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Linear prediction is a mathematical operation where future values of a digital signal is estimated as a linear function of previous samples.

In digital signal processing linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory. In system analysis (a subfield of mathematics), linear prediction can be viewed as a part of mathematical modelling or optimization.

The most common representation is

x'(n) = sum_i=1^p a_i x(n-i)

where x'(n) is the estimated signal value, x(n) the previous values, and a_i the predictor coefficients. The error generated by this estimate is

e(n) = x(n) - x'(n)

where x(n) is the true signal value and x'(n) the estimated value. These equations are valid for all types of (one-dimensional) linear prediction. The differences are found in the way the parameters a_i are chosen.

The most common choice in optimisation of parameters a_i is the root mean square criterion which is also called the autocorrelation criterion. In this method we minimise the expected value of the squared error E{e^2(n)}, which yields the equation

sum_i=1^p a_i R(i-j) = -R(j), for 1 <= j <= p,

where R is the autocorrelation of signal x(n) defined as R(i) = E{x(n)x(n-i)}.

The above equations are called the normal equations or Yule-Walker equations. In matrix form the equations can be equivalently written as

R*a = r, where autocorrelation matrix R is a symmetric Toeplitz matrix with elements r_i,j = R(i-j), vector r is the autocorrelation vector r_j = R(j), and vector a is the parameter vector of a_i.

Optimisation of the parameters is a wide topic and a large number of other approaches have been proposed. Still, the autocorrelation method is the most common and it is used, for example, for speech coding in the GSM standard.

Solution of the matrix equation R*a = r is computationally a relatively expensive process. The Gauss algorithm for matrix inversion is probably the oldest solution but this approach does not efficiently use the symmtric properties of R and r. A faster algorithm is the Levinson recursion proposed by N. Levinson in 1947, which recursively calculates the solution. Later, Delsarte et al proposed an improvement to this algorithm called the split Levinson recursion which requires about half the number of multiplications and divisions. It uses a special symmetrical property of parameter vectors on subsequent recursion levels.