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
:
where is the predicted signal value, the previous observed values, and the predictor coefficients. The error generated by this estimate is
:
where is the true signal value.
These equations are valid for all types of (one-dimensional) linear prediction. The differences are found in the way the parameters are chosen.
For multi-dimensional signals the error metric is often defined as
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where the index ''i'' ranges from 0 to ''p'', and ''R'' is a (''p'' + 1) × (''p'' + 1) matrix.
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 ''Ra'' = ''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 symmetry of ''R'' and ''r''. A faster algorithm is the
Levinson Recursion proposed by
Norman 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.
- G. U. Yule. On a method of investigating periodicities in disturbed series, with special reference to wolfer’s sunspot numbers. Phil. Trans. Roy. Soc., 226-A:267–298, 1927.
- J. Makhoul. Linear prediction: A tutorial review. Proceedings of the IEEE, 63 (5):561–580, April 1975.
- M. H. Hayes. Statistical Digital Signal Processing and Modeling. J. Wiley & Sons, Inc., New York, 1996.
