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Linear prediction

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The prediction model Edit

The most common representation is

\widehat{x}(n) = -\sum_{i=1}^p a_i x(n-i)\,



where x_n is the true signal value.


For multi-dimensional signals the error metric is often defined as

e(n) = ||x(n) - \widehat{x}(n)||\,

where ||.|| is a suitable chosen vector norm.

Estimating the parameters Edit


\sum_{i=1}^p a_i R(i-j) = -R(j),





Ra = -r,\,

where the autocorrelation matrix R is a symmetric, circulant matrix with elements ri,j = R(ij), vector r is the autocorrelation vector rj = R(j), and vector a is the parameter vector.

Another, more general, approach is to minimize

e(n) = x(n) - \widehat{x}(n) = x(n) + \sum_{i=1}^p a_i x(n-i) = \sum_{i=0}^p a_i x(n-i)

where we usually constrain the parameters a_i with a_0=1 to avoid the trivial solution. This constraint yields the same predictor as above but the normal equations are then



Optimisation of the parameters is a wide topic and a large number of other approaches have been proposed.


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.

References Edit

Original Edit

  • 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.

Overview Edit

  • J. Makhoul. Linear prediction: A tutorial review. Proceedings of the IEEE, 63 (5):561–580, April 1975.

External links Edit

See also Edit

de:Linear prediction]]
it:Equazioni di Yule-Walker]]

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