Linear prediction

Linear prediction is a mathematical operation where future values of a discrete-time signal are 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

where ${\displaystyle {\widehat {x}}(n)}$ is the predicted signal value, ${\displaystyle x(n-i)}$ the previous observed values, and ${\displaystyle a_{i}}$ the predictor coefficients. The error generated by this estimate is

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