# 线性预测

## 预测模型

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

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

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

## 预测参数

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

$\ R(i) = E\{x(n)x(n-i)\}\,$

$Ra = -r,\,$

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

$\ Ra = [1, 0, ... , 0]^{\mathrm{T}}$

## 参考文献

• 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.