This is opposite from the convention found in $[7].$ The notable difference is that the erro

signal $e[n]$ must now be written with a plus sign:

$e[n]=x[n+r]-$hat$(x)[n]$

$=x[n+r]+{\displaystyle \underset{k=1}{\overset{p}{}}}{a}_{k}x[n-k]$

Thus the error signal $e[n]$ can be interpreted as the output of an LTI system with transfer

function $A(z)$ and input $x[n].$

PROJECT $1$: LINEAR PREDICTION OF DATA

In this project, linear prediction is applied to synthetic signals and to real data. The real data

are from the Dow$-$Jones Industrial Average sampled weekly for about $94$ years. With such

a long sequence, the linear prediction method can be designed over one section of the data

and then tested over other sections to evaluate its effectiveness as a predictor. Performancr

of the method on real data should illustrate some of the limits of the method imposed by

the inherent assumption that the data fit an all$-$pole model. The book by Marple $[3]$ also

contains an interesting data set$-$sunspot numbers for the years $1845-1978.$

Hints

You may find the MATLAB function convmt $x$ useful, along with the backslash $($ Function for Linear Prediction

Wrile $1$wo MATLAB func$1$ions to compule the prediction error fi lter coefficients $\{$a$,$ : k $=$

I $.2,...,$ P$\},$ one for $1$he "autocorrelation$$ method and $1$he $01$her for lhe "covariance $..$ melhod.

Each func$1$ion should accept three input arguments: a vector of da$1$a $($x$),1$he order of the

prediclor $($ P$),$ and $1$he predic$1$ion distance $($ r $).$ The outpul arguments should include the filter

coefficienls $($a $),$ the error sequence $($e$),$ and an index variable containing lhe sample numbers

at which the error signal was computed $($ I $).$ An example function shell is as follows: