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# AR model

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### AR model

1. 1. AR MODEL ORDER PRESENTED BY:- Sarbjeet Singh NITTTR- Chandigarh
2. 2. IntroductionIn the model-based approach, the spectrum estimation procedure consists of two steps.(i) We estimate the parameters{ak}and{bk} of the model.(ii) From these estimates, we compute the power spectrum estimate.There are three types of models;-AR ModelMA ModelARMA Model
3. 3.  Of these three models the AR model is by far the most widely used. Reasons are twofold:-(i) The AR model is suitable for representing spectra with narrow peaks.(ii) The AR model result in very simple linear equations
4. 4. What is AR ModelA model which depends only on the previous outputs of the system is called an autoregressive model (AR).Note that:- AR model is based on frequency-domain analysis and should be windowed. (We use the hamming.) AR model has only poles while the MA model has only zeros.
5. 5. One of the most important consideration is the choice of the number of terms in the AR model, this is known as its order p.If a model with too low an order, We obtain a highly smoothed spectrum.If a model with too high an order, There is risk of introducing spurious low-level peaks in the spectrum.
6. 6. The AR-model of a random process in discrete time is defined by the following expression:where a1,a2…..,ap coefficients of the recursive filter;p is the order of the model; Є(t) are output uncorrelated errors.
7. 7. The model parameters are found by solving a set of linear equation obtained by minimizing the mean squared error.The characteristic of this error is that it decreases as the order of the AR model is increased.Much work has been done by various researchers on this problem and many experimental results have been given:-
8. 8. Two of the better known criteria for selection the model order have been proposed by Akaike – (1969,1974.) Known as (FPE) criterion. = estimated variance of the linear prediction error. N = number of samples.
9. 9. The second criterion proposed by Akaike (1974),called the AIC,AIC(p)= decreases & therefore also decreases as the order of the AR model is increased. increases with increases in p.
10. 10. Difference between FPE & AIC(i) FPE (p) Is recommended for longer data records.(ii) AIC (p) Is recommended for short data records. .
11. 11. An alternative information criterion, proposed by Rissanen (1983),is based on selecting the order that minimizes the description length :-
12. 12. A fourth criterion has been proposed by Parzen(1974).This is called the CAT function & defined asThe order P is selected to minimize CAT(p)
13. 13. The experimental results, just indicate that the model-order selection criteria do not yields definitive results.The FPE(p) criterion tends to underestimate the model order.The AIC criterion is statistically inconsistent as N→∞.The MDL information criterion is statistically consistent.
14. 14. Other experimental results indicate that:-For small data lengths, the order of the AR model should be selected in the range N/3 to N/2 for good results.The computational complexity of sequential estimation method is generally proportional to p, the order of the AR process
15. 15. ReferencesProakis John G. , “ Digital Signal Processing “ 3rd edition