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# Arma model

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### Arma model

1. 1. ARMA model Presented by:- Sarbjeet SinghNITTTR- Chandigarh
2. 2. Parametric methods for powerspectrum estimation Itis a model based approach. In these methods a model for the signal generation can be constructed with a number of parameters. Parameters can be estimated from the observed data. From the estimated parameters the power density spectrum can be computed.
3. 3. Introduction to model basedapproach In the model based approach, the estimation procedure consists of two steps:-Step 1:- estimate the parameters {ak} and {bk} of the model.Step 2:- from these estimates compute the power spectrum estimate.
4. 4. Types of modelThere are three types of models:- AR (Autoregressive) model MA (Moving average) model ARMA (Autoregressive moving average) model
5. 5. ARMA model It is a tool for understanding and predicting the future values in the series. It consists of two parts, an autoregressive (AR) part and a moving average (MA) part. It is usually referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part.
6. 6. ARMA model Itrequires fewer model parameters for the spectrum estimation. This model is appropriate when the signal has been corrupted by noise.
7. 7. Calculation of modelparameters Consider a data sequence x(n) generated by AR model. Let the output is corrupted by additive white noise. The Z-transform of the autocorrelation of the signal is:-
8. 8.  Relationship between autocorrelation and model parameters for ARMA(p,q) process
9. 9.  Matrix representation
10. 10.  Matrix representation for m > p+q
11. 11. It may be represented as:-On minimizing, the result is:-
12. 12.  Fromthe AR model parameters, A(Z) can be estimated by:- This yields the sequence
13. 13.  Theestimated ARMA power spectrum is given by:-