ARMA model

Presented by:-

Sarbjeet Singh

NITTTR- Chandigarh

Parametric methods for power spectrum estimation

It is 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.

Introduction to model based approach

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.

Types of model

There are three types of models:-

AR (Autoregressive) model MA (Moving average) model ARMA (Autoregressive moving average)

model

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.

ARMA model

It requires fewer model parameters for the spectrum estimation.

This model is appropriate when the signal has been corrupted by noise.

Calculation of model parameters

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:-

Relationship between autocorrelation and model parameters for ARMA(p,q) process

Matrix representation

Matrix representation for m > p+q

It may be represented as:-

On minimizing, the result is:-

From the AR model parameters, A(Z) can be estimated by:-

This yields the sequence

The estimated ARMA power spectrum is given by:-