Background Information

Contents Home The Problem [The ARMA Model] Autocorrelation What we did The Data Conclusions A Word on Decoding Acknowledgements Who we are

The topic of linear signal prediction is a very high level topic that usually requires a great deal of knowledge in the areas of statistics and control theory. In this section we will try to reduce the material to an understandable form so that no knowledge about control theory and a only a very basic background in statistics will be required. The heart of linear prediction lies in a new way of representing the point on a signal that you want to predict. This new representation is called parametric modeling. Since we want to make things easy for ourselves we will select a linear model with finite complexity called an ARMA(autoregressive moving average) model. This model states that if we want to predict an output to a system called X(n) with unknown input U(n) then where A*(k) and B*(k) are just weighting coefficients just as in Fourier analysis. Three Major components of an ARMA process 1. The unknown input U(n) is a white process that is not noise but is the part that causes the random behavior of our system. 2. The averaging (MA) section works on the present and finite past values of the input and thus creates a moving average. This is an FIR filter. 3. The recursive (AR) section works only on past values of the output. This is an IIR filter.    To find the transfer function X(n)/U(n) of the ARMA model we take the Z transform of the above equation and derive    If you set the numerator to one in the above equation you will notice that our transfer function can only contain poles. This all pole transfer function model is called an AR process because it just contains the recursive part of the equation. Likewise if you set the denominator to one you will notice that the transfer function can only contain zeros. This is called an MA process because it only consists of the moving average section.    It can be derived through the use of autocorrelation functions(review if needed from a basic statistics class) that both the ARMA and MA processes are non-linear due to the fact that their transfer functions contain A(k) and B(k) multiplied by another B(k). These methods are both hard and expensive. The AR process just consists of a set of linear equations, which are easy and cheap to solve. Thus this is the method we will decide to use when we make an attempt at linear prediction and it will work due to a theorem that states that any ARMA or MA process with a continuous power spectral density can be represented uniquely by an AR model of infinite order(thus an AR model of a very large order.

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