## Autoregressive Models

The autoregressive model is one of a group of linear prediction formulas that attempt to predict an output y[n] of a system based on the previous outputs ( y[n-1],y[n-2]...) and inputs ( x[n], x[n-1], x[n-2]...).

Deriving the linear prediction model involves determining the coeffiecients a1,a2,.. and b0,b1,b2,... in the equation:

ye[n] (estimated) = a1*y[n-1] + a2*y[n-2]... + b0*x[n] + b1*x[n-1] + ...

Note the REMARKABLE similarity between the prediction formula and the difference equation used to describe discrete linear time invariant systems. Calculating a set of coefficients that give a good prediciton ye[n] is tantamount to determining what the system is, within the constraints of the order chosen.

A model which depends only on the previous outputs of the system is called an autoregressive model (AR), while a model which depends only on the inputs to the system is called a moving average model (MA), and of course a model based on both inputs and outputs is an autoregressive-moving-average model (ARMA). Note that by definition, the AR model has only poles while the MA model has only zeros.

Several methods and algorithms exist for calculating the coefficients of the AR model, all of which are implemented by the matlab command 'ar'. We use the default setting ('forward-backward') to calculate the AR model for the vocal tract, with the following justifications:

The simplest model for the vocal tract, consisting of linked cylindrical tubes, produces an all-pole transfer function.

Only the outputs of the system are available to us.

Note that the AR model is based on frequency-domain analysis and should be windowed. (We use the hamming.)

The order of the system: We are using the AR model to determine the characteristics of the vocal system and from this system model evaluate the formants, or resonant frequencies of the vocal system.(i.e. the peaks in the frequency response) One conjugate pole pair is required to produce each formant, and one formant is expected in each 1kHz band or so. Therefore the order of the model is a function of the sampling frequency: fs/2 + 2 (the added 2 being the 'empirically determined adjustment factor')

All our autoregressive matlab techniques are in the function formants.m.

There is TONS of material about autoregressive models. Check out your library...