## 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...