6.3 The autoregressive model and formant analysis

The autoregressive model

Interpreting this signal first begins with determining an actual equation for the signal. The best way to dothat is by using an autoregressive model. An autoregressive model is simply a model used to find an estimation of a signal based onprevious input values of the signal. The actual equation for the model is as follows:

The model consists of three parts: a constant part, an error or noise part, and the autoregressive summation. Theactual summation represents the fact that the current value of the input depends only on previous values of the input. The variable prepresents the order of the model. The higher the order of the system, the more accurate a representation it will be. Therefore,as the order of the system approaches infinity, we get almost an exact representation of our input system.

This system looks almost exactly like a differential equation. In fact, this equation can be used to findthe transfer function for the signal.

Finding the formants

Once you have the transfer function, you merely need to get your enveloped syllables and pass them throughthis transfer function. Once you take the frequency response of the transfer function, you can get a very nice plot as itsoutput (Figure 1).

This gives us something we can actually interpret. Specifically, you can clearly see the formants of thevowel–that is, you can see the peak values of the frequency response. These peaks are what differentiate vowel sounds from oneanother. For instance, looking at these vowel sounds, all from the same person, there is a clear discrepancy in theirappearances (see Sample Formants).

Examining the first two formants, there are clear differences between where they occur and their magnitude ineach vowel sound. These peak values will also be different from person to person, even for the same vowel. For instance, comparethe sound‘a’(as in cat) for each member of the group (see Speaker Vowel Comparisons).

Even though the structure of the frequency responses are similar, the vowel sounds each have slightlydifferent formants, both in the frequency at which they occur and the height that they attain. So finally, we have some way toanalyze our signal. All that remains is the final step–comparing these formants to the formants of the whole group.