The HMM formalism

HMMs themselves are quite general models and, although they were originally developed for speech recognition [36], they have been used for many tasks in speech and language technology, and are now in fact one of the fundamental techniques in biology (to such 

The input to an ASR system is a sequence of frames of speech, known as observations 

The frames are processed so as to remove phase and source information. The most common representation is based on mel-scale cepstral coefficients (MFCCs), which were inttoduced in Section 12.5.7. Hence each observation oj is a vector of continuous values. For each phone we build a probabilistic model, which tells us the probability of 

So far in the book, all the probabilities we have considered have been of discrete events, for example, the probability that we will observe a word w, in a particular context. In all the cases we have so far considered, this effectively means keeping a unique record of each event (e.g. the probability of observing word w, in the context of word wy) no general properties of the process have been modelled. We often find, however, that the probabilities with which events occur are not arbitrary rather, we find that the probabilities with which events occur can be described by a parameterised function. The advantage of this is that we can effectively summarise the probabilistic behaviour of an event with a small munber of parameters. For continuous variables, such functions are called probability density fimctions (pdfs). The integral, or total area under the curve, always stuns to exactly 1, in just the same way as the probabilities for all possible events in a discrete system stun to 1. 

The Gaussian has many interesting mathematical properties and because of these and the fact that a large range of natural phenomena seem to belong to this distribution (e.g. the range of peoples heights in a population), it is often seen as the most naturaf or elementary probability distribution. 

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For synthesis therefore, we are required to make some modifications to ASR-style HMMs to ensure good quality speech. MFCCs on their own are not sufticient to generate speech, as we need (as a minimum) to include FO information as well. Given a vector of MFCCs and an FO value, a number of techniques, detailed in Section 14.6 can be used to generate the waveform. While we can include an FO value directly in the observation vector Ot but it is more common to make use of an additional aspect of the HMM formalism known as streams. Each stream has a mixture of Gaussians observation pdfs as before. The addition is that we can now have a number of these observation pdfs, and this is particularly useful when we want to generate observations... 

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