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This is a question that came to mind as a result of a previous question Hidden Markov Models - Distinct Observation Symbols and subsequent answer from @pichenettes.

One approach to speech recognition is to use Hidden Markov Models (HMM) to identify patterns in speech. Both discrete HMM models and continuous HMM models have been used, but the continuous HMM approach seems to produce better results.

In continuous approach, the “input” to the HMM models is a series of vectors constructed from frames of processed speech. So speech as processed in blocks of some time length resulting in vectors of some fixed dimension say N, with the size of N being directly related to the frame length.

My question is this:

Is there a relationship between the frame length (or alternatively the dimension of the vectors, N) and the number of states resulting in the HMM models used for recognition?

Also is there a relationshiop between the number of HMM models required to predic speech and the size of N?

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The dimension of the frame's feature vector N depends on the frequency range and the number of frequency bins in the frame. For example for 16khz audio signal it's common to take 39 feature vectors created from 40 cepstrum values. For 8khz it's enough to have 20 cepstrum values.

Feature dimension is not equivalent to the frame length. Frame length could be rather arbitrary as long as you catch sound variation in time. Frame length is determined by how fast your speech signal changes. It's enough to have 10ms frame length, though 20ms and 5ms work almost equally well. If you have frame of 30ms you will have issues since some speech sounds change faster. Thrilled r for example.

Number of distinct HMM models (tied phone detectors for example) depends on the diversity of the speech signal, on the number of distinct sounds (temporal audio patterns) there. Each distinct sound ideally has to be better modelled with it's own HMM model. The number of states depends on the size of the training database and on the vocabulary of the database, not on the feature extraction parameters. The more words you want to recognize the more HMM models you need.

There is no direct relation between both feature vector size and length of the frame and the number of states in HMM models.

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Thanks for the explanation. I was thinking larger feature vectors would lead to greater diversity, but this is only true up to a point. At some point, even though you have greater dimension, there will be vectors that contain overlapping information so increased the vector size is essential a waist. I guess an important problem is optimizing the vector size which must be a function of the signal diversity. –  B Z Feb 22 '13 at 17:02
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To add here, it's proven that speech recognition benefits from multiresolution features, most conventional systems use delta features which consider signal frame of 10ms and 90ms at the same feature vector. However, I believe rarely a large frame is considered. –  Nikolay Shmyrev Feb 23 '13 at 6:21
    
@NikolayShmyrev Would it be right in saying that 13 MFCC values (for each block) would not be enough to train a HMM? –  Phorce Feb 23 '13 at 17:52
    
"not enough" is not really applicable here, but delta and delta-delta features make system more accurate. –  Nikolay Shmyrev Feb 23 '13 at 18:44
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