The performance of a speech recognition system can be greatly enhanced by adding time derivatives to the basic static parameters. In HTK, these are indicated by attaching qualifiers to the basic parameter kind. The qualifier _D indicates that first order regression coefficients (referred to as delta coefficients) are appended, and the qualifier _A indicates that second order regression coefficients (referred to as acceleration coefficients) are appended. The _A qualifier cannot be used without also using the _D qualifier.
The delta coefficients are computed using the following regression formula
where 
is a delta coefficient at time t computed in terms of the
corresponding static coefficients 
to 
. The value
of 
is set using the configuration parameter
DELTAWINDOW . The same formula
is applied to the delta coefficients to obtain acceleration coefficients except
that in this case the window size is set by
ACCWINDOW . Since
equation 5.16 relies on past and future speech parameter values, it
some modification is needed at the beginning and end of the speech. The
default behaviour is to replicate the first or last vector as needed to fill
the regression window.
In older version 1.5 of HTK and earlier, this end-effect problem was solved by using simple first order differences at the start and end of the speech, that is
and
where T is the length of the data file. If required, this older behaviour can be restored by setting the configuration variable V1COMPAT to true in HPARM.
For some purposes, it is useful to use simple differences throughout. This can be achieved by setting the configuration variable SIMPLEDIFFS to true in HPARM. In this case, just the end-points of the delta window are used, i.e.
When delta and acceleration coefficients are requested, they are computed for all static parameters including energy if present. In some applications, the absolute energy is not useful but time derivatives of the energy may be. By including the _E qualifier together with the _N qualifier, the absolute energy is suppressed leaving just the delta and acceleration coefficients of the energy.
