The amplitude envelope Sk(t) and phase of Xk(t) are determined using the amplitude and phase spectra. Since , we must find the input phase . It can be determined by applying three physical constraints, derived from regularities (ii) and (iv), as shown below[7].
Regularity (ii) means that ``a single sound tends to change its properties smoothly and slowly (gradualness of change)'' [6].
First constraint, considered as ``slowness'', is
dAk(t)/dt=Ck,R(t), where
Ck,R(t) is an R-th-order differentiable polynomial.
By applying it to Eq. (), and solving the resulting linear differential equation, we obtain
Second constraint, considered as ``smoothness'', is that, in the bound (t=Tr) between pre-segment (
)
and post-segment (
),
Regularity (iv) means that ``many changes take place in an acoustic event that affect all the components of the resulting sound in the same way and at the same time'' [6].
Third constraint, considered as this regularity, is
(23) |
Hence, the above computational process can be summarized as follows: (a) a general solution of is determined using physical constraint 1; (b) candidates of Ck,0 that can uniquely determine , is determined using physical constraint 2; (c) an optimal Ck,0 is determined using physical constraint 3; and (d) can be uniquely determined by the optimal Ck,0.