If τ is too large, there will be no any correlation between adjacent components of each phase point in the phase space.
The embedding delay τ obtained by this method not only maintains a lower correlation but also has a non-complete independence between adjacent components of each phase point in the phase space.
where y k, k = 1,2,…, M- m-1)τ is a M- m-1nsional row vector, and it represents a phase point in phase space, the eM- m-1ng dimension m represents the minimum dimension number of phase space, and the embedding delay τ descrises the distance between adjacent components of each phase point.
If τ is too small, the difference between the adjacent components of each phase point in the phase space will become too small.
To verify whether the nearest neighbor of each phase point is false or not, the relative change of Euclidean distance between phase point and its nearest neighbor is denoted as f d (k) when the dimension number of phase space is changed to d + 1 from d, i.e., f d (k) is defined by [14] f d k = D 2 d + 1 k - D 2 d k D 2 d k 1 2 = x k + τd - x NN k + τd D d k. (6).
The (linear or non-linear) relationship between phase point y5 and its adjacent phase points, such as y3 and y4, can be used to predict the component x 7) included in phase point y5.
In another case, as shown in Fig. 4b, the adjacent samples' real phase, like point 5 and 6, whose difference is smaller than 2π but larger than π, so that the phase difference between their measured phase (point 5 ' and 6 ') is below π.
Thus, once the phase space of LF-MDCT coefficients is reconstructed, the relationship between the phase points and MDCT coefficients can be described by a non-linear model.
Let the Euclidean distance between phase points y k and y k NN be D d (k).
When the dimension number of phase space is increased to d + 1 from d, the Euclidean distance between phase points y k and y k NN is denoted by Dd + 1(k).
