Fig. 1. Two real-valued neurons which are equivalent to a complex-valued neuron.

Fig. 2. An example of the decision boundary of the 1-12-1 Complex-BP network learned with the learning pattern 1. The meanings of the numerals in Fig. 2 are as follows. 1: Real part OFF(0.0–0.5), Imaginary part OFF, 2: Real part ON(0.5–1.0), Imaginary part OFF, 3: Real part OFF, Imaginary part ON, and 4: Real part ON, Imaginary part ON. The decision boundary for the real part (i.e., the boundary that the region “1+3” and the region “2+4” form) and that for imaginary part (i.e., the boundary that the region “1+2” and the region “3+4” form) intersect orthogonally.

Fig. 3. An example of the decision boundary of the 2-14-2 Real-BP network learned with the learning pattern 1. The numbers 1–4 have the same meanings as those of Fig. 2. The decision boundary for the real part (i.e., the boundary that the region “1+3” and the region “2+4” form) and that for imaginary part (i.e., the boundary that the region “1+2” and the region “3+4” form) do not intersect orthogonally.

Fig. 4. Comparison of the angles of the decision boundaries (the average and the standard deviation). (a) Learning pattern 1, (b) Learning pattern 2, (c) Learning pattern 3. The definition of network structures is as follows. Case 1: the 1-3-1 Complex-BP and 2-4-2 Real-BP networks, Case 2: the 1-6-1 Complex-BP and 2-7-2 Real-BP networks, Case 3: the 1-9-1 Complex-BP and 2-11-2 Real-BP networks, and Case 4: the 1-12-1 Complex-BP and 2-14-2 Real-BP networks.

Fig. 5. Comparison of the generalization ability (the average and the standard deviation). (a) Learning pattern 1, (b) Learning pattern 2, (c) Learning pattern 3. The definition of network structures is the same as that of Fig. 4.

Fig. 6. Comparison of the learning speed (the average and the standard deviation). (a) Learning pattern 1, (b) Learning pattern 2, (c) Learning pattern 3. The definition of network structures is the same as that of Fig. 4.

Table 1. Learning pattern 1

Table 2. Learning pattern 2

Table 3. Learning pattern 3

Table 4. The number of parameters in the Real-BP and Complex-BP networks