Dual-Input Dual-Output Feedforward Neural Network (DIDO-FFNN)
The second architecture proposed to model complex static systems is based on splitting the complex data stream into two components, which are then processed separately using two real-valued feedforward neural networks (RVFFNNs) as shown in Figure 7.5 and proposed in  and . The RVFFNN is similar to a typical complex FFNN but only takes real values for inputs. In the polar based architecture, the complex input signal Cin is first decomposed into its polar (Ain, Oin) components. In architecture, the magnitude Ain and phase Oin of the input signal feed the input of the first and second real valued NNs, respectively. Similarly, in the Cartesian based architecture of this model, the signal is first decomposed into its Cartesian components (Iin, Qin). Then, the real or in-phase Iin and imaginary or quadrature-phase Qin parts of the input signal feed the input of the first and second real-valued NNs, respectively. Finally, the outputs of both real-valued FFNNs are recombined to construct the complex output signal. Thus, this can be seen as a dual-input dual-output (DIDO) polar or Cartesian based architecture.
The DIDO polar based FFNN architecture utilizes two uncoupled NNs that attempt to capture the AM/AM (amplitude modulation to amplitude modulation) and AM/PM (amplitude modulation to phase modulation) responses separately. The main drawback of this topology is the asynchronous convergence of the separate phase and amplitude FFNNs, where both NNs do not converge to an optimal model at same time, leading to over- or under-training of one NN. In the Cartesian based DIDO-FFNN architecture, separate real valued FFNNs are used to model the in-phase (I) and quadrature-phase (Q) components of the system’s output signal. This approach takes advantage of the availability of the I and Q components, but it is also prone to asynchronous convergence between the I and Q sub-models of the NN.
Figure 7.5 Block diagram of dual-input dual-output feedforward neural network applied on the polar components
Dual-Input Dual-Output Coupled Cartesian Based Neural Network (DIDO-CC-NN)
To avoid asynchronous convergence of the DIDO polar or Cartesian based FFNN, the DIDO-CC-NN was proposed in . As depicted in Figure 7.6, the structure of this model decomposes the complex input signal Cin into its Cartesian components (Iin, Qin), which are then simultaneously fed to two separate RVFFNNs. The major difference with the previous models is that in this case, both Cartesian components in the input layer are coupled with both NNs. Finally, the outputs of two RVFFNNs are recombined to obtain the complex output signal.
Figure 7.6 Dual-input dual-output coupled Cartesian based neural network
The three architectures described in this section have been found to be effective, to various extents, for the forward modeling of systems with strong static nonlinearity; they all fall short of expectations when the system exhibits strong dynamics and memory effects.