# Complex Time-Delay Neural Network (CTDNN)

The complex time-delay neural network (CTDNN) is a simplified version of previously described dynamic model where the feedback path of the model is omitted and make this model a complex non-recurrent FFNN model. This model is shown in Figure 7.8.

The complex input *x(k)* is delayed *p* times with tapped delay line making *p +* 1 inputs. Unlike the previous architecture, this architecture doesnâ€™t depend on the previous values of the output. The input-output relationship of the CTDNN model can be obtained as:

where the input vector of the NN at instant *k* is:

# Real Valued Time-Delay Recurrent Neural Network (RVTDRNN)

Real-valued time-delay recurrent neural network (RVTDRNN) model was introduced from the inspiration of CTDRNN where the complex signal is decomposed into two real valued components *(I, Q)* and fed to two real valued TDRNNs (time-delay recurrent neural networks). The major difference between the RVTDRNN and the CTDRNN is that the RVTDRNN is structured to take advantage of the availability of *I* and *Q* components of the complex baseband signal waveforms. Furthermore, the training process of the RVTDRNN becomes significantly faster with the use of real weights instead of the complex weights as in the TDRNN.

**Figure 7.8 **Block diagram of complex time delay neural network (CTDNN)

In this architecture, shown in Figure 7.9, the input and output signalsâ€™ in-phase and quadrature components, (*I _{in}, Q*

_{in}) and

*(I*are delayed with TDLs for

_{out}, Q_{out}),*p*and

*q*times, respectively. Therefore, the order of the input vector for RVTDRNN at any moment of the training sequence is

*2*

*(p +*1 + q)-by-l including past samples of the input and output signals [12]. Here,

*p*and

*q*are the memory orders of the input and feedback signals, respectively. The input expression is given as,

The in-phase and quadrature-phase components of the output signal, *I _{out}* and

*Q*respectively; are given by:

_{out},

where *f** _{1}* and

*f*

*are activation functions modeled by RVRNN (real valued recurrent neural network).*

_{2}Figure 7.9 Block diagram of a three-layer real valued time delay recurrent neural network (RVTDRNN)