# EVM Loss: A Loss Function for Training Neural Networks in Communication Systems
## 3 Results  
We test the proposed method with three cases, starting with an MSE loss on a split-complex network as a baseline. We then extend this to show a gain of utilizing concatenated-complex networks, and then the further gain of implementing the proposed 'EVM loss' system.  
Figure 4 shows the received constellations for these three tested approaches. The results show notable reductions in variance as one first converts from a split-complex network to a concatenated-complex network, owing to the extra indirect phase information being retained within the network. A further notable gain is then seen when converting from MSE loss in the concatenated-complex network to the proposed EVM loss, demonstrating the effectiveness of the approach.  
The first test was performed for the OFDM data transmitted over the VLC link. As can be observed from Figure 5, the highest EVM was reported by the split-complex approach. The reason for this was eluded to previously, by creating a neural network for both the real and imaginary constituents of the input signal individually, one deprives the system of information that is embedded in the relationship between the two components of the input signal. Communication system channels that affect the phase of the signal being transmitted alter this relationship, which, when divided into real and imaginary signals  
Figure 4: Received constellations after neural network processing for (**a**) mean square error (MSE) loss w/Split Complex, (**b**) MSE loss w/Concatenated Complex, (**c**) EVM loss w/Concatenated Complex for spectrally efficient frequency division multiplexing (SEFDM) with \(\alpha=0.6\).  
Figure 3: Illustration of error vector magnitude (EVM).  
and trained separately is not taken into consideration. This is rectified by modifying a single neural network to allow for both the real and imaginary constituents as input. The result of this allows the neural network to exploit the phase relationship during training, meaning a lower overall error vector magnitude. Finally, the lowest error vector magnitude calculated across all subcarriers is achieved when the neural network uses the proposed EVM loss method as the objective function rather than the typical mean square error.  
This demonstrates that the mean square error is not an optimal objective function when training neural networks in a communication systems context. Whilst the two other neural networks have minimized their mean square error objective functions as optimally as possible, the fact still remains that when being used in a communication system the training metric and the evaluation metric no longer align, leading to a degradation of the overall system performance. This can be seen when considering the implication of the EVM loss function. The MSE loss function calculates a per neuron MSE loss, where the neuron has implicit access to the available information from its real/imaginary counterpart via the hidden layer neurons, thus the real/imaginary neurons are loosely coupled. The EVM loss function also calculates a per neuron loss, however as well as the implicit information from the network, the loss function itself explicitly examines the available information from its real/imaginary counterpart, forcing a tight coupling. These same trends can again be seen in Figures 6 and 7, where SEFDM was evaluated with \(\alpha=0.8\) and \(\alpha=0.6\) being used as the transmitted physical layer waveform respectively.  
Figure 5: Error vector magnitudes across the subcarriers for the three neural network receiver approaches for orthogonal frequency division multiplexing (OFDM).