# EVM Loss: A Loss Function for Training Neural Networks in Communication Systems
## 2 Materials and Methods  
To verify the effectiveness of the EVM loss approach, we test it using two physical layer waveforms, namely orthogonal frequency division multiplexing (OFDM) and spectrally efficient frequency division multiplexing (SEFDM), with data that was obtained experimentally using a wireless visible light communication (VLC) link. It should be noted that this technique is medium-independent and was tested on a VLC link due to availability, but also works for radio links. The test setup can be found in Figure 1. \(2^{16}-1\) bits were generated and mapped to the quadrature phase shift keying (QPSK) constellation. After serial-to-parallel conversion, the size of the FFT was set to generate 64 subcarriers. The symbols were then converted back from parallel to serial and transmitted over the VLC link before the inverse process was applied. The decision for placing the neural network after the FFT owes to the property of neural networks known as the 'universal approximation theorem' [13]. The theorem states that a neural network can learn to approximate any function, however since the FFT/IFFT pairing is already exactly known and computationally efficient, we do not incorporate this into the task of the neural network. The experiment was conducted with NI USRP-RIOs and as such were controlled via LabVIEW software. The neural network was implemented in Python, and the final analysis into the EVM results was done in MATLAB. This work, without loss of generality, will not consider FEC and uses hard decision decoding. As EVM is inversely proportional to the square of the signal-to-noise ratio (SNR), the methodology proposed here of lowering  
Figure 1: The block diagram of the system used in the experiment. the EVM when used before FEC will allow for exploitation of higher SNR within a soft decision decoder. OFDM is a well known and researched multi-carrier modulation scheme, whereas SEFDM is a more recently proposed non-orthogonal signal, first introduced in [4]. SEFDM employs the fractional Fourier transform (FrFT) as opposed to the standard Fourier transform (FFT) in OFDM. The use of the FrFT compresses the input signal in the frequency domain beyond the orthogonality limit of \(1/T\), where \(T\) is the symbol period. Whilst this improves spectral efficiency, it also introduces deterministic inter-carrier interference (ICI) which is detrimental to the recoverability of the signals on each subcarrier. SEFDM can be described mathematically as shown in Equation (1) [14]:  
\[x(t)=\frac{1}{\sqrt{N}}\sum_{m=-\infty}^{\infty}\sum_{n=0}^{N-1}s_{m,n}\exp \left[\frac{j2\pi nm\alpha}{N}\right]. \tag{1}\]  
where \(s_{m,n}\) is the complex-valued symbol on the \(m^{\text{th}}\) SEFDM symbol on the \(n^{\text{th}}\) subcarrier. \(N\) is the total number of subcarriers and \(\alpha\) is the bandwidth compression factor, which defines the frequency spacing of the subcarriers. In the OFDM case, \(\alpha=1\) and the subcarrier spacing is set to multiples of \(1/T\), where \(T\) is the symbol period. For further information including detailed mathematics of the signal, refer to [14,15]. The deterministic ICI can be modelled as a correlation matrix \(\mathbf{C}\) as shown in Equation (2) [15]:  
\[\mathbf{C}_{(l,n)}=\left\{ \begin{array}{ll}1,&l=n\\ \frac{1}{N}\left[\frac{1-\exp\left[j2\pi(l-n)\right]}{1-\exp\left[ \frac{2\pi(l-n)}{N}\right]}\right],&l\
eq n\end{array} \right. \tag{2}\]  
where \(l\) and \(n\) represent two subcarriers indices to be compared. \(\mathbf{C}\) reduces to an identity matrix for OFDM. Neural networks are able to generalize any input-output sequence given sufficient neurons in the hidden layers [16]. In principle, this means that given a sufficient signal-to-noise ratio, the neural network should be able to estimate \(\mathbf{C}\), for any value of \(\alpha\). In this work, we test \(\alpha\geq 0.6\), meaning a bandwidth saving up to 40%.