## 2 Principle  
### Principle of CAP Modulation and Conventional CAP Receiver  
CAP modulation is one kind of advanced multi-dimension and multi-level modulation format, which achieves a high data rate and high spectral efficiency with limited bandwidth [28; 29]. CAP modulation can adopt any order m-QAM signal for flexible coding to achieve high-order modulation, and FIR filters are used for pulse shaping, so its signal bandwidth is close to the minimum bandwidth of Nyquist theory. Compared with traditional QAM and OFDM modulation, electrical or optical complex-to-real-value conversion and discrete Fourier transform (DFT) are no longer required, which means that the system complexity can be greatly reduced. Therefore, it is a promising modulation scheme for bandwidth-limited and low-complexity required UVLC systems. At the transmitting end, the original bit sequence is firstly mapped to M-order complex symbols. When using GS technology, the mapping methods are different, thus obtaining constellation points of different shapes. For example, 64 QAM has square constellation points, while 64 APSK has four circles. Then, complex symbols are up-sampled according to the minimum number of samples per symbol, which avoids spectrum aliasing. After that, the symbols are separated into the in-phase and quadrature components, which are represented by \(s_{I}(t)\) and \(s_{Q}(t)\) respectively. Finally, they are sent into a pair of orthogonal shaping filters and added together to generate CAP-modulated signals that will be con verted into analog signals through a digital-to-analog converter (DAC). The signals can be expressed as  
\[s(t)=s_{I}(t)\bigotimes f_{I}(t)-s_{Q}(t)\bigotimes f_{Q}(t) \tag{1}\]  
Here, \(\otimes\) represents a convolution operation. \(f_{I}(t)\) and \(f_{Q}(t)\) are the corresponding impulse response of orthogonal shaping filters in the time domain, which can be expressed as  
\[ \begin{cases}f_{I}(t)=\mathrm{g}(t)\cos(2\pi f_{c}t)\\ f_{Q}(t)=\mathrm{g}(t)\sin(2\pi f_{c}t)\end{cases}  \tag{2}\]  
In Formula (2), \(f_{c}\) is the central frequency of shaping filters, which can be expressed as \(f_{c}=(1+\alpha)/2T+\Delta f\). \(T\), \(\Delta f\), and \(\alpha\) are the symbol period, frequency offset and roll-off coefficient, respectively. \(\mathrm{g}(t)\) is baseband impulse response which is usually root raised cosine filter. It can be expressed as  
\[\mathrm{g}(t)=\frac{T\sin[\pi(1-\alpha)t/T]+4\alpha t\cos[\pi(1+\alpha)t/T]}{ \pi t\Big{[}1-(1-(4\alpha t/T))^{2}\Big{]}} \tag{3}\]  
At the receiving end, the received CAP waveform suffers from severe distortions due to serious linear or nonlinear damage and noise interference. So linear or nonlinear waveform-level equalization algorithms are usually utilized for a received signal after being sampled and quantified by an oscilloscope, such as LMS [15] or Volterra series-based algorithms [18] and so on. Then the signals pass through a pair of matched filters to obtain the in-phase and quadrature components, which can be expressed as  
\[ \begin{cases}r_{I}(t)=r(t)\bigotimes m_{I}(t)\\ r_{Q}(t)=r(t)\bigotimes m_{Q}(t)\end{cases}  \tag{4}\]  
Here, \(m_{I}(t)=f_{I}(-t)\) and \(m_{Q}(t)=f_{Q}(-t)\) are corresponding impulse response of the corresponding matched filters. I and Q signal are combined with being complex symbols after matched filtering and down-sampling, which are actually received constellation points. To further mitigate inter-symbol interference (ISI) and linear or nonlinear distortions, symbol-level post-equalizations are widely applied, including zero-forcing, LMS, DD-LMS, RLS, Volterra series and even neural network-based post equalization algorithms mentioned above. Finally, the original bit data can be retrieved from de-mapping.