# Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems
## 2 Principle  
### Fractional Fourier Transform  
The FRFT is a generalized form of the FT in the fractional domain. If the FT is considered a \(90^{\circ}\) counterclockwise rotation from the time axis to the frequency axis, the FRFT can be viewed as a rotation of the signal by an angle \(\alpha=p\pi/2\) counterclockwise on the time axis [21], where \(p\) represents the order of the FRFT. There are several ways to define the FRFT, each offering a different physical interpretation [28]. The fundamental definition of the FRFT, from the perspective of a linear integral transform, is as follows:  
\[\mathrm{X}_{p}(u)\ =\ \{F^{p}[x(t)]\}(u)\ =\ \int_{-\infty}^{\infty}K_{p}(u,t)x(t)\ dt \tag{1}\]where \(F^{p}\) denotes the FRFT operator, and \(K_{p}(u,t)\), the kernel of the FRFT, is defined as  
\[K_{p}(u,t)\ =\ \left\{ \begin{array}{ll}\sqrt{1-j\cot\alpha}\exp[j\pi(u^{2} \cot\alpha-2 ut\csc\alpha+t^{2}\cot\alpha)],&\alpha\
eq n\pi\\ \delta(u-t),&\alpha\ =\ 2n\pi\\ \delta(u+t),&\alpha\ =\ (2n+1)\pi\end{array} \right. \tag{2}\]  
Note that when \(p\ =\ 1\), the FRFT reduces to the FT [29]. Equation (1) can be rewritten as  
\[X_{p}(u)\ =\ \left\{ \begin{array}{ll}\sqrt{\frac{1-j\cot\alpha}{2\pi}}\int_{- \infty}^{\infty}\exp(j\frac{u^{2}+t^{2}}{2}\cot\alpha-jut\csc\alpha)x(t)\ dt,&\alpha\
eq n\pi\\ x(t),&\alpha\ =\ 2n\pi\\ x(-t),&\alpha\ =\ (2n+1)\pi\end{array} \right. \tag{3}\]  
Additionally, the inverse FRFT (IFRFT) of order \(p\) is equivalent to the FRFT of order \(-p\) and is expressed as  
\[x(t)\ =\ \int_{-\infty}^{\infty}X_{p}(u)K_{-p}(u,t)\ du \tag{4}\]  
In recent years, researchers have proposed various discrete algorithms for the FRFT. The current discrete fractional Fourier transform (DFRFT) algorithms can be categorized into three main types: sampling-type DFRFT [30; 31], eigenvector decomposition-type DFRFT [32; 33], and linear combination-type DFRFT [34]. In this thesis, the sampling-type discretization algorithm proposed by [PERSON] is utilized [31]. This algorithm has computational complexity comparable to that of the FT, with the computation for \(N\) point data being \(O(N\log_{2}N)\). Additionally, it ensures the validity of the transform kernel matrix, which strictly satisfies invertibility. When applying this algorithm for modulation and demodulation at the transmitter and receiver of the FRFT-OFDM scheme, no errors will arise from the algorithm itself. By applying this algorithm, the DFRFT can be expressed in matrix form as  
\[\mathbf{X}_{p}\ =\ \mathbf{F}_{p}\mathbf{x} \tag{5}\]  
where \(\mathbf{x}\ =\ [x(0),x(1),\cdots,x(N-1)]^{T}\) and \(\mathbf{X}_{p}\ =\ [X_{p}(0),X_{p}(1),\cdots,X_{p}(N-1)]^{T}\) are the vectors before and  
In IM/DD VLC systems, it is essential for OFDM signals to be positive real numbers. To achieve this, we modify the structure of the FRFT algorithm to ensure that the output signals are real and implement unipolarity by adding a DC bias to the transmitter. In FT-OFDM schemes, Hermitian conjugate symmetry is often applied to signals before the FT to guarantee that the transformed output signals remain real numbers. However, since Hermitian conjugate symmetry is specific to the FT, a new form of the FRFT is required. In [35], a method was proposed to derive a discrete fractional cosine (sine) transform variant that maintains real-valuedness and decentralization, along with most of the properties needed for FRFT matrices. We have adopted the RPFRFT method introduced in [35; 36] to meet the signal requirements in our FRFT-OFDM scheme. Let \(\mathbf{F}_{p}\) be a complex-valued DFRFT matrix with size \(N/2\), where N is even, and let \(\mathbf{x}~{}=~{}[x_{1},x_{2},\cdots,x_{N}]^{T}\) be a real signal of length \(N\). We construct a complex vector \(\mathbf{\hat{x}}~{}=~{}[x_{1}+jx_{N/2+1},x_{2}+jx_{N/2+2},\cdots,x_{N/2}+jx_{N}]^{T}\) of length \(N/2\) from \(\mathbf{x}\)[37]. By performing a DFRFT on \(\mathbf{\hat{x}}\), we obtain a complex vector,  
\[