# Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels
## 2 System Model  
### Signal Model of UP-OCDM  
In this section, we will introduce the signal model of UP-OCDM first. We consider a bank of baseband continuous chirp signals with chirp rate equaling to \(a=-\frac{N}{T^{2}}\), where \(N\) is the number of subcarriers, and \(T\) is the symbol duration. Therefore, the chirp signals can be expressed as  
\[\begin{split} c_{k}(t)&=e^{i\pi\alpha(t-k\frac{T}{N} )^{2}}\\ &=e^{-j\pi\frac{N}{T^{2}}(t-k\frac{T}{N})^{2}},0\leq t<T.\end{split} \tag{1}\]  
Notice that the chirp waveforms are mutually orthogonal,  
\[\int_{0}^{T}e^{-j\pi\frac{N}{T^{2}}(t-m\frac{T}{N})^{2}}e^{j\pi\frac{N}{T^{2}}( t-k\frac{T}{N})^{2}}dt=\delta(m-k). \tag{2}\]  
Thus, similar to the OFDM modulation schemes, the modulated transmission signal is  
\[s(t)=\sum_{k=0}^{N-1}d_{k}c_{k}(t),0\leq t<T, \tag{3}\]  
where \(d_{k}\) is the symbol transmitted by the \(k\)th subcarrier, and is drawn from a finite constellation (e.g., 4-QAM, 16-QAM, 64 QAM). We assume a sampling frequency of \(f_{\rm s}=\frac{N}{T}\), the baseband discrete signal is  
\[s(n)=\sum_{k=0}^{N-1}d_{k}e^{-j\frac{n}{N}(n-k)^{2}},0\leq t<T. \tag{4}\]  
Equivalently, the matrix form of (4) is  
\[\mathbf{s}=\mathbf{\Phi}\mathbf{d}, \tag{5}\]  
where \(\mathbf{s}=[s_{0},s_{1},\ldots,s_{N-1}]^{T}\in\mathbb{C}^{N}\) is the modulated signal, and \(\mathbf{d}=[d_{0},d_{1},\ldots,d_{N-1}]^{T}\in\mathbb{C}^{N}\) is the symbol vector. And  
\[\begin{split}\Phi(m,n)&=e^{-j\pi\frac{N}{N}(m-n)^{ 2}}\\ &=e^{-j\pi\frac{N^{2}}{N}}e^{j2\pi\frac{N}{N}}e^{-j\pi\frac{N^{2} }{N}}.\end{split} \tag{6}\]  
Define the phase matrix as  
\[\mathbf{\Gamma}=\mathcal{D}([1,e^{-j\pi\frac{N^{2}}{N}-e^{-j\pi(\frac{N-1}{N})^{2 }}}]). \tag{7}\]  
Therefore, the transform matrix \(\mathbf{\Phi}\) can be rewritten as  
\[\mathbf{\Phi}=\mathbf{\Gamma}\mathbf{F}^{H}\mathbf{\Gamma}, \tag{8}\]  
where \(\mathbf{F}\) is the Fourier transform matrix. It is implied in (8) that, with the transmit symbol vector \(\mathbf{d}\), the baseband transmit signal \(\mathbf{s}\) in UP-OCDM can be obtained within the following three steps:  
(1) Multiplying the symbol vector \(\mathbf{d}\) by the phase matrix \(\mathbf{\Gamma}\),  
(2) Performing IFFT algorithm,  
(3) Multiplying the same phase matrix \(\mathbf{\Gamma}\).  
Note that compared with OCDM, the number of complex parameters to be stored by UP-OCDM for phase matrices is halved of that in OCDM, this is because OCDM needs to multiply two different matrices while UP-OCDM uses two uniform phase matrices denoted as \(\mathbf{\Gamma}\). We will also prove in the next subsection that the transform matrix of UP-OCDM is circulant so that low complexity equalizationcan be achieved based on this property. Meanwhile, the computation complexities of modulation remain the same in both themes. The comparison between modulation procedures of UP-OCDM and OCDM is given in Figure 1.  
Speaking of the spectral efficiency, the bandwidth of the chirp subcarrier in UP-OCDM is \(B=N\Delta f=\frac{N}{T}\). Therefore, the analog UP-OCDM signal is likely to occupy a bandwidth of \(B+\frac{N}{T^{2}}T=2B\). Whereas, it is proved in Reference [8] that, if each chirp signal has a bandwidth of \(B=\frac{N}{T}\), the spectra of each chirp subcarrier can be folded into the baseband from \(-\frac{B}{2}\) to \(\frac{B}{2}\) with a sampling rate of exactly \(f_{s}=B=\frac{N}{T}Hz\). Therefore, by applying up-sampling, the spectrum of UP-OCDM can be fit into the OFDM spectrum, that is, occupying the same bandwidth of \(B\).