# Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels
## 1 Introduction  
In the last two decades, high-speed data communication over dispersive channels has been widely investigated. One of the most successful modulation schemes is orthogonal frequency division multiplexing (OFDM) due to its ability to transmit information symbols on orthogonal subcarriers [1]. However, because the subcarriers used in OFDM are cosine signals, which are relatively sensitive to Doppler spread, when the channel is doubly selective, the performance of OFDM systems will suffer serious degradation. A chirp signal is a signal whose frequency increases or decreases with time. Due to the good temporal resolution of the autocorrelation function and the insensitivity to Doppler effects, chirp signals are extensively used in radar and sonar applications [2; 3; 4]. In the areas of communication, chirp spread spectrum (CSS) techniques are widely applied for applications between mobile nodes [5; 6]. In order to obtain a higher data rate, a common way is to transmit a bank of chirp signals withoutinterference in one period. Based on this principle, Fractional Fourier Transform-Orthogonal Frequency Division Multiplexing (FrFT-OFDM) is proposed in Reference [7], which is able to generate orthogonal chirp signals via fractional Fourier transform (FrFT). It is shown in Reference [7] that the performance of FrFT-OFDM is better than OFDM over varying Doppler spread factors. However, FrFT-OFDM systems require an estimation of the angle parameter \(\alpha\), which corresponds with the Doppler spread factor. The estimation will lead to computational overhead and is hard to realize in fast varying channels. In Reference [8], a novel multicarrier modulation system called orthogonal chirp division multiplexing (OCDM) has been proposed. It is shown that when the channel is time-invariant, OCDM can achieve better performance with similar peak-to-average power ratio (PAPR) and spectral efficiency when compared with the well studied orthogonal frequency division multiplexing (OFDM) technology [8]. In the meantime, the receiver structure is simpler than FrFT-OFDM because no extra parameter estimation is needed. OCDM has already been applied for optical and underwater communications [9; 10; 11; 12; 13; 14]. In Reference [9], OCDM is firstly applied for high-speed coherent optical communication, then an underwater acoustic communication system based on the underloaded OCDM system is proposed in Reference [10]. However, in OCDM, the use of different phase matrices might lead to extra storage overhead. More importantly, since the emergence of OCDM is relatively new, most existing research has focused on their performance over frequency-selective channels so far [8; 9; 13; 14]. When the channel is doubly selective, the performance of OCDM systems is mainly investigated in underwater acoustic communication (UAC) applications with experimental channels recorded in field tests [10; 11; 15]. Phase correcting methods are applied in the proposed UAC systems to deal with Doppler spreads by estimating and compensating the Doppler phase shift in each block, which will lead to extra computational costs. To date, low complexity equalization algorithms for OCDM based systems over doubly selective channels have not been investigated yet. In this paper, we propose an OCDM based modulation scheme termed uniform phase-Orthogonal Chirp Division Multiplexing (UP-OCDM) for high-speed communication over doubly selective channels. By using uniform phase matrices, UP-OCDM can reduce the storage requirement of OCDM while maintaining the time-frequency diversity gain and high spectral efficiency. Then we prove that like OCDM, the transform matrix of UP-OCDM is circulant. Based on the circulant structure, two low complexity equalization methods are proposed for UP-OCDM over doubly selective channels. The first one is proposed in the frequency domain. We derive that the equivalent frequency-domain channel matrix of UP-OCDM system can be approximated as a band matrix.