Practical Pulse-Shaping Waveforms for Reduced-Cyclic-Prefix OTFS

In this paper we model $M\times N$ orthogonal time frequency space modulation (OTFS) over a $P$-path doubly-dispersive channel with delays less than $\tau_{\max}$ and Doppler shifts in the range $(\nu_{\min},\nu_{\max})$. We first derive in a simple matrix form the input--output relation in the delay--Doppler domain for practical (e.g., rectangular) pulse-shaping waveforms, next generalize it to arbitrary waveforms. This relation extends the original OTFS input--output approach, which assumes ideal pulse-shaping waveforms that are bi-orthogonal in both time and frequency. We show that the OTFS input--output relation has a simple sparse structure that enables one to use low-complexity detection algorithms. {\color{black} Different from previous work, only a single cyclic prefix (CP) is  added at the end of the OTFS frame, significantly reducing the overhead, without incurring any penalty from the loss of bi-orthogonality of the pulse-shaping waveforms. } Finally, we compare the OTFS performance with different pulse-shaping waveforms, and show that the reduction of out-of-band power may introduce nonuniform channel gains for the transmitted symbols, thus impairing the overall error performance.OTFS, circulant matrices, delay--Doppler domain.