Practical pulse-shaping waveforms for reduced-cyclic-prefix OTFS

In this paper we model <formula><tex>$M \times N$</tex></formula> orthogonal time frequency space modulation (OTFS) over a <formula><tex>$P$</tex></formula>-path doubly-dispersive channel with delays less than <formula><tex>$\tau_{\max}$</tex></formula> and Doppler shifts in the range <formula><tex>$(\nu_{\min},\nu_{\max})$</tex></formula>. 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. 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.