The Transmitted Signals of OTFS and VOFDM Are Exactly the Same

Xiang-Gen Xia, University of Delaware, Newark, DE 19716, USA

April 25, 2022

Abstract

This short note connects OTFS and VOFDM systems and explains why VOFDM is good to time varying channels with Doppler spread, which explains why OTFS is so as well.

Main Description

It is not hard to check that the transmitted signals in either discrete or continuous format of OTFS [8] and single antenna vector OFDM (VOFDM) [1, 2] are the same. For the discrete format, for convenience, let a rectangular pulse be used. Then, the transmitted sequences of OTFS and VOFDM are the same. If a continuous pulse is used and a single math formula is to represent VOFDM, it is the same as that of OTFS, i.e.,  formula (5) in [8]. Note that the cyclic prefix for VOFDM does not have to be a multiple of vectors and it can be a truncated sequence of length not smaller than the channel length [4, 5].

Recently it has been claimed that OTFS is good to deal with Doppler spread for time-varying channels. From the VOFDM point of view, it has been shown in [1, 2, 3, 4] that VOFDM can achieve multipath diversity and/or signal space diversity, even with the MMSE linear receiver in a vectorized subchannel [4]. This is because in VOFDM, at transmit side, a vector of information symbols is DFT (or IDFT) transformed and then, at receive side, the information symbols in this vector are demodulated together. Since in VOFDM, a vectorized channel matrix is pseudo-circulant, it can be simultaneously diagonalized by DFT/IDFT matrix with a phase shift diagonal matrix, see formula (4.1) in [2]. Then, this DFT (or IDFT) of a vector of information symbols is similar to the precoding in single antenna systems to collect signal space diversity to combat wireless fading [6] or diagonal space-time block coding in MIMO systems to collect spatial diversity [7].  We believe that this is the main reason why OTFS (or VOFDM) is good to deal with the channel variance due to the Doppler spread.

For more details about single antenna VOFDM, please see [2,4,5].

References

[1] X.-G. Xia, “Precoded OFDM systems robust to spectral null channels and vector OFDM systems with reduced cyclic prefix length,” Proceedings of ICC, vol. 2, pp. 1110-1114, New Orleans, LA, USA, June 18-22, 2000.

[2] X.-G. Xia, “Precoded and vector OFDM robust to channel spectral nulls and with reduced cyclic prefix length in single transmit antenna systems,” IEEE Transactions on Communications, vol. 49, no. 8, pp. 1363–1374, August 2001.

[3] H. Zhang and X.-G. Xia, "Iterative decoding and demodulation for single antenna vector OFDM systems," IEEE Transactions on Vehicular Technology, vol. 35, no. 4, pp. 1447-1454, July 2006.

[4] Y. Li, I. Ngebani, X.-G. Xia, and A. Host-Madsen, “On performance of vector OFDM with linear receivers,” IEEE Transactions on Signal Processing, vol. 60, no. 10, pp.5268-5280, October 2012.

[5] https://www.eecis.udel.edu/~xxia/VOFDM_PPT.pdf

[6] J. Boutros and E. Viterbo, "Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel," IEEE Transactions on Information Theory, vol. 44, pp. 1453-1467,  Jul. 1998.

[7] M. O. Damen, K. A. Meraim, and J.-C. Belfiore, "Diagonal algebraic space-time block codes," IEEE Transactions on Information Theory, vol. 48, pp. 628-636, Mar. 2002.

[8] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), San Francisco, CA, USA, Mar. 2017, pp. 1–6.