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• I. Abstract and Introduction
• II. Related Work
• III. Modeling of Mobile Channels
• IV. Channel Discretization
• V. Channel Interpolation and Extrapolation
• VI. Numerical Evaluations
• VII. Conclusions, Appendix, and References

\ Abstract—The OTFS (Orthogonal Time Frequency Space) is widely acknowledged for its ability to combat Doppler spread in time-varying channels. In this paper, another advantage of OTFS over OFDM (Orthogonal Frequency Division Multiplexing) will be demonstrated: much reduced channel training overhead. Specifically, the sparsity of the channel in delay-Doppler (D-D) domain implies strong correlation of channel gains in timefrequency (T-F) domain, which can be harnessed to reduce channel training overhead through interpolation. An immediate question is how much training overhead is needed in doublydispersive channels? A conventional belief is that the overhead is only dependent on the product of delay and Doppler spreads, but we will show that it’s also dependent on the T-F window size. The finite T-F window leads to infinite spreading in D-D domain, and aliasing will be inevitable after sampling in T-F domain. Two direct consequences of the aliasing are increased channel training overhead and interference. Another factor contributing to channel estimation error is the inter-symbol-carrier-interference (ISCI), resulting from the uncertainty principle. Both aliasing and ISCI are considered in channel modelling, a low-complexity algorithm is proposed for channel estimation and interpolation through FFT. A large T-F window is necessary for reduced channel training overhead and aliasing, but increases processing delay. Fortunately, we show that the proposed algorithm can be implemented in a pipeline fashion. Further more, we showed that data-aided channel tracking is possible in D-D domain to further reduce the channel estimation frequency, i.e., channel extrapolation. The impacts of aliasing and ISCI on channel interpolation error are analyzed. The spectral efficiency of OTFS and OFDM will be compared by considering the channel estimation error and ISCI. These discussions will shed light on the design of communications systems over doubly-dispersive channels.

I. INTRODUCTION

A. The Legendary Success of OFDM

\ From the ADSL (Asymmetrical Digital Subscriber Line) and DAB (Digital Audio Broadcasting) in the 1990s to WiFi and LTE (Long Term Evolution) in the 2000s, we have witnessed the legendary success of the orthogonal frequency division multiplexing (OFDM) in wireless communications [1]. In 5G and WiFi 7, we are still using this technique for multiplexing and multiple access, i.e., the orthogonal frequency division multiple access (OFDMA). The huge success of OFDM is built upon a very simple channel model, i.e., a linear time-invariant (LTI) model. It is hard to believe that the various characteristics of wireless channels, such as dispersion, reflection, multi-path effect, etc. can be very accurately described by an LTI model. With such a simple model, we can use Fourier transform for signal modulation/demodulation on different sub-carriers, because complex sinusoids are eigenfunctions of LTI systems. That is to say, when we transmit a sinusoid at f Hz, the receiver will also receive a sinusoid of the identical frequency, although the amplitude and phase will change for sure. In other words, the Doppler effect is totally ignored in the modeling! In mobile channels, different paths can have drastically different Doppler frequency shifts (i.e., Doppler spread), and it is very challenging, if not impossible, to compensate for them individually. What makes it worse, different frequency components (or sub-carriers) have different Doppler shifts for ultra-wideband signals, i.e., Doppler migration in frequency domain.

\ B. Will the legend continue?

\ A natural question is how did OFDM achieve the great success while ignoring such a fundamental characteristic of mobile channels? As a matter of fact, the basic idea of OFDM was proposed in 1966 [2], but its application to mobile communications was not clear until 1985 [3]. As we will see in Section III on channel modeling, the Doppler effect leads to channel variation in time domain. However, for a short time window, the channel is almost static, i.e., the quasi-static channel model, or block fading channel model [4], [5]. When OFDM is combined with the multiple-input-multiple-output (MIMO) technique, we need to estimate the channel state information (CSI) for coherent data detection and spatial multiplexing/multiple access. For the quasi-static channel model, we need to re-estimate the CSI in each frame, and the amount of resources required for channel estimation is proportional to the number of transmit antennas and the delay spread [6], [7]. An immediate question is how frequently do we need to estimate the CSI? This question can be answered by computing the coherence time, which is inversely proportional to the Doppler spread [4]. In 5G NR, the channel estimation can be as frequent as four times per slot (each slot contains 14 OFDM symbols) [8]. For reliable communications, the coherence time should be ten times larger than the delay spread [9], [10]. From another perspective, the Doppler spread leads to inter-carrier-interference (ICI), and the sub-carrier spacing should be at least 100 times the Doppler spread to suppress the ICI to the level of -30 dB, and the bit-error-rate (BER) performance degradation is inevitable [11], i.e., error floors will be observed at medium to high SNR regime. Generally, the Doppler spread grows proportionally with carrier frequency and device speed. Naturally, when the device moves faster, the frame length should be reduced, while the amount of resources required for channel estimation remains unchanged, leading to larger overhead. It will eventually become impossible to acquire the CSI in real-time with an affordable cost. This is a fundamental limit of OFDM, and it is deeply rooted in the LTI channel model. Such a problem has been manifesting itself in various forms, and one of the most famous one is the pilot contamination issue in massive MIMO [5].

\ C. The Promises of OTFS

\ The above-mentioned problems of OFDM come from the fact that LTI models cannot capture the time-variant characteristics of the mobile channels. Then can we solve this problem by using a linear time-variant (LTV) channel model instead? The orthogonal time frequency space (OTFS) modulation is one of the possible answers to the above mentioned challenges [12]–[14]. Other efforts include the affine frequency division multiplexing (AFDM) [15], the orthogonal chirp division multiplexing (OCDM) [16], [17], the orthogonal delay-Doppler division multiplexing (ODDM) [18]. In spite of the different waveforms (i.e., signals), all these techniques are based on an LTV channel model in the delay-Doppler (D-D) domain (i.e., systems). In this paper, we will take OTFS as an example to unveil the huge potentials of signaling techniques over doublydispersive channels, in terms of spectral efficiency.

\ By employing the D-D domain channel model, OTFS can overcome the shortcomings of OFDM at a price of slightly increased complexity and processing delay. The fundamental reason is that the mobile channel changes much slower in the D-D domain, i.e., the longer geometric coherence time [19]–[21]. Then we can estimate the CSI in a much reduced frequency with less cost. From another perspective, although the complex gains of different paths change over time due to the Doppler effect, the way they change can be described by a small number of parameters. It is then possible to estimate these parameters with reduced cost. In the following section, the related work on OTFS will be reviewed.

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:::info This paper is available on arxiv under CC BY-NC-ND 4.0 license.

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1. Z. Gong is with the IOT Thrust, HKUST (Guangzhou), Guangzhou, Guangdong 511453, China; and the Department of ECE, HKUST, Hong Kong SAR, China (E-mail: gongzijun@ust.hk)

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2. F. Jiang and X. Tao are with the Department of Broadband Communications, Pengcheng Laboratory, Shenzhen 518055, China. X. Tao is also with National Engineering Research Center of Mobile Network Technologies, Beijing University of Posts and Telecommunications, Beijing 100876, China (Email: jiangf02@pcl.ac.cn, taoxf@bupt.edu.cn).

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3. Y. Song is with the Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, A1B 3X5, Canada (E-mail: yuhuis@mun.ca).

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4. Prof. C. Li is with the School of Engineering Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada (E-mail: cheng li 5@sfu.ca)

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