Table of Links
- 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
Table of Links
- 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
II. RELATED WORK
A. Relation Between OFDM and OTFS
\ The term OTFS was first proposed in [12], and the model was developed in [13], [22] and [23]. The underlying idea of OTFS is that although the Doppler effect leads to phase shift in each path, but the shifting rates of the phases are almost constant in a relatively long period (i.e., linear phase shift over time caused by the Doppler effect)! As a result, we will have time to estimate the channel parameters even in high-mobility scenarios, as we will see later with examples. Another way to understand this is that the mobile channels are generally sparse in the D-D domain, in the sense that the product of delay spread and Doppler spread is much smaller than 1. Thus, there are only a small number of parameters to estimate in the D-D domain. As the name suggests, OTFS involves signal modeling/processing in three dimensions, i.e., time, frequency and space. The spatial dimension in OTFS comes from the Doppler effect. That is, when a mobile device moves, the Doppler shift leads to a constant phase shift between adjacent time slots on each path. Geometrically, we can synthesize a virtual antenna array by jointly processing the received signal at different time slots, similar to the concept of SAR (Synthetic Aperture Radar).
\ In spite of the difference on mathematical motivations, OTFS is very closely related to OFDM. In [24], Dr. Xia pointed out that the two-stage OTFS is basically the vector OFDM (VOFDM) he proposed in 2001 [25]. The modulated signals are indeed very similar, but the motivation of VOFDM was to avoid the spectral nulls and reduce CP length through precoding. In [26], the authors showed that the OTFS is equivalent to the asymmetric OFDM (A-OFDM) for static multipath channels, a special case of the single-carrier frequency domain equalization (SC-FDE) systems.
\ B. Diversity Gain
\ One of the major motivations of OTFS is the potential diversity gain [13]. More than two decades ago, the authors in [27] already pointed out that the temporal variation of wireless channels induced by the Doppler spread can actually be exploited to harness diversity gain in spread-spectrum communications, and the maximum diversity gain is proportional to the product of delay spread and Doppler spread. In [28], the authors took one more step and showed that this conclusion holds for not only spread-spectrum systems but also other wideband communications systems in general.
\ In the pioneer paper on OTFS [12], the authors showed that full diversity gain can be obtained with OTFS modulation/demodulation. The Doppler shift is dependent on angle of arrival, and different arriving paths have different Doppler shifts. With OTFS, we can isolate different paths in the space domain (or Doppler domain), and align their phases. By doing so, we can stop the multi-path components from being destructively combined, and thus eliminate the fading in TF domain. A diversity gain can thus be obtained. In [29], experiments were conducted for high-speed railway systems at 371 km/h, and a diversity gain of OTFS was verified for a carrier frequency of 450 MHz. The OTFS can be combined with MIMO to further enjoy the spatial diversity gain [30], and proper space-time coding is necessary [31].
\ C. Channel Estimation for OTFS
\ Similar to OFDM, channel estimation is indispensable in OTFS systems, for equalization, multiplexing and multiple access. In [32], the authors advocate the D-D domain channel modeling by emphasizing the predictability of the wireless hannels in this domain. With OTFS modulation, the authors showed that the equivalent baseband channel in D-D domain is predicable and non-fading, given that the crystallization condition holds. The authors investigated model-based and model-free channel estimation methods [33]. For the modelbased case, they basically assume that there is a small number of distinguishable paths in the D-D domain, and estimate the parameters of each path. For the model-free case, a continuous D-D profile is considered. The model-based scenario is widely considered, and typical algorithms include OMP (Orthogonal Matching Pursuit) [34]–[36] and Bayesian learning with expectation-maximization [37].
\ The pilot can be transmitted in different ways. In [12], channel estimation of MU-MIMO (Multi-User-MIMO) systems is considered, and each transmit antenna will sequentially transmit an impulse in the D-D domain. The spacing should be larger than the delay spread and Doppler spread, so that inter-antenna interference can be avoided. In such a case, the required resources for channel estimation is proportional to the product of transmit antenna number (or user number), delay spread and Doppler spread. The data and pilot can be isolated in time domain [38], or D-D domain [21]. It is even possible to superimpose the pilot on the data in T-F domain [20], and an MMSE (Minimum Mean Square Error) channel estimator is developed for single-input-single-output (SISO) communications systems.
\ D. Contributions and Notations
\ In existing work, most of the papers are trying to justify the superiority of OTFS over OFDM by showing that OTFS harnesses the diversity gain in D-D domain, like Dr. Hadani did in his pioneer papers on OTFS. However, we will demonstrate the necessity (not just superiority) of OTFS (or more generally speaking, signaling techniques designed for doubly-dispersive channels) in highly dynamic channels by showing that OFDM is consuming a significant amount of resources on channel estimation, i.e., the spectral efficiency perspective. There are already papers comparing OTFS and OFDM on spectral efficiency. For example in [39] and [40], the authors compared the achievable rates of OTFS and OFDM in LTV channels, and argued that OTFS has higher efficiency due to shorter cyclic prefix (CP). In [41] and [42] the spectral efficiency was derived by considering the ISI and ICI, under the assumption of perfect CSI. In this paper, we will consider channel estimation error during the comparison, and show that OTFS has much improved spectral efficiency due to the reduced channel training overhead. The major contributions are summarized below.
\ • We derived the discrete baseband channel model from the continuous channel response. In this process, the implicit approximations/assumptions of the D-D domain channel model will be unveiled, one of which is that the product of bandwidth and frame length must be upper bounded.
\ • The minimum amount of resources required for channel estimation is discussed in the context of general WeylHeisenberg systems, and a low-complex channel estimation algorithm based on the fast Fourier transform (FFT) is presented to recover the 2D channel response in TF domain with a small number of training symbols. A pipelined algorithm is proposed to reduce the processing delay, and data-aided channel extrapolation is explored to further reduce channel training overhead.
\ • Apart from noise, two other sources of channel estimation error are unveiled: aliasing in the DD domain resulting from confined time and bandwidth, and ISCI (Intersymbol-carrier-Interference) due to channel dispersion. These factors are generally ignored in existing work by assuming the bi-orthogonality (no ISCI) and finite DD spreading (no aliasing). By increasing the time and bandwidth, we can suppress the impact of aliasing, but suffer more from the ISCI.
\ • Comprehensive theoretical and numerical results are presented to compare the spectral efficiencies of OTFS and OFDM. Different from existing work, channel estimation overhead and error are considered in the performance evaluation. These discussions shed light on the design of signaling techniques in doubly-dispersive channels.
\ In the next section, channel modeling will be presented, and channel responses in T-F and D-D domains will be connected. Following that, the channel will be discretized in Section IV, laying the foundation for channel estimation and reconstruction in Section V. Simulations are presented in Section VI, while Section VII concludes the paper.
\ 
\
:::info Authors:
(1) Zijun Gong, Member, IEEE;
(2) Fan Jiang, Member, IEEE;
(3) Yuhui Song, Student Member, IEEE;
(4) Cheng Li, Senior Member, IEEE;
(5) Xiaofeng Tao, Senior Member, IEEE.
:::
:::info This paper is available on arxiv under CC BY-NC-ND 4.0 license.
:::
\
Table of Links
- 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
II. RELATED WORK
A. Relation Between OFDM and OTFS
\ The term OTFS was first proposed in [12], and the model was developed in [13], [22] and [23]. The underlying idea of OTFS is that although the Doppler effect leads to phase shift in each path, but the shifting rates of the phases are almost constant in a relatively long period (i.e., linear phase shift over time caused by the Doppler effect)! As a result, we will have time to estimate the channel parameters even in high-mobility scenarios, as we will see later with examples. Another way to understand this is that the mobile channels are generally sparse in the D-D domain, in the sense that the product of delay spread and Doppler spread is much smaller than 1. Thus, there are only a small number of parameters to estimate in the D-D domain. As the name suggests, OTFS involves signal modeling/processing in three dimensions, i.e., time, frequency and space. The spatial dimension in OTFS comes from the Doppler effect. That is, when a mobile device moves, the Doppler shift leads to a constant phase shift between adjacent time slots on each path. Geometrically, we can synthesize a virtual antenna array by jointly processing the received signal at different time slots, similar to the concept of SAR (Synthetic Aperture Radar).
\ In spite of the difference on mathematical motivations, OTFS is very closely related to OFDM. In [24], Dr. Xia pointed out that the two-stage OTFS is basically the vector OFDM (VOFDM) he proposed in 2001 [25]. The modulated signals are indeed very similar, but the motivation of VOFDM was to avoid the spectral nulls and reduce CP length through precoding. In [26], the authors showed that the OTFS is equivalent to the asymmetric OFDM (A-OFDM) for static multipath channels, a special case of the single-carrier frequency domain equalization (SC-FDE) systems.
\ B. Diversity Gain
\ One of the major motivations of OTFS is the potential diversity gain [13]. More than two decades ago, the authors in [27] already pointed out that the temporal variation of wireless channels induced by the Doppler spread can actually be exploited to harness diversity gain in spread-spectrum communications, and the maximum diversity gain is proportional to the product of delay spread and Doppler spread. In [28], the authors took one more step and showed that this conclusion holds for not only spread-spectrum systems but also other wideband communications systems in general.
\ In the pioneer paper on OTFS [12], the authors showed that full diversity gain can be obtained with OTFS modulation/demodulation. The Doppler shift is dependent on angle of arrival, and different arriving paths have different Doppler shifts. With OTFS, we can isolate different paths in the space domain (or Doppler domain), and align their phases. By doing so, we can stop the multi-path components from being destructively combined, and thus eliminate the fading in TF domain. A diversity gain can thus be obtained. In [29], experiments were conducted for high-speed railway systems at 371 km/h, and a diversity gain of OTFS was verified for a carrier frequency of 450 MHz. The OTFS can be combined with MIMO to further enjoy the spatial diversity gain [30], and proper space-time coding is necessary [31].
\ C. Channel Estimation for OTFS
\ Similar to OFDM, channel estimation is indispensable in OTFS systems, for equalization, multiplexing and multiple access. In [32], the authors advocate the D-D domain channel modeling by emphasizing the predictability of the wireless hannels in this domain. With OTFS modulation, the authors showed that the equivalent baseband channel in D-D domain is predicable and non-fading, given that the crystallization condition holds. The authors investigated model-based and model-free channel estimation methods [33]. For the modelbased case, they basically assume that there is a small number of distinguishable paths in the D-D domain, and estimate the parameters of each path. For the model-free case, a continuous D-D profile is considered. The model-based scenario is widely considered, and typical algorithms include OMP (Orthogonal Matching Pursuit) [34]–[36] and Bayesian learning with expectation-maximization [37].
\ The pilot can be transmitted in different ways. In [12], channel estimation of MU-MIMO (Multi-User-MIMO) systems is considered, and each transmit antenna will sequentially transmit an impulse in the D-D domain. The spacing should be larger than the delay spread and Doppler spread, so that inter-antenna interference can be avoided. In such a case, the required resources for channel estimation is proportional to the product of transmit antenna number (or user number), delay spread and Doppler spread. The data and pilot can be isolated in time domain [38], or D-D domain [21]. It is even possible to superimpose the pilot on the data in T-F domain [20], and an MMSE (Minimum Mean Square Error) channel estimator is developed for single-input-single-output (SISO) communications systems.
\ D. Contributions and Notations
\ In existing work, most of the papers are trying to justify the superiority of OTFS over OFDM by showing that OTFS harnesses the diversity gain in D-D domain, like Dr. Hadani did in his pioneer papers on OTFS. However, we will demonstrate the necessity (not just superiority) of OTFS (or more generally speaking, signaling techniques designed for doubly-dispersive channels) in highly dynamic channels by showing that OFDM is consuming a significant amount of resources on channel estimation, i.e., the spectral efficiency perspective. There are already papers comparing OTFS and OFDM on spectral efficiency. For example in [39] and [40], the authors compared the achievable rates of OTFS and OFDM in LTV channels, and argued that OTFS has higher efficiency due to shorter cyclic prefix (CP). In [41] and [42] the spectral efficiency was derived by considering the ISI and ICI, under the assumption of perfect CSI. In this paper, we will consider channel estimation error during the comparison, and show that OTFS has much improved spectral efficiency due to the reduced channel training overhead. The major contributions are summarized below.
\ • We derived the discrete baseband channel model from the continuous channel response. In this process, the implicit approximations/assumptions of the D-D domain channel model will be unveiled, one of which is that the product of bandwidth and frame length must be upper bounded.
\ • The minimum amount of resources required for channel estimation is discussed in the context of general WeylHeisenberg systems, and a low-complex channel estimation algorithm based on the fast Fourier transform (FFT) is presented to recover the 2D channel response in TF domain with a small number of training symbols. A pipelined algorithm is proposed to reduce the processing delay, and data-aided channel extrapolation is explored to further reduce channel training overhead.
\ • Apart from noise, two other sources of channel estimation error are unveiled: aliasing in the DD domain resulting from confined time and bandwidth, and ISCI (Intersymbol-carrier-Interference) due to channel dispersion. These factors are generally ignored in existing work by assuming the bi-orthogonality (no ISCI) and finite DD spreading (no aliasing). By increasing the time and bandwidth, we can suppress the impact of aliasing, but suffer more from the ISCI.
\ • Comprehensive theoretical and numerical results are presented to compare the spectral efficiencies of OTFS and OFDM. Different from existing work, channel estimation overhead and error are considered in the performance evaluation. These discussions shed light on the design of signaling techniques in doubly-dispersive channels.
\ In the next section, channel modeling will be presented, and channel responses in T-F and D-D domains will be connected. Following that, the channel will be discretized in Section IV, laying the foundation for channel estimation and reconstruction in Section V. Simulations are presented in Section VI, while Section VII concludes the paper.
\
\
:::info Authors:
(1) Zijun Gong, Member, IEEE;
(2) Fan Jiang, Member, IEEE;
(3) Yuhui Song, Student Member, IEEE;
(4) Cheng Li, Senior Member, IEEE;
(5) Xiaofeng Tao, Senior Member, IEEE.
:::
:::info This paper is available on arxiv under CC BY-NC-ND 4.0 license.
:::
\