Fundamentals of Delay-Doppler Communications: Practical Implementation and Extensions to OTFS

The recently proposed orthogonal time frequency space (OTFS) modulation, which is a typical Delay-Doppler (DD) communication scheme, has attracted significant attention thanks to its appealing performance over doubly-selective channels. In this paper, we present the fundamentals of general DD communications from the viewpoint of the Zak transform. We start our study by constructing DD domain basis functions aligning with the time-frequency (TF)-consistency condition, which are globally quasi-periodic and locally twisted-shifted. We unveil that these features are translated to unique signal structures in both time and frequency, which are beneficial for communication purposes. Then, we focus on the practical implementations of DD Nyquist communications, where we show that rectangular windows achieve perfect DD orthogonality, while truncated periodic signals can obtain sufficient DD orthogonality. Particularly, smoothed rectangular window with excess bandwidth can result in a slightly worse orthogonality but better pulse localization in the DD domain. Furthermore, we present a practical pulse shaping framework for general DD communications and derive the corresponding input-output relation under various shaping pulses. Our numerical results agree with our derivations and also demonstrate advantages of DD communications over conventional orthogonal frequency-division multiplexing (OFDM).

Reduced-Complexity Cross-Domain Iterative Detection for OTFS Modulation via Delay-Doppler Decoupling

In this paper, a reduced-complexity cross-domain iterative detection for orthogonal time frequency space (OTFS) modulation is proposed, which exploits channel properties in both time and delay-Doppler domains. Specifically, we first show that in the time domain effective channel, the path delay only introduces interference among samples in adjacent time slots, while the Doppler becomes a phase term that does not affect the channel sparsity. This ``band-limited'' matrix structure motivates us to apply a reduced-size linear minimum mean square error (LMMSE) filter to eliminate the effect of delay in the time domain, while exploiting the cross-domain iteration for minimizing the effect of Doppler by noticing that the time and Doppler are a pair of Fourier dual. The state (MSE) evolution was derived and compared with bounds to verify the effectiveness of the proposed scheme. Simulation results demonstrate that the proposed scheme achieves almost the same error performance as the optimal detection, but only requires a reduced complexity.

On the Pulse Shaping for Delay-Doppler Communications

In this paper, we study the pulse shaping for delay-Doppler (DD) communications. We start with constructing a basis function in the DD domain following the properties of the Zak transform. Particularly, we show that the constructed basis functions are globally quasi-periodic while locally twisted-shifted, and their significance in time and frequency domains are then revealed. We further analyze the ambiguity function of the basis function, and show that fully localized ambiguity function can be achieved by constructing the basis function using periodic signals. More importantly, we prove that time and frequency truncating such basis functions naturally leads to delay and Doppler orthogonalities, if the truncating windows are orthogonal or periodic. Motivated by this, we propose a DD Nyquist pulse shaping scheme considering signals with periodicity. Finally, our conclusions are verified by using various orthogonal and periodic pulses.

A Novel ISAC Transmission Framework based on Spatially-Spread Orthogonal Time Frequency Space Modulation

In this paper, we propose a novel integrated sensing and communication (ISAC) transmission framework based on the spatially-spread orthogonal time frequency space (SS-OTFS) modulation by considering the fact that communication channel strengths cannot be directly obtained from radar sensing. We first propose the concept of SS-OTFS modulation, where the key novelty is the angular domain discretization enabled by the spatial-spreading/de-spreading. This discretization gives rise to simple and insightful effective models for both radar sensing and communication, which result in simplified designs for the related estimation and detection problems. In particular, we design simple beam tracking, angle estimation, and power allocation schemes for radar sensing, by utilizing the special structure of the effective radar sensing matrix. Meanwhile, we provide a detailed analysis on the pair-wise error probability (PEP) for communication, which unveils the key conditions for both precoding and power allocation designs. Based on those conditions, we design a symbol-wise precoding scheme for communication based only on the delay, Doppler, and angle estimates from radar sensing, without the a priori knowledge of the communication channel fading coefficients, and also introduce the power allocation for communication. Furthermore, we notice that radar sensing and communication requires different power allocations. Therefore, we discuss the performances of both the radar sensing and communication with different power allocations and show that the power allocation should be designed leaning towards radar sensing in practical scenarios. The effectiveness of the proposed ISAC transmission framework is verified by our numerical results, which also agree with our analysis and discussions.