arXiv subjects

Nishant Mehrotra

Publications and source records attributed to Nishant Mehrotra.

15 recordsLinked to original sources

arXiv2026

Zak-OTFS: A Predictable Physical Layer for Communications and Sensing

This tutorial derives the mathematical foundations of what it means for a carrier waveform to be predictable and non-selective. We focus on Zak-OTFS, where each carrier waveform is a pulse in the delay-Doppler (DD) domain, formally a quasi-periodic localized function with specific periods along delay and Doppler. Viewed in the time domain, the Zak-OTFS carrier is realized as a pulse train modulated by a tone (termed a pulsone). We start by providing physical intuition, describing what it means for the Zak-OTFS carrier waveforms to be geometric modes of the Heisenberg-Weyl (HW) group of discrete delay and Doppler shifts that define the discrete-time communication model. In fact, we show that these geometric modes are common eigenvectors of a maximal commutative subgroup of our discrete HW group. When the channel delay spread is less than the delay period, and the channel Doppler spread is less than the Doppler period, we show that the Zak-OTFS input-output (I/O) relation is predictable and non-selective. Given the I/O response at one DD point in a frame, it is possible to predict the I/O response at all other points, without recourse to some mathematical model of the channel. While it may be intuitive that geometric modes of the HW group are predictable and non-selective wireless carriers, this is not a requirement. We provide a necessary and sufficient condition that depends on the ambiguity properties of the basis of carrier waveforms. In fact, we show that the structure of a pulse train modulated by a Hadamard matrix is common to several families of waveforms proposed for 6G, including Zak-OTFS, AFDM, OTSM and ODDM.

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arXiv2026

Delay-Doppler Channel Estimation using Arbitrarily Modulated Data Transmissions

Conventional delay-Doppler (DD) communication and sensing systems require transmitting pilot frames at every channel coherence time interval in order to keep track of channel variations at the cost of spectral efficiency. In this paper, we propose an approach to utilize data transmissions that modulate arbitrary waveforms with zero-mean, unit average energy symbols for DD channel estimation without requiring pilot transmissions in every coherence time interval. Numerical evaluation over practical doubly-selective channel models demonstrate $\sim 1.8 \times$ improvement in uncoded spectral efficiency with our proposed data-based approach over conventional pilot-based approaches across various $6$G modulation schemes.

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arXiv2026

Real-Time and Scalable Zak-OTFS Receiver Processing on GPUs

Orthogonal time frequency space (OTFS) modulation offers superior robustness to high-mobility channels compared to conventional orthogonal frequency-division multiplexing (OFDM) waveforms. However, its explicit delay-Doppler (DD) domain representation incurs substantial signal processing complexity, especially with increased DD domain grid sizes. To address this challenge, we present a scalable, real-time Zak-OTFS receiver architecture on GPUs through hardware--algorithm co-design that exploits DD-domain channel sparsity. Our design leverages compact matrix operations for key processing stages, a branchless iterative equalizer, and a structured sparse channel matrix of the DD domain channel matrix to significantly reduce computational and memory overhead. These optimizations enable low-latency processing that consistently meets the 99.9-th percentile real-time processing deadline. The proposed system achieves up to 906.52 Mbps throughput with a DD grid size of (16384,32) using 16QAM modulation over 245.76 MHz bandwidth. Extensive evaluations under a Vehicular-A channel model demonstrate strong scalability and robust performance across CPU (Intel Xeon) and multiple GPU platforms (NVIDIA Jetson Orin, RTX 6000 Ada, A100, and H200), highlighting the effectiveness of compute-aware Zak-OTFS receiver design for next-generation (NextG) high-mobility communication systems.

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arXiv2025

Increasing Data Rate through Shaping on Wireless Channels Subject to Mobility and Delay Spread

This letter describes how to improve performance of cellular systems by combining non-equiprobable signaling (shaping) with low-density parity check (LDPC) coding for an orthogonal frequency division multiplexing system. We focus on improving performance at the cell edge, where the 5G standard specifies a suite of LDPC codes with different rates that are applied to 4-QAM. We employ the method of shaping on rings which adds to the transmission rate as it shapes the input distribution. We double the size of the $4$-QAM constellation by introducing a second shell of signal points, and we implement non-equiprobable signaling through a shaping code which selects the high energy shell less frequently than the low energy shell. We describe how to combine coding and shaping by integrating shaping into the calculation of log-likelihood ratios (LLRs) necessary for decoding LDPC codes. We employ rate $1/2$ LDPC coding and select the rate of the shaping code to match that of rate $3/4$ LDPC coding using $4$-QAM. We present simulation results for a representative Veh-A channel showing gains of $4$ dB at a bit error rate (BER) of $10^{-3}$. When we choose an LDPC code from the 5G suite to match the BER performance of rate $1/2$ LDPC coding with shaping we show that transmission rate can be improved by $20 $%.

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arXiv2025

A Design Framework that Unifies 6G Modulation Schemes for Double Selectivity

There is significant recent interest in designing new modulation schemes for doubly-selective channels with large delay and Doppler spreads, where legacy modulation schemes based on time-frequency signal representations underperform. Multiple modulation schemes, e.g., in the delay-Doppler, chirp, time-sequency, and other domains, have been proposed in the literature for this purpose, with varying implementation details. In this letter, we establish that all previously proposed modulation schemes for doubly-selective signaling are instances of a single family of complex Hadamard-modulated pulse trains. When the delay and Doppler spread of the doubly-selective channel is limited to a certain support, all modulation schemes in this waveform family offer equivalent, full diversity achieving performance with no symbol fading and low channel estimation overhead. The existence of this waveform family also enables flexible, multi-waveform co-existence -- allowing a common transceiver architecture to generate multiple waveforms in the family, that may each be flexibly allocated to different users and services.

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arXiv2025

Pulse Shaping Filter Design for Integrated Sensing & Communication with Zak-OTFS

Zak-OTFS provides a framework for integrated sensing & communication (ISAC) in high delay and Doppler spread environments. Pulse shaping filter design enables joint optimization of sensing and communication performance. For sensing, a localized pulse shaping filter enables input-output (I/O) relation estimates close to the physical scattering channel. For communication, orthogonality of the pulse shape on the information lattice prevents inter-symbol interference, and no time and bandwidth expansion enables full spectral efficiency. A filter simultaneously meeting all three objectives is ideal for ISAC. Existing filter designs achieve two, but not all three objectives. In this work, we design pulse shaping filters meeting all three objectives via the Isotropic Orthogonal Transform Algorithm. The proposed filters have improved spectral efficiency, data detection and sensing performance over existing filter choices.

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arXiv2025

Instantaneous Polarimetry with Zak-OTFS

Polarimetry, which is the ability to measure the scattering response of the environment across orthogonal polarizations, is fundamental to enhancing wireless communication and radar system performance. In this paper, we utilize the Zak-OTFS modulation to enable instantaneous polarimetry within a single transmission frame. We transmit a Zak-OTFS carrier waveform and a spread carrier waveform mutually unbiased to it simultaneously over orthogonal polarizations. The mutual unbiasedness of the two waveforms enables the receiver to estimate the full polarimetric response of the scattering environment from a single received frame. Unlike existing methods for instantaneous polarimetry with computational complexity quadratic in the time-bandwidth product, the proposed method enables instantaneous polarimetry at near-linear complexity in the time-bandwidth product. Via numerical simulations, we show ideal polarimetric target detection and parameter estimation results with the proposed method, with improvements in computational complexity and greater clutter resilience over comparable baselines.

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arXiv2025

Discrete Radar based on Modulo Arithmetic

Zak-OTFS is modulation scheme where signals are formed in the delay-Doppler (DD) domain, converted to the time domain (DD) for transmission and reception, then returned to the DD domain for processing. We describe how to use the same architecture for radar sensing. The intended delay resolution is $\frac{1}{B}$ where $B$ is the radar bandwidth, and the intended Doppler resolution is $\frac{1}{T}$ where $T$ is the transmission time. We form a radar waveform in the DD domain, illuminate the scattering environment, match filter the return, then correlate with delay and Doppler shifts of the transmitted waveform. This produces an image of the scattering environment, and the radar ambiguity function expresses the blurriness of this image. The possible delay and Doppler shifts generate the continuous Heisenberg-Weyl group which has been widely studied in the theory of radar. We describe how to approach the problem of waveform design, not from the perspective of this continuous group, but from the perspective of a discrete group of delay and Doppler shifts, where the discretization is determined by the intended delay and Doppler resolution of the radar. We describe how to approach the problem of shaping the ambiguity surface through symplectic transformations that normalize our discrete Heisenberg-Weyl group. The complexity of traditional continuous radar signal processing is $\mathcal{O}\big(B^2T^2\big)$. We describe how to reduce this complexity to $\mathcal{O}\big(BT\log T\big)$ by choosing the radar waveform to be a common eigenvector of a maximal commutative subgroup of our discrete Heisenberg-Weyl group. The theory of symplectic transformations also enables defining libraries of optimal radar waveforms with small peak-to-average power ratios.

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arXiv2025

Low-Complexity Equalization of Zak-OTFS in the Frequency Domain

4G/5G wireless standards use orthogonal frequency division multiplexing (OFDM) which is robust to frequency selectivity. Equalization is possible with a single tap filter, and low-complexity equalization makes OFDM an attractive physical layer. However the performance of OFDM degrades with mobility, since Doppler spreads introduce inter-carrier interference (ICI) between subcarriers and they are no longer orthogonal. Zak-transform based orthogonal time frequency space (Zak-OTFS) modulation has been shown to be robust to doubly selective channels. Zak-OTFS signals are formed in the delay-Doppler (DD) domain, converted to time domain (TD) for transmission and reception, then returned to the DD domain for processing. The received signal is a superposition of many attenuated copies since the doubly selective channel introduces delay and Doppler shifts. The received symbols are more difficult to equalize since they are subject to interference along both delay and Doppler axes. In this paper, we propose a new low-complexity method of equalizing Zak-OTFS in the frequency domain (FD). We derive the FD system model and show that it is unitarily equivalent to the DD system model. We show that the channel matrix in the FD is banded, making it possible to apply conjugate gradient methods to reduce the complexity of equalization. We show that complexity of FD equalization is linear in the dimension of a Zak-OTFS frame. For comparison the complexity of naive MMSE equalization is cubic in the frame dimension. Through numerical simulations we show that FD equalization of Zak-OTFS achieves similar performance as equalization in DD domain.

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arXiv2025

Improving the Spectral Efficiency of Zak-OTFS via Mutually Unbiased Bases

Orthogonal signaling or Nyquist signaling limits the number of information symbols transmitted in bandwidth $B$ and time $T$ to be $BT$, the time-bandwidth product. Transmitting more than $BT$ symbols leads to loss of orthogonality. The standard approach is to reduce the symbol interval and resolve the resulting inter-symbol interference. This requires changing the sampling frequency and possibly the sampling clock. This paper shows that it is possible to improve spectral efficiency on doubly spread channels without changing the sampling frequency. The idea is to superimpose the information symbols using mutually unbiased bases (MUB) while maintaining the original spacing. We carry this out in the delay-Doppler domain using Zak-transform based orthogonal time frequency space (Zak-OTFS) modulation as it allows construction of MUB. We also construct a precoder that mitigates the effect of the doubly-spread channel. This simplifies receiver processing to detection in Gaussian noise since each basis appears to the other as Gaussian noise. This reduction makes it possible to use trellis coded modulation to further improve the bit-error performance. Numerical results demonstrate that the proposed signaling scheme using MUB achieves good bit-error performance.

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arXiv2025

Differential Communication in Channels with Mobility and Delay Spread using Zak-OTFS

Zak-transform based orthogonal time frequency space (Zak-OTFS) is a delay-Doppler (DD) domain modulation scheme in which the signal processing is carried out in the DD domain. The channel when viewed in the DD domain is predictable. However, even with Zak-OTFS, pilots need to be sent periodically, albeit at a lower rate. In this paper, we propose a differential communication scheme for Zak-OTFS systems that alleviates the need for periodic pilot transmission. Towards this, we analytically show that the detected data can be used as a pilot and that the channel estimate obtained from the detected data can enable further detection enabling the "differential" aspect of the communication. Specifically, we leverage the prediction capability of the DD channel in Zak-OTFS to use the channel estimate (obtained from detected data symbols treated as pilots) in the previous instant to detect data in the next instant and propagate this forward. The advantages are two fold. First, it allows the data symbols to enjoy higher energy since the energy that would otherwise be required for pilot symbols can also be allocated to data symbols. Second, it allows for full spectral efficiency compared to point or embedded pilots. Comparison with the full spectral efficiency achieving spread pilot scheme shows that the proposed method achieves better bit-error rate at lower complexity.

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arXiv2025

Zak-OTFS with Spread Carrier Waveforms

Zak-OTFS (orthogonal time frequency space) modulation is a communication framework that parameterizes the wireless channel in the delay-Doppler (DD) domain, where the parameters map directly to physical attributes of the scatterers that comprise the scattering environment. As a consequence, the channel can be efficiently acquired and equalized. The Zak-OTFS carrier is a pulse in the DD domain, and the Zak transform converts it to a pulse train modulated by a tone (pulsone) in the time domain. The pulsone waveform is localized rather than spread, and it suffers from high peak-to-average power ratio (PAPR). We describe how to transform the orthonormal basis of Zak-OTFS pulsones into an orthonormal basis of spread carrier waveforms with low PAPR (only $6.58$ dB) that support communication in the presence of mobility and delay spread. This transformation is realized by a unitary transform based on the discrete affine Fourier transform. Unlike other spread modulations that achieve low PAPR by spreading information across a wider bandwidth (thus reducing the spectral efficiency), the proposed spread carrier-based Zak-OTFS achieves full spectral efficiency like pulsone-based Zak-OTFS, with $5.6$ dB lower PAPR per basis element. We demonstrate uncoded bit error rate (BER) similar to pulsone-based Zak-OTFS, and improved BER performance over competing methods based on OFDM and OTFS in high mobility & delay spread environments.

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arXiv2025

Low-Complexity Frequency Domain Equalization of Zak-OTFS in Doubly-Spread Channels

We communicate over wireless channels by first estimating and then equalizing the effective channel. In Zak-OTFS (orthogonal time frequency space) modulation the carrier waveform is a pulse in the delay-Doppler (DD) domain, formally a quasi-periodic localized function with specific periods along delay and Doppler. When the channel delay spread is less than the delay period, and the channel Doppler spread is less than the Doppler period, the response to a single Zak-OTFS carrier provides an image of the scattering environment and can be used to predict the effective channel at all other carriers. This makes DD domain channel estimation straightforward, and there is no loss in spectral efficiency since it is possible to design data and pilot signals that are mutually unbiased. However, equalization in the DD domain has high complexity ${\mathcal O}(M^3N^3)$ where $M$, $N$ are respectively the number of delay and Doppler bins in an OTFS frame, and $MN$ is the number of information symbols. We demonstrate that equalization in the frequency domain (FD) reduces complexity to only ${\mathcal O}(M^2 N^2)$ by taking advantage of the banded structure of the effective FD channel. We also derive a low-complexity method to reconstruct the effective FD channel from the estimated DD domain effective channel.

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arXiv2025

Zak-OTFS for Mutually Unbiased Sensing and Communication

Waveforms with ideal ambiguity functions are fundamental to integrated sensing and communication, to active sensing (radar), and to uplink multiple access. We describe a general method of constructing waveforms using the discrete Zak transform (DZT) to convert sequences of length $MN$ in the time domain to waveforms in the delay-Doppler (DD) domain, each of which is defined by an $M\times N$ quasi-periodic array. The DZT preserves inner products, and we show that phase coded waveforms used in radar (CAZAC sequences) determine noise-like waveforms in the DD domain, each with low Peak to Average Power Ratio. In a Zak-OTFS communication system, we show that these waveforms are mutually unbiased with respect to every carrier and use them to integrate sensing and communication as spread pilots. We view each waveform as a linear combination of Zak-OTFS carriers and show that the self-ambiguity function is supported on a discrete line in the integers modulo $MN$. The sidelobes are significantly lower than the original CAZAC sequence, and the advantage of discrete support is better localization/resolution in delay and Doppler compared with standard methods based on chirps or tones. We show that the absolute value of the cross-ambiguity function for pairs of waveforms in the same family is small and constant. This property makes the waveforms ideal preambles in the 2-step RACH protocol introduced in Release 15, 3GPP to enable grant-free multiple access. The characteristics of the cross-ambiguity function make it possible to simultaneously detect multiple preambles in the presence of mobility and delay spread.

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arXiv2022

Consensus ADMM-Based Distributed Simultaneous Imaging & Communication

This paper takes the first steps toward enabling wireless networks to perform both imaging and communication in a distributed manner. We propose Distributed Simultaneous Imaging and Symbol Detection (DSISD), a provably convergent distributed simultaneous imaging and communication scheme based on the alternating direction method of multipliers. We show that DSISD achieves similar imaging and communication performance as centralized schemes, with order-wise reduction in computational complexity. We evaluate the performance of DSISD via 2.4 GHz Wi-Fi simulations.

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