arXiv subjects

Saif Khan Mohammed

Publications and source records attributed to Saif Khan Mohammed.

At least 37 records · Page 2Linked to original sources

arXiv2024

Zak-OTFS: Pulse Shaping and the Tradeoff between Time/Bandwidth Expansion and Predictability

The Zak-OTFS input/output (I/O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which we refer to as the crystallization condition. When the crystallization condition is satisfied, we describe how to integrate sensing and communication within a single Zak-OTFS subframe by transmitting a pilot in the center of the subframe and surrounding the pilot with a pilot region and guard band to mitigate interference between data symbols and pilot. At the receiver we first read off the effective channel taps within the pilot region, and then use the estimated channel taps to recover the data from the symbols received outside the pilot region. We introduce a framework for filter design in the delay-Doppler (DD) domain where the symplectic Fourier transform connects aliasing in the DD domain (predictability of the I/O relation) with time/bandwidth expansion. The choice of pulse shaping filter determines the fraction of pilot energy that lies outside the pilot region and the degradation in BER performance that results from the interference to data symbols. We demonstrate that Gaussian filters in the DD domain provide significant improvements in BER performance over the sinc and root raised cosine filters considered in previous work. We also demonstrate that, by limiting DD domain aliasing, Gaussian filters extend the region where the crystallization condition is satisfied. The Gaussian filters considered in this paper are a particular case of factorizable pulse shaping filters in the DD domain, and this family of filters may be of independent interest.

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arXiv2024

Zak-OTFS and LDPC Codes

Orthogonal Time Frequency Space (OTFS) is a framework for communications and active sensing that processes signals in the delay-Doppler (DD) domain. It is informed by 6G propagation environments, where Doppler spreads measured in kHz make it more and more difficult to estimate channels, and the standard model-dependent approach to wireless communication is starting to break down. We consider Zak-OTFS where inverse Zak transform converts information symbols mounted on DD domain pulses to the time domain for transmission. Zak-OTFS modulation is parameterized by a delay period $τ_{p}$ and a Doppler period $ν_{p}$, where the product $τ_{p}ν_{p}=1$. When the channel spread is less than the delay period, and the Doppler spread is less than the Doppler period, the Zak-OTFS input-output relation can be predicted from the response to a single pilot symbol. The highly reliable channel estimates concentrate around the pilot location, and we configure low-density parity-check (LDPC) codes that take advantage of this prior information about reliability. It is advantageous to allocate information symbols to more reliable bins in the DD domain. We report simulation results for a Veh-A channel model where it is not possible to resolve all the paths, showing that LDPC coding extends the range of Doppler spreads for which reliable model-free communication is possible. We show that LDPC coding reduces sensitivity to the choice of transmit filter, making bandwidth expansion less necessary. Finally, we compare BER performance of Zak-OTFS to that of a multicarrier approximation (MC-OTFS), showing LDPC coding amplifies the gains previously reported for uncoded transmission.

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arXiv2023

OTFS -- Predictability in the Delay-Doppler Domain and its Value to Communication and Radar Sensing

In our first paper [2] we explained why the Zak-OTFS input-output (I/O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which we refer to as the crystallization condition. We argued that a communication system should operate within the crystalline regime. It is well known that it is possible to identify a linear time varying (LTV) channel if and only if it is under-spread. The crystallization condition is more restrictive than the under-spread condition, so identification is always possible. In the crystalline regime, we show that Zak-OTFS pilot sequences minimize the complexity of identifying the effective DD domain channel filter. We demonstrate that the filter taps can simply be read off from the response to a single Zak-OTFS pilot. In general, we provide an explicit formula for reconstructing the Zak-OTFS I/O relation from a finite number of received pilot symbols in the delay-Doppler (DD) domain. This reconstruction formula makes it possible to study predictability of the Zak-OTFS I/O relation for a sampled system that operates under finite duration and bandwidth constraints. We analyze reconstruction accuracy for different choices of the delay and Doppler periods, and of the pulse shaping filter. Reconstruction accuracy is high when the crystallization condition is satisfied, implying that it is possible to learn directly the I/O relation without needing to estimate the underlying channel. This opens up the possibility of a model-free mode of operation, which is especially useful when a traditional model-dependent mode of operation (reliant on estimation of the underlying physical channel) is out of reach (for example, when the channel comprises of unresolvable paths, or exhibits a continuous delay-Doppler profile such as in presence of acceleration). Our study clarifies the

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arXiv2023

OTFS -- A Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain

Orthogonal time frequency space (OTFS) is a framework for communication and active sensing that processes signals in the delay-Doppler (DD) domain. This paper explores three key features of the OTFS framework, and explains their value to applications. The first feature is a compact and sparse DD domain parameterization of the wireless channel, where the parameters map directly to physical attributes of the reflectors that comprise the scattering environment, and as a consequence these parameters evolve predictably. The second feature is a novel waveform / modulation technique, matched to the DD channel model, that embeds information symbols in the DD domain. The relation between channel inputs and outputs is localized, non-fading and predictable, even in the presence of significant delay and Doppler spread, and as a consequence the channel can be efficiently acquired and equalized. By avoiding fading, the post equalization SNR remains constant across all information symbols in a packet, so that bit error performance is superior to contemporary multi-carrier waveforms. Further, the OTFS carrier waveform is a localized pulse in the DD domain, making it possible to separate reflectors along both delay and Doppler simultaneously, and to achieve a high-resolution delay-Doppler radar image of the environment. In other words, the DD parameterization provides a common mathematical framework for communication and radar. This is the third feature of the OTFS framework, and it is ideally suited to intelligent transportation systems involving self-driving cars and unmanned ground/aerial vehicles which are self/network controlled. The OTFS waveform is able to support stable and superior performance over a wide range of user speeds.

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arXiv2021

Low Complexity Channel Estimation for OTFS Modulation with Fractional Delay and Doppler

We consider the problem of accurate channel estimation for OTFS based systems with few transmit/receive antennas, where additional sparsity due to large number of antennas is not a possibility. For such systems the sparsity of the effective delay-Doppler (DD) domain channel is adversely affected in the presence of channel path delay and Doppler shifts which are non-integer multiples of the delay and Doppler domain resolution. The sparsity is also adversely affected when practical transmit and receive pulses are used. In this paper we propose a Modified Maximum Likelihood Channel Estimation (M-MLE) method for OTFS based systems which exploits the fine delay and Doppler domain resolution of the OTFS modulated signal to decouple the joint estimation of the channel parameters (i.e., channel gain, delay and Doppler shift) of all channel paths into separate estimation of the channel parameters for each path. We further observe that with fine delay and Doppler domain resolution, the received DD domain signal along a particular channel path can be written as a product of a delay domain term and a Doppler domain term where the delay domain term is primarily dependent on the delay of this path and the Doppler domain term is primarily dependent on the Doppler shift of this path. This allows us to propose another method termed as the two-step method (TSE), where the joint two-dimensional estimation of the delay and Doppler shift of a particular path in the M-MLE method is further decoupled into two separate one-dimensional estimation for the delay and for the Doppler shift of that path. Simulations reveal that the proposed methods (M-MLE and TSE) achieve better channel estimation accuracy at lower complexity when compared to other known methods for accurate OTFS channel estimation.

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arXiv2021

Spectral Efficiency of OTFS Based Orthogonal Multiple Access with Rectangular Pulses

In this paper we consider Orthogonal Time Frequency Space (OTFS) modulation based multiple-access (MA). We specifically consider Orthogonal MA methods (OMA) where the user terminals (UTs) are allocated non-overlapping physical resource in the delay-Doppler (DD) and/or time-frequency (TF) domain. To the best of our knowledge, in prior literature, the performance of OMA methods have been reported only for ideal transmit and receive pulses. In [20] and [21], OMA methods were proposed which were shown to achieve multi-user interference (MUI) free communication with ideal pulses. Since ideal pulses are not realizable, in this paper we study the spectral efficiency (SE) performance of these OMA methods with practical rectangular pulses. For these OMA methods, we derive the expression for the received DD domain symbols at the base station (BS) receiver and the effective DD domain channel matrix when rectangular pulses are used. We then derive the expression for the achievable sum SE. These expressions are also derived for another well known OMA method where guard bands (GB) are used to reduce MUI (called as the GB based MA methods) [19]. Through simulations, we observe that with rectangular pulses the sum SE achieved by the method in [21] is almost invariant of the Doppler shift and is higher than that achieved by the methods in [19], [20] at practical values of the received signal-to-noise ratio.

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arXiv2020

Low Complexity Precoding and Detection in Multi-user Massive MIMO OTFS Downlink

We consider the problem of degradation in performance of multi-carrier multi-user massive MIMO systems when channel induced Doppler spread is high. Recently, Orthogonal Time Frequency Space (OTFS) modulation has been shown to be robust to channel induced Doppler spread. In OTFS based systems, information symbols are embedded in the delay-Doppler (DD) domain where they are jointly modulated to generate the time-domain transmit signal. Due to the multi-path delay and Doppler shifts, the DD domain information symbols need to be jointly demodulated at the receiver. For multi-carrier based communication (e.g., Orthogonal Frequency Division Multiplexing (OFDM)), massive MIMO systems have been shown to achieve high spectral and energy efficiency with low complexity multi-user precoding in the downlink. Extending the same to OTFS based downlink multi-user massive MIMO systems is challenging due to the requirement for joint demodulation of all information symbols at the user terminal (UT). In this paper, we solve this problem by proposing a novel OTFS based multi-user precoder at the base station (BS) and a corresponding low complexity detector (LCD) at the user terminals (UTs), which allows for separate demodulation of each DD domain information symbol at the UT. The complexity of the proposed precoder increases only linearly with increasing number BS antennas Q and the number of UTs. We show, through analysis, that as Q increases (with total transmitted power decreased linearly with Q), the proposed low complexity detector achieves a sum SE close to that achieved with optimal joint demodulation at each UT. Numerical simulations confirm our analysis and also show that the SE and error rate performance of the proposed OTFS based massive MIMO precoder and LCD detector is significantly more robust to channel induced Doppler spread when compared to OFDM based multi-user massive MIMO systems.

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arXiv2020

OTFS Based Random Access Preamble Transmission For High Mobility Scenarios

We consider the problem of uplink timing synchronization for Orthogonal Time Frequency Space (OTFS) modulation based systems where information is embedded in the delay-Doppler (DD) domain. For this, we propose a novel Random Access (RA) preamble waveform based on OTFS modulation. We also propose a method to estimate the round-trip propagation delay between a user terminal (UT) and the base station (BS) based on the received RA preambles in the DD domain. This estimate (known as the timing advance estimate) is fed back to the respective UTs so that they can advance their uplink timing in order that the signal from all UTs in a cell is received at the BS in a time-synchronized manner. Through analysis and simulations we study the impact of OTFS modulation parameters of the RA preamble on the probability of timing error, which gives valuable insights on how to choose these parameters. Exhaustive numerical simulations of high mobility scenarios suggests that the timing error probability (TEP) performance of the proposed OTFS based RA is much more robust to channel induced multi-path Doppler shift when compared to the RA method in Fourth Generation (4G) systems.

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arXiv2020

Derivation of OTFS Modulation from First Principles

Orthogonal Time Frequency Space (OTFS) modulation has been recently proposed to be robust to channel induced Doppler shift in high mobility wireless communication systems. However, to the best of our knowledge, none of the prior works on OTFS have derived it from first principles. In this paper, using the ZAK representation of time-domain (TD) signals, we rigorously derive an orthonormal basis of approximately time and bandwidth limited signals which are also localized in the delay-Doppler (DD) domain. We then consider DD domain modulation based on this orthonormal basis, and derive OTFS modulation. To the best of our knowledge, this is the first paper to rigorously derive OTFS modulation from first principles. We show that irrespective of the amount of Doppler shift, the received DD domain basis signals are localized in a small interval of size roughly equal to the inverse time duration along the Doppler domain and of size roughly equal to the inverse bandwidth along the delay domain (time duration refers to the length of the time-interval where the TD transmit signal has been limited). With sufficiently large time duration and bandwidth, there is little interference between information symbols modulated on different basis signals, which allows for joint DD domain equalization of all information symbols. This explains the inherent robustness of DD domain modulation to channel induced Doppler shift when compared with Orthogonal Frequency Division Multiplexing (OFDM). The degree of localization of the DD domain basis signals is inversely related to the time duration of the transmit signal, which explains the trade-off between robustness to Doppler shift and latency.

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arXiv2020

Time-Domain to Delay-Doppler Domain Conversion of OTFS Signals in Very High Mobility Scenarios

In Orthogonal Time Frequency Space (OTFS) modulation, information symbols are embedded in the delay-Doppler (DD) domain instead of the time-frequency (TF) domain. n order to ensure compatibility with existing OFDM systems (e.g. 4G LTE), most prior work on OTFS receivers consider a two-step conversion, where the received time-domain (TD) signal is firstly converted to a time-frequency (TF) signal (using an OFDM demodulator) followed by post-processing of this TF signal into a DD domain signal. In this paper, we show that the spectral efficiency (SE) performance of a two-step conversion based receiver degrades in very high mobility scenarios where the Doppler shift is a significant fraction of the communication bandwidth (e.g., control and non-payload communication (CNPC) in Unmanned Aircraft Systems (UAS)). We therefore consider an alternate conversion, where the received TD signal is directly converted to the DD domain. The resulting received DD domain signal is shown to be not the same as that obtained in the two-step conversion considered in prior works. The alternate conversion does not require an OFDM demodulator and is shown to have lower complexity than the two-step conversion. Analysis and simulations reveal that even in very high mobility scenarios, the SE achieved with the alternate conversion is invariant of Doppler shift and is significantly higher than the SE achieved with two-step conversion (which degrades with increasing Doppler shift).

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arXiv2018

Timing Advance Estimation and Beamforming of Random Access Response in Crowded TDD Massive MIMO Systems

Timing advance (TA) estimation at the base station (BS) and reliable decoding of random access response (RAR) at the users are the most important steps in the initial random access (RA) procedure. However, due to the limited availability of physical resources dedicated for RA, successful completion of RA requests would become increasingly difficult in high user density scenarios, due to contention among users requesting RA. In this paper, we propose to use the large antenna array at the massive MIMO BS to jointly group RA requests from different users using the same RA preamble. We then beamform the common RAR of each detected user group onto the same frequency resource, in such a way that most users in the group can reliably decode the RAR. The proposed RAR beamforming therefore automatically resolves the problem of collision between multiple RA requests on the same RA preamble, which reduces the RA latency significantly as compared to LTE. Analysis and simulations also reveal that for a fixed desired SINR of the received RAR, both the required per-user RA preamble transmission power and the total RAR beamforming power can be decreased roughly by 1.5 dB with every doubling in the number of BS antennas.

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arXiv2017

Achievable Rate Region of the Zero-Forcing Precoder in a 2 X 2 MU-MISO Broadcast VLC Channel with Per-LED Peak Power Constraint and Dimming Control

In this paper, we consider the 2 X 2 multi-user multiple-input-single-output (MU-MISO) broadcast visible light communication (VLC) channel with two light emitting diodes (LEDs) at the transmitter and a single photo diode (PD) at each of the two users. We propose an achievable rate region of the Zero-Forcing (ZF) precoder in this 2 X 2 MU-MISO VLC channel under a per-LED peak and average power constraint, where the average optical power emitted from each LED is fixed for constant lighting, but is controllable (referred to as dimming control in IEEE 802.15.7 standard on VLC). We analytically characterize the proposed rate region boundary and show that it is Pareto-optimal. Further analysis reveals that the largest rate region is achieved when the fixed per-LED average optical power is half of the allowed per-LED peak optical power. We also propose a novel transceiver architecture where the channel encoder and dimming control are separated which greatly simplifies the complexity of the transceiver. A case study of an indoor VLC channel with the proposed transceiver reveals that the achievable information rates are sensitive to the placement of the LEDs and the PDs. An interesting observation is that for a given placement of LEDs in a 5 m X 5 m X 3 m room, even with a substantial displacement of the users from their optimum placement, reduction in the achievable rates is not significant. This observation could therefore be used to define "coverage zones" within a room where the reduction in the information rates to the two users is within an acceptable tolerance limit.

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arXiv2016

Information Theoretic Performance of Periodogram-based CFO Estimation in Massive MU-MIMO Systems

In this paper, we study the information theoretic performance of the modified time-reversal maximum ratio combining (TR-MRC) receiver (presented in [9]) with the spatially averaged periodogram-based carrier frequency offset (CFO) estimator (proposed in [7]) in multi-user massive MIMO systems. Our analysis shows that an $\mathcal{O}(\sqrt{M})$ array gain is achieved with this periodogram-based CFO estimator, which is same as the array gain achieved in the ideal/zero CFO scenario ($M$ is the number of base station antennas). Information theoretic performance comparison with the correlation-based CFO estimator for massive MIMO systems (proposed in [6]) reveals that this periodogram-based CFO estimator is more energy efficient in slowly time-varying channels.

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arXiv2016

Impact of Frequency Selectivity on the Information Rate Performance of CFO Impaired Single-Carrier Massive MU-MIMO Uplink

In this paper, we study the impact of frequency-selectivity on the gap between the required per-user transmit power in the residual CFO scenario (i.e. after CFO estimation/compensation at the base-station (BS) from [6]) and that in the ideal/zero CFO scenario, for a fixed per-user information rate, in single-carrier massive MU-MIMO uplink systems with the TR-MRC receiver. Information theoretic analysis reveals that this gap decreases with increasing frequency-selectivity of the channel. Also, in the residual CFO scenario, an $\mathcal{O}(\sqrt{M})$ array gain is still achievable ($M$ is the number of BS antennas) in frequency-selective channels with imperfect channel estimates.

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arXiv2016

Information Rate Performance of Massive MU-MIMO Uplink with Constant Envelope Pilot-based Frequency Synchronization

In this paper, we consider a constant envelope (CE) pilot-based low-complexity technique for frequency synchronization in multi-user massive MIMO systems. Study of the complexity-performance trade-off shows that this CE-pilot-based technique provides better MSE performance when compared to existing low-complexity high-PAPR pilot-based CFO (carrier frequency offset) estimator. Numerical study of the information rate performance of the TR-MRC receiver in imperfect CSI scenario with this CE-pilot based CFO estimator shows that it is more energy-and-spectrally efficient than existing low-complexity CFO estimator in massive MIMO systems. It is also observed that with this CE-pilot based CFO estimation, an $\mathcal{O}(\sqrt{M})$ array gain is achievable.

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arXiv2016

Constant Envelope Pilot-Based Low-Complexity CFO Estimation in Massive MU-MIMO Systems

In this paper we consider a constant envelope pilot signal based carrier frequency offset (CFO) estimation in massive multiple-input multiple-output (MIMO) systems. The proposed algorithm performs spatial averaging on the periodogram of the received pilots across the base station (BS) antennas. Our study reveals that the proposed algorithm has complexity only linear in $M$ (the number of BS antennas). Further our analysis and numerical simulations also reveal that with fixed number of users and a fixed pilot length, the minimum required transmit pilot power decreases as $\frac{1}{\sqrt{M}}$ with increasing $M$, while maintaining a fixed desired mean squared error (MSE) of CFO estimation.

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arXiv2016

How Much Bandpass Filtering is Required in Massive MIMO Basestations?

In this paper, we study the impact of aliased out-of-band interference signals on the information sum-rate of the maximum ratio combining receiver in massive multiple-input multiple-output (MIMO) uplink, with both perfect and imperfect channel estimates, in order to determine the required out-of-band attenuation in RF bandpass filters (BPFs). With imperfect channel estimates, our study reveals that as the number of base-station (BS) antennas ($M$) increases, the required attenuation at the BPFs increases as $\mathcal{O}(\sqrt{M})$ with $M \to \infty$, provided the desired information sum-rate (both in the presence and in the absence of AOOBIs (aliased out-of-band interferers)) remains fixed. This implies a practical limit on the number of BS antennas due to the increase in BPF design complexity and power consumption with increasing $M$.

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arXiv2016

Impact of Underlaid Multi-antenna D2D on Cellular Downlink in Massive MIMO Systems

In this paper, we consider a massive multiple-input multiple-output (MIMO) downlink system underlaid with a network of multi-antenna D2D user equipments (UEs). Each D2D transmitter (Tx) uses all its antennas to beamform information towards its desired D2D receiver, which uses only a single antenna for reception. While beamforming at the D2D Tx reduces D2D interference to the neighbouring cellular UEs (CUEs), the cellular-to-D2D interference is also negligible due to highly directional beamforming at the massive MIMO base station. For the above proposed system, we analyze the average per-user spectral efficiency (SE) of CUEs ($R^{c,d}$) as a function of the D2D area spectral efficiency (ASE). Our analysis reveals that for a fixed D2D ASE ($R_0^{(d)}$) and fixed number of D2D antennas ($N$), with increasing density of D2D Txs ($λ$), $R^{c,d}$ increases (for sufficiently large $λ$) and approaches a fundamental limit $R_{\infty}^{c,d}$ as $λ\to \infty$. Also, $R_{\infty}^{c,d}$ depends on $R_{0}^{(d)}$ and $N$, only through the ratio $\frac{R_{0}^{(d)}}{N-1}$, i.e, for a given fundamental limit $R_{\infty}^{c,d}$, the D2D ASE can be approximately doubled with every doubling in $N$.

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