Generalized Adaptive Spreading Modulation: A Novel Waveform for Integrated Sensing and Communication Oriented Vehicular Applications

This study aims to present a comparative analysis of existing waveforms for integrated sensing and communication (ISAC) in vehicular environments. A novel multicarrier framework called generalized adaptive spreading modulation (GASM) is proposed for ISAC-enabled vehicular environments. The GASM waveform offers symbol spreading in both time and frequency domains with tunable spreading parameters, which allows the proposed GASM-based waveform to adapt according to the rapid time-frequency variations of the fading channel. This helps to combat the most common system impairments, such as carrier frequency offset (CFO) and symbol timing offset (STO). The GASM scheme is the generalization of various existing waveforms, such as orthogonal frequency-division multiplexing (OFDM), fractional Fourier transform-based OFDM (FrFT-based OFDM), and orthogonal chirp division multiplexing (OCDM). The proposed GASM-based ISAC system is evaluated in terms of average bit error rate (ABER) for the communication and ambiguity function (AF) for sensing capabilities. The performance of the GASM-based ISAC system is found superior as compared to the existing waveforms, i.e., OFDM, FrFT-based OFDM, OCDM, generalized frequency division multiplexing (GFDM), and orthogonal time frequency space (OTFS) modulation.

This study aims to present a comparative analysis of existing waveforms for Integrated sensing and communication (ISAC) in vehicular environments.Further, a novel multicarrier framework based on Generalized Adaptive Spreading Modulation (GASM) is proposed for ISAC-enabled vehicular environments.The GASM waveform offers symbol spreading in both time and frequency domains with tunable spreading parameters, which allows the proposed GASM-based waveform to adapt according to the rapid time-frequency variations of the fading channel and combat the most common system impairments such as carrier frequency offset (CFO) and symbol timing offset (STO).The GASM scheme is the generalization of various existing waveforms such as Orthogonal frequencydivision multiplexing (OFDM), Fractional Fourier Transform based OFDM (FrFT-based OFDM), and Orthogonal Chirp Division Multiplexing (OCDM).The proposed GASM-based ISAC system is evaluated in terms of average bit error rate (ABER) for communication and ambiguity function (AF) for range and Doppler capabilities.The performance of GASMbased ISAC system is compared with existing waveforms, i.e., OFDM, FrFT-based OFDM, OCDM, Generalized Frequency Division Multiplexing (GFDM), and Orthogonal Time Frequency Space (OTFS) modulation.It is observed from the simulation results that the proposed GASM scheme outperforms all existing waveforms due to its superior competence in handling fading and synchronization errors.

I. ISAC FOR VEHICLES: MOTIVATION AND USER CASES
R ECENT years have witnessed the rise of smart era wherein every system is aiming to become autonomous and self-sustaining for improving the human lifestyle.This booming intelligence in technology is suffusing the transport infrastructure to materialize futuristic applications such as D. Singh and T. Myllylä are affiliated with Research Unit of Health Sciences and Technology, University of Oulu, Finland (e-mail: {daljeetsingh.thapar@gmail.com/daljeet.singh,teemu.myllyla}@oulu.fi).
H. D. Joshi and A. K. Singh are affiliated with the Department of Electronics and Communication Engineering, Thapar Institute of Engineering and Technology, Patiala, Punjab 147001, India (e-mail: {hemdutt.joshi,aksingh}@thapar.edu).
W. Anwar is with Vodafone Chair Mobile Communications Systems, Technical University of Dresden, Dresden, Germany (e-mail: waqar.anwar@tudresden.de).
M. Magarini is with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan, Italy (e-mail: maurizio.magarini@polimi.it).pedestrian detection, lane change caution, traffic sign observance, emergency braking, and finally fully autonomous driving which were just a dream in the past.The incubation of autonomous driving capabilities in vehicles has proven to enhance the driving experience by reducing vehicle accidents and liberating thousands of humans from the tedious task of driving [3].

A. Motivation
In order to inculcate such competence in vehicles and make driving autonomous, the current vehicles are equipped with Advanced Driver Assistance System (ADAS) that harness the information from onboard devices and sensors such as long and short-range radars for relative ranging and Doppler estimation, global positioning system for the absolute position of a vehicle, Light detection and ranging (LiDAR) for object detection, cameras for lane detection, traffic signal, and other visual monitoring of other vehicles, Sonars for determining surrounding obstacles [4].In order to support privacy, future ADAS systems majorly rely on the echolocation information given by radars for ambient cognition.Moreover, the information collected solely by ADAS of vehicles can only raise them from level zero with no automation to level three with conditional automation or level four with high automation [5].
The practical driving environments are labyrinthine in nature and in order to reach level five with full automation wherein the vehicle performs all driving tasks under all conditions and no human attention or interaction is required, the vehicle can not rely only on sensing information from ADAS but also requires supplementary support from surrounding infrastructure through communication.Such advanced vehicular systems involve the integration of communication and sensing capabilities to use the same equipment, spectrum, and signals for both sensing and communication purposes.This integration improves the overall efficiency of the system in terms of spectral usage, hardware utilization, and information processing.Therefore, it is natural to connect the operations of communication and sensing in vehicular environments, which motivates the recent research theme of Integrated Sensing and Communications (ISAC) [6].

B. Use cases of ISAC in vehicular environments
An ISAC-oriented vehicular environment includes interactions between vehicle-to-vehicle (V2V), vehicle-to-pedestrian (V2P), and vehicle-to-infrastructure (V2I) for safe and efficient transportation.These interactions improve the overall performance of vehicular infrastructure on three different levels: (i).Firstly, individual vehicles can access local and global road information in real-time to enhance driving safety and encourage autonomous driving capabilities; (ii).Secondly, assists in intelligent traffic management and personalized traffic control; (iii).Finally, enables the driver to relax and enjoy the journey without the necessity to focus on the road thus elevating the driving experience.
Several use cases and system Key Performance Indicators (KPIs) required for these vehicular interactions involving ISAC have been presented recently in Cellular-Vehicle-to-Everything (C-V2X) white paper by 5G Automotive Association [2] and 3GPP feasibility study on ISAC in Release-19 [1].Some of the relevant use cases with original terminology and serial number are: 5.2: Pedestrian/animal intrusion detection on a highway, 5.7: Sensing for railway intrusion detection, 5.8: Sensing Assisted Automotive Maneuvering and Navigation, 5.11: sensing at crossroads with/without obstacle, 5.26: Accurate sensing for automotive maneuvering and navigation service, 5.28: Vehicles Sensing for ADAS, 5.30: Sensing for automotive maneuvering and navigation service when not served by random access network and 5.31: Blind spot detection [1].
Figure 1 showcases a scenario generated especially to replicate the practical fully autonomous vehicular environment based on the ISAC system.The use cases suggested by 3GPP are included in each case.Case 1 subsumes more generic use cases 5.8, 5.26, and 5.28 as well as other applications such as cooperative traffic gap, infrastructure-assisted environment perception, and teleoperated driving support wherein all vehicles are working in autonomous mode using ADAS and assistive automotive maneuvering and navigation systems [2].Case 2 is a more complex situation of blind spot detection (5.31) and vehicle decision assist in which vehicle B is trying to overtake vehicle A but is cautioned by A about incoming vehicle C.
Infrastructure-assisted environment perception is visualized in case 3 wherein an animal on the road and a vulnerable road user (VRU), such as a pedestrian or cyclist (P) standing near the road are detected by ADAS of vehicle A but not by vehicle B due to signal blockage.Therefore, Drone (DR) gives this sensing information to vehicle B through Base Station BS as per the use case 5.2 of [1].Further, in Case 4, Vehicle C does not have sufficient information on traffic and cannot connect to BS or DR.Therefore, Vehicle C requests the assistance of sensing data from Vehicle A and B to maneuver safely through traffic which covers use case 5.30 of [1].Finally, in case 5, Vehicle D wants to take a left turn on a cross-road but the information given by ADAS of D is not enough for maneuvering safely.Therefore, Vehicle D requests assistance from Vehicle E and F to turn safely which covers the use case 5.11 of [1].It should be noted that the ISAC system undertaken in this study is applicable to both driven and autonomous vehicles (AV) and is not limited to these use cases only.

II. REVIEW OF EXISTING WAVEFORMS FOR ISAC
The waveform design is the backbone of every ISAC system wherein the choice of a particular waveform influences its overall performance.An appropriate waveform should be able to achieve the tasks of supporting high data rates as well as provide accurate sensing capabilities in vehicular environments.Section II-A provides an overview of existing waveforms that have been proposed in ISAC system till now.Further, the requirements and challenges while designing a waveform for the ISAC system are presented in Sec.II-B.

A. Traditional ISAC waveforms
Academicians and researchers have worked a lot in the last two decades to find the optimized waveform for ISAC system.To date, most of the co-design ISAC systems have focused on utilizing the existing waveforms and the associated modulation schemes [7].The Joint Radar-Communication (JRC) system implements communication as a secondary function on a radar platform and utilizes chirp sequence, phase-modulated continuous wave (PMCW), Frequency-Modulated Continuous-Wave (FMCW), and its variations [8].This JRC system achieves satisfactory sensing performance but fails to accomplish the goal of high data rate communication which is not suitable for real-time vehicular applications wherein, the data from several different commodities i.e.V2V, V2P, and V2I is collected and processed.
On the other hand, the Joint Communications-Radar (JCR) system, implements secondary radar functions on a standardized communication system by utilizing digital multi-carrier schemes such as OFDM, FrFT-based OFDM, GFDM, DFTs-OFDM, OCDM, and OTFS, etc., thus achieving decent communication performance but sacrificing the sensing [9].Therefore, there is a very strong motivation to design a novel unified waveform that does not favor one or the other by default but rather adapts according to the application requirements.

B. Challenges in designing ISAC waveform
There are two main challenges while designing a waveform for an ISAC-based vehicular system, which are hereafter discussed.
1) Tunable parameter for communication sensing trade-off: The first challenge is the trade-off between the communication and sensing performance of ISAC systems.The reason behind this is that even though radar and communication systems work on similar frequencies, they differ completely in waveform design due to the unique KPIs [7].The performance of sensing primly depends on the following parameters: radar maximum unambiguous range and range resolution; target velocity and velocity resolution; latency, missed detection, and false alarm rate.Similarly, for performance evaluation of the communication link, the following metrics are used: spectral and energy efficiency; and robustness in terms of BER and packet error rate (PER) [1].Moreover, both systems operate under different channel conditions and have different time-frequency characteristics of the signal.Therefore, the ISAC system should have tunable parameters to optimize the radar and communication performance to fulfill the Quality of Service (QoS) requirements.
2) Catering fading and system impairments: Another key requirement for ISAC systems is that the designed waveform should suit the rapidly varying channel conditions in time and frequency that result in dual time-frequency selective (TFS) fading, also known as doubly selective fading scenarios which are common in vehicular environments, and generate large delay and Doppler errors.Figure 2 represents the time-frequency representation of existing MCSs along with the proposed GASM waveform (to be introduced in Sec.IV).It is clearly evident from Fig. 2 that the performance of waveform achieving constant spacing of ∆ f in the frequency domain (OFDM) or ∆ t in the time domain (DFT-s-OFDM) degrades drastically for such a channel [10].
Further, by achieving symbol spacing in two dimensions, a higher diversity gain can be realized, resulting in improved system performance compared to techniques having no spreading or spreading in only one dimension [10].This is evident from the popularity of GFDM, OTFS, and OCDM systems that achieve two-dimensional symbol spacing [10].However, GFDM has constant ∆ t and ∆ f in the time-frequency domain, OTFS has constant ∆ t and ∆ f in the delay-Doppler domain, and OCDM also has constant 2-dimensional spreading in chirp domain as shown in Fig. 2. But, none of the above waveforms offers two-dimensional adaptive spacing (∆ ti and ∆ fi ) in the time-frequency domain, which can be optimized as per system requirements and channel conditions.On the other hand, the proposed GASM waveform presents two-dimensional spreading with adaptive spacing parameters ∆ ti and ∆ fi as shown in Fig. 2. Using optimum values of these adaptive spacing parameters a nonuniform symbol spreading can be achieved with the proposed GASM.
In order to achieve the full potential of ISAC, the utilized waveform should be flexible in both time as well as frequency domains with tunable parameters.This becomes even more important from the aspect of future 6G systems, which are expected to be equipped with ultra-high data rate communication and onboard sensing for smart operation.However, they will face extremely dense simultaneous communication and sensing environments in very fast and highly frequency selective fading conditions.Thus, it becomes inevitable that the waveform design for an ISAC system should be highly adaptable and suitable for all these conditions.In the next section, we present the comprehensive system model for ISAC in a vehicular environment.

III. ISAC ARCHITECTURE IN VEHICULAR ENVIRONMENT
The system model of the ISAC in a vehicular environment is presented in Fig. 3. Referring to the fully autonomous vehicular environment shown in Fig. 1, all the vehicles are assumed to be equipped with an ISAC module capable of performing two tasks i.e. (i) communicating the data available with the ISAC module to the ISAC modules of other vehicles, and (ii) sensing the targets in the vicinity of vehicle by processing the reflections of transmitted signal from multiple targets.To accomplish these tasks, the ISAC module is subdivided into two modules ISAC transmitter and ISAC receiver as shown in Fig. 3.For simplification, we have considered the ISAC module in In-Band Full-Duplex (IBFD) mode working with Single-Input Single-Output (SISO) configuration 1 .The following subsections describe the assumptions undertaken while making the system model, transmitter, and receiver model of the ISAC system undertaken in this work.

A. Assumptions
• The ISAC nodes of the system are assumed to be operating in the IBFD mode.This infers the transmission and reception of signals simultaneously in the same frequency band resulting in higher throughput.• The transmitting and receiving antennas at ISAC nodes are in Quasi-Colocation (QCL) of type A (QCL-A) i.e. signal transmitted from both antennas experience similar channel conditions.• The change in the channel between transmitter and receiver ISAC as well as transmitter ISAC and targets is induced by Doppler shift, i.e., the change in location and velocity of scattered signals from a target is negligible.These assumptions are not compulsory but are very straightforward and hold true for most vehicular environments.

B. Transmitter Model
The detailed block diagram depicting the transmitter of an ISAC module is given in Fig. 3.The transmitted signal consists of F frames each of duration t F .Each frame consists of B data blocks of duration t B and one header block of duration t H i.e. t F = Bt B + t H .Each data block consists of one ISAC symbol carrying N complex data symbols, and G cyclic prefix symbols.For the generation of one ISAC symbol, the input bit stream is first converted into complex data symbols {X k } using either M-QAM or M-PSK modulation scheme.Then these serial complex data symbols are converted into N parallel complex data symbols using a serialto-parallel converter (S/P).These N parallel symbols are then processed using the ISAC modulator.In order to present a comparative analysis of existing waveforms for ISAC, the ISAC modulator is divided into sub-blocks each representing a unique waveform-specific transmitter for OFDM, FrFT-based OFDM, DFT-s-OFDM, GFDM, OTFS, OCDM, and proposed GASM.The N parallel symbols are fed to any one of the subblocks depending on the chosen waveform design.The output of ISAC modulator is then converted into serially arranged ISAC symbols {x n } using a parallel-to-serial converter (P/S).Thereafter, the cyclic prefix of length G is added to each ISAC symbol, which generates a data block of size N + G.After that, this data block containing the baseband ISAC samples is converted into a bandpass equivalent using a Digital-to-Analog Converter (DAC) and I/Q modulation.

C. Receiver Model
At the receiver side, the received bandpass signal y(t) consists of two parts: (i) the communication signal transmitted from ISAC module of other vehicles (x C (t)), and (ii) the echo of the signal transmitted from ISAC module of the current vehicle after being reflected from the target (x R (t)).In order to process these parts of the received signal, the ISAC receiver is subdivided into two blocks: radar processing block and communication processing block as presented in Fig. 3.The received signal y(t) is first converted into its baseband equivalent using a low noise amplifier (LNA), Band Pass filter, and I/Q demodulator.After this, the resultant is given to the matched filter block that utilizes the transmitted ISAC data block {x n } embedded with CP to process the received signal.The output of the matched filter block is then fed to the delay and Doppler estimation block that calculates the Doppler spread and time delay of the received signal with respect to the transmitted signal.This delay and Doppler information are further utilized by the target parameter detection block to generate the Range Profile of the targets.The group of these three blocks, i.e., matched filter, delay-Doppler estimation, and target parameter detection blocks are collectively termed as 'Radar Processing Block'.
It is worth noting that the range profile of the target is generated using the received baseband signal having both sensing as well as communication signals.In order to extract x C (t) from the baseband received signal y(t), it is first important to estimate the echo of the signal transmitted from the ISAC module of the current vehicle after being reflected from the target (x R (t)).Therefore, the information on delay and Doppler of targets as provided by the delay and Doppler estimation block is utilized by the delay and modulator block to generate an estimate of the received sending signal xR .Thereafter, an estimated communication signal xc is generated by extracting xR from the baseband received signal.After this, the cyclic prefix is removed from each block resulting in received ISAC samples {z n }, which are then fed to the S/P converter.The output of the S/P converter then fed to the ISAC demodulator that consists of waveform-specific receivers.The output of ISAC demodulator is then converted into serially arranged estimated communication symbols { Xk } using a parallel-to-serial converter (P/S).These estimated communication symbols are finally converted into an estimated bit stream using a suitable demodulation scheme.

IV. NOVEL WAVEFORM DESIGN FOR ISAC SYSTEM
Motivated by the challenges and requirements listed in Sec.II-B and ISAC system described in Sec.III, a novel waveform design is presented in this section whose synthesis is shown in Fig. 3.The novel waveform design is named Generalized Adaptive Spreading Modulation (GASM) and has the ability to control symbol spreading in both time as well as frequency domains independently.
Firstly, the complex data symbols generated using a typical modulation scheme (M-QAM/M-PSK) are passed through linear frequency spreading.Thereafter, these linear frequencymodulated signals are transformed into the time domain using the Fast Fourier Transform (FFT).Next, these symbols are processed by linear time spreading that results in further spreading of the signal in the time domain and gives samples of the proposed GASM waveform.Due to the rapid timefrequency variations in the channel, the traditional waveforms do not converge to an optimal solution for achieving the best possible overall system performance in terms of sensing as well as communication.The proposed GASM-based ISAC waveform can be optimized in terms of σ t and σ f to achieve optimum symbol spacing ∆ t and ∆ f in both time and frequency domains to best suit the system KPI requirements.The objective of this optimization is to maximize the instantaneous Signal Interference plus Noise Ratio (SINR) γ i at a given channel instance i, which ultimately affects both the sensing and communication capabilities of the system.This optimization is achieved by choosing ∆ t and ∆ f as per channel state information at the transmitter (CSIT).The process of generating the proposed GASM waveform can be mathematically expressed as linear time spreading where ∆ t = 1 f sample is the sampling interval in time domain, and f sample is the sampling frequency, parameters σ t and ∆ f are the time spreading factor and sampling interval of the resultant signal respectively.It is interesting to note that the computation of ∆ f is not as straightforward as ∆ t .The wellknown Shannon and Nyquist theorems cannot be applied as the modulated samples are not linearly periodic.The sampling interval of the resultant signal is calculated as ∆ f = 2πc N ∆t by using the generalized sampling theorem relating sampling frequency with σ t and σ f as [11], with constant c as the highest common factor that can divide σ t and σ f .

V. SPECIAL CASES OF PROPOSED GASM WAVEFORM
As discussed in Sec.II-B, the ISAC waveform needs to have tunable spacing parameters ∆ t and ∆ f so that the spreading of the waveform can be optimized in the time and frequency domain as given in (1) to achieve an optimum trade-off between communication and sensing performance with change in system requirements.In the following, some special cases of the proposed GASM waveform are presented.

A. OFDM
With careful observation of (1), it can be observed that the proposed ISAC waveform can be converted into OFDM by substituting σ t = σ f = 0.

B. FrFT-based OFDM
The FrFT-based OFDM can be implemented from the proposed GASM by substituting σ t = σ f = cot(ϕ).

C. OCDM
The OCDM can also be implemented as a special case of the proposed waveform by substituting σ t = σ f = Φ(constant similar to the chirp rate).Now, it is clear that OFDM, FrFTbased OFDM, and OCDM are special cases of the proposed waveform, and therefore the values of σ t and σ f can be suitably tuned to replicate the performance of these MCSs.

VI. PERFORMANCE COMPARISON
In this section, the performance analysis of the proposed GASM-based ISAC system is presented.For simplicity, the radar signal is assumed to face multipath-free propagation of reflections from targets and the sensing signal estimator block can perfectly remove the radar signal from the received communication signal.The performance of the proposed GASMbased ISAC system is compared with other standard MCSs: OFDM, FrFT-based OFDM, GFDM, OTFS, and OCDM.It is to be noted that this analysis does not include pure radarbased waveforms such as chirp sequence (CS), digital phase modulated continuous wave (PMCW), and its variants due to their very low data rate which is not suitable for practical ISAC applications.The system parameters utilized for performance analysis are given in Table I.These system parameters are inspired by the use cases and service level requirements presented in C-V2X white paper [2] and 3GPP feasibility study on ISAC in Release-19 [1].
In order for the results to be coherently analyzed and to ensure fair comparison, the ISAC system considered in the simulations utilizes uniform energy and bit allocation techniques.Moreover, keeping in mind the practical constraints regarding signal processing, the signals are processed in the time-frequency domain and zero forcing channel equalization method is utilized at the receiver [12].The communication performance of the proposed ISAC system is evaluated in VI-A

A. Communication Performance
In this section, the proposed GASM-based ISAC system along with other MCSs are evaluated for their communication performance.As per the requirements listed in [1], [2], a data rate of 300-1400 bytes per message for periodic kinematics and position information and 8-35 MBps for camera and Lidar streaming is required which is fulfilled by all the MCS listed in Table I.The ABER performance of ISAC system is evaluated under AWGN, frequency selective fading (FSF), and timefrequency selective (TFS) or doubly selective fading channels with CFO and STO.Fig. 4a shows the ABER v/s SNR plot for the proposed GASM-based ISAC system under perfect synchronization (ϵ = 0, θ = 0), with only CFO (ϵ = 0.1, θ = 0), (ϵ = 0.2, θ = 0) and with STO (ϵ = 0, θ = 0.3) cases over AWGN and frequency-selective Rayleigh fading channel with 2-taps equal power delay profile channel.It can be visualized from Fig. 4a that the ABER performance of the proposed GASM-based ISAC system matches with other state-of-the-art MCSs i.e.OFDM, FrFT-based OFDM [13], GFDM [14], and OCDM [12].This justifies that GASM performs at par when compared with standard modulation schemes under AWGN channel conditions.However, this trend does not hold under a more challenging frequency selective fading scenario wherein the proposed GASM scheme outperforms all other MCSs due to its superior capability to handle fading variations and synchronization errors.It is clearly observed from Fig. 4a that the ABER performance of GASM is better than other standard MCSs [13]- [15], especially in the practical SNR range.For example, at fixed SNR = 30 dB, the ABER of GASM, OFDM, FrFT-based OFDM, GFDM, and OCDM are 5.35 × 10 −4 , 4.05 × 10 −3 , 1.24 × 10 −3 , 1.04 × 10 −3 and 5.21 × 10 −3 respectively.
In order to study the effect of CFO and STO in more detail, the ABER of the proposed system is calculated for ϵ = [0 − 5] with different values of θ = 0, 0.3, 0.4, and 0.5 at a constant SNR of 20 dB in Fig. 4b.The results are compared with OFDM using the same system parameters as of GASM.It can be observed from Fig. 4b that in all presented cases, the performance of GASM is better than OFDM system [13].An interesting observation from Fig. 4b is that the performance of OFDM is very poor and reaches an error floor when the system is exposed to high timing offset errors (θ = 0.3, 0.4, 0.5) etc. but the performance of GASM based ISAC system is satisfactorily much better [13].Therefore, the resilience of the proposed GASM scheme against impairments is evident under complex practical scenarios from Fig. 4b.

B. Radar Performance
This section presents the sensing performance of the ISAC system with proposed scheme and its comparison with other standard MCSs.For this purpose, the ambiguity function (AF) of the ISAC waveform is evaluated to study its range and Doppler capabilities.The efficiency of a typical radar system to separate closely located targets with respect to their distance or velocity heavily depends on the transmitted waveform and is visible in the shape of its AF.The results are calculated for two scenarios, i.e., for long-range radar (LRR) with a maximum unambiguous range of 300 m and range resolution of 75 cm, and for short-range radar environment with a maximum unambiguous range of 100 m and range resolution of 20 cm [1], [2].A more detailed list of simulation parameters such as target velocity, velocity resolution, latency, missed detection rate, and false alarm rate is given in Table I.Fig. 5a presents the ambiguity function v/s Doppler plots with zero delay shift in LRR environment for BPSK modulated GASM data and compared with other MCSs.It is well known that the presence of dominant side lobes affects the sensing performance of radar because it creates interference during the matched filter operation.This results in a decreased probability of detection and higher false alarm rates, especially with higher Doppler shifts associated with relative radial velocities of targets, and can be computed numerically as the mean side lobe level (µ).It can be visualized from Fig. 5a that the ambiguity function curve of the proposed GASM-based waveform has lower side lobe levels as compared to other MCSs.The mean side lobe level of ambiguity function v/s Doppler (µ doppler ) for proposed GASM is only 0.122 while for OFDM, FrFT-based OFDM, GFDM, OTFS, and OCDM are 0.1873, 0.1852, 0.3713, 0.2383, and 0.2962 respectively thus achieving a percentage improvement of 53.52, 51.80, 204.34, 95.32, and 142.78 percent respectively as compared with other MCSs.Further, the corresponding plot of ambiguity function v/s delay with zero Doppler shift in LRR environment for BPSK modulated GASM data is shown in Fig. 5b.A trend similar to Fig. 5a is also visible in Fig. 5b wherein the proposed GASM-based waveform shows the sharpest main lobe with minimum mean side lobe level (µ delay ).The value of (µ doppler ) for proposed GASM is only 0.0625 while for OFDM, FrFT-based OFDM, GFDM, OTFS, and OCDM are 0.1873, 0.1852, 03713, 0.2383, 0.2962.

VII. CONCLUSION
In this paper, a novel multi-carrier framework based on GASM is proposed and analyzed for the ISAC-based vehicular systems.The standard MCSs such as OFDM, FrFT-based OFDM, and OCDM are derived as special cases of the proposed GASM.The proposed GASM-based ISAC system is evaluated for both communication and sensing performance in fading channels with CFO and STO and compared with existing MCSs.It is observed from the simulation results that the proposed GASM-based ISAC system has superior performance in terms of ABER under frequency-selective and doubly selective fading environments.Further, the GASMbased system has higher resilience to cater to the effect of synchronization errors such as timing and frequency offset thus resulting in better overall system performance when compared to existing MCSs.On a similar note, the proposed GASM system sensing showcases better sensing performance in terms of range and Doppler calculation from AF of the transmitted waveform.The ability to fine-tune the symbol spreading in both time and frequency domains to best suit the fading scenario is the prime reason for such superior performance of GASM.This justifies the utilization of the proposed GASM scheme in current and future vehicular systems equipped with ISAC.

Fig. 1 .
Fig. 1.Scenario replicating different use cases of the practical fully autonomous vehicular environment based on 3GPP feasibility study on ISAC in Release-19 [1] and C-V2X white paper [2].

Fig. 2 .
Fig. 2. Time-frequency representation of various waveforms along with the proposed GASM waveform.

Fig. 4 .
Fig. 4. Simulation results of proposed GASM-based ISAC system with existing MCSs for communication performance: (a) ABER v/s SNR over AWGN and frequency-selective Rayleigh fading channel with 2-taps equal power delay profile and (b) ABER v/s ϵ over frequency-selective Rayleigh fading channel at SNR = 20dB.

Fig. 5 .
Fig. 5. Simulation results of proposed GASM-based ISAC system with existing MCSs for sensing performance with a maximum unambiguous range of 300 m, range resolution of 75 m, and BPSK modulated data: (a) Ambiguity function v/s delay with zero Doppler shift and (b) Ambiguity function v/s Doppler with zero delay shift.