Design and analysis of CP-free OFDM PDMA transmission system

The paragraph introduces a proposed CP-free OFDM PDMA downlink transmission system. The main focus of the system is to address the capacity limitations caused by the overhead of cyclic prefix in traditional PDMA systems. The transmitter utilizes a pattern mapping unit before CP-free OFDM modulation to enhance system capacity and frequency efficiency. Decision feedback equalization (DFE) is employed in the receiver to eliminate intersymbol interference. The output signals from the DFE are then passed through a CP restoration unit to convert a linear-shifted signal into a cyclic-shifted signals. To assess the system’s performance, simulations are conducted, investigating different key parameters such as overload rate, channel condition, and signal-to-noise ratio. The results indicate that, compared to CP OFDM PDMA systems, the proposed CP-free PDMA system significantly enhances system capacity under the same overload rate. Additionally, bit error rate is also evaluated during the simulations. Overall, the paragraph provides an overview of the proposed CP-free OFDM


Introduction
The surge in the number of access terminals in mobile communication networks, driven by the development of social intelligence, has created a pressing challenge in utilizing limited wireless resources to accommodate more terminals and enhance system throughput in mobile communications.Non-orthogonal multiple access (NOMA) has been investigated as a potential solution for upcoming networks like beyond 5G and 6G networks.The advantages of NOMA are recognized for meeting the performance requirements of 6G networks, particularly in terms of extensive connectivity [1,2].NOMA is a technique for multiple access that enables simultaneous sharing of frequency and temporal resources among users [3].It accomplishes this through power domain multiplexing, where users are allocated different power levels based on their individual channel conditions.As a result, users with weaker signal strengths can utilize the shared resources alongside users with stronger signals simultaneously.PDMA represents a new form of NOMA technology, founded on the comprehensive optimization of multi-user communication systems as proposed following early SAMA technology research.PADM leverages PDMA patterns to segregate user signals at the transmitter, enabling their realization across multiple resource domains [4,5].For multi-user detection, the terminal employs the generalized successive interference cancelation (SIC) detection algorithm [6].Recent investigations have explored more sophisticated receivers for PDMA.In [7], an iterative detection and decoding algorithm is proposed, while [8] introduces a joint PDMA receiver algorithm that integrates the SIC detection algorithm with the minimum mean square error (MMSE) detection algorithm and associates cyclic redundancy check (CRC) to mitigate error propagation in the SIC receiver [9].The allocation of PDMA overload to physical resources is elucidated through a pattern matrix.The principles and procedures of the PDMA pattern matrix are expounded in [10], which delineates diverse pattern design criteria tailored to distinct 5G uplink service requisites.The value of the PDMA pattern matrix hinges on the trade-off between the desired multiplexing factor and the system implementation complexity [11].Additionally, it is generally observed that the multi-user pattern design methods vary based on the characteristics of different signal domains.Notably, for a PDMA system overloading J users on K sub-carriers, its pattern matrix is denoted as where p j ( j ∈ { 1,2,…, J}) is a K-dimensional column vector representing the pattern vector of user j.
For example, a pattern matrix with four sub-carriers and six users can be expressed as where j-th column of the pattern matrix P represents the pattern vector of user j, and the k-th row indicates the overloaded user situation on the k-th sub-carrier.
In PDMA pattern design, a combination of PDMA and multiple-input multipleoutput (MIMO) was proposed in [12].The paper also presents a receiver detection algorithm for the MIMO-PDMA scheme and analyzes the performance of MIMO-PDMA in both power and code domains.In [13], various PDMA multi-user power allocation methods are discussed, including global search power allocation, fixed power allocation, and partial power allocation.However, the specific implementation process and performance analysis of these methods were not provided.To optimize the throughput of all users, a scheme of joint pattern allocation and power allocation based on the water-filling algorithm was proposed in [14].One of the advantages of PDMA technology is its ability to design non-orthogonal feature patterns using multi-dimensional domains.This allows for a wider application range and high flexibility, as there is no restriction on the number of users sharing the same resource.PDMA can fully utilize the benefits of multi-dimensional domain processing. (1) (2) P [4,6] = 111100 110100 111010 100001 Orthogonal frequency division multiplexing (OFDM) technology originated in the mid-1950s.In an OFDM system, a high-rate data stream is divided into several lower-rate sub-streams.The data information is then carried by a set of orthogonal sub-carriers after undergoing an inverse fast Fourier transform (IFFT).OFDM is widely used in various wireless systems due to its ability to effectively handle the delay spread of wireless channels [15].To completely mitigate intersymbol interference (ISI) in OFDM systems, an adequate cyclic prefix (CP) must be inserted at the beginning of every OFDM block before transmission.At the receiver, demodulation of the original OFDM symbol is successful upon removal of the CP.However, the use of CP introduces additional bandwidth consumption, leading to a loss in spectral efficiency, energy efficiency, and transmission rate.This issue becomes more critical in scenarios with long-delay extension, where the CP overhead becomes significant and reduces the spectral efficiency of OFDM systems.Addressing this challenge, a CP-free OFDM transmission scheme is proposed in [16].This scheme incorporates decision feedback equalization (DEF) and CP restoration units, utilizing a symbol cyclic-shift equalization algorithm to eliminate both ISI and inter-carrier interference (ICI).[17] proposes a minimum mean square error iterative receiver based on parallel interference cancelation for CP-free OFDM signal reception.Moreover, an AI-aided CP-free OFDM system, utilizing artificial intelligence methods for channel estimation and signal detection, is investigated in [18].The impact of removing CP in massive multiple-input multiple-output (MIMO) OFDM systems is discussed in [19][20][21].In [22], a CP-free OFDM signal detection method is realized in the receiver.However, it is important to note that this method may sacrifice real-time performance of signal processing.[23] proposes DNN-based estimation in offline training and detection in online testing of downlink power domain multi-user NOMA-OFDM symbols robust against various channel impairments.
In this article, we propose a CP-free orthogonal frequency division multiplexing (OFDM) pulse division multiple access (PDMA) downlink system.The transmitter consists of PDMA encoding and CP-free OFDM modulation.The receiver includes CP restoration, OFDM demodulation, decision feedback equalizer (DFE), and PDMA multiuser detection.DFE is a commonly used method to eliminate intersymbol interference (ISI) in frequency-selective fading channels [24][25][26].In [27], a sparse code division multiple access uplink access scheme based on time-domain precoding was proposed, which significantly improves system capacity by eliminating CP overhead.CP-free OFDM transmission allows for a reduction in bandwidth consumption, resulting in increased spectral efficiency.By combining CP-free OFDM with PDMA technology, the system's capacity can be further enhanced.PDMA enables multiple users to simultaneously access the downlink, thereby increasing the number of supported users and improving system capacity.The use of DFE in the receiver enhances the system's ability to recover the transmitted data accurately.
The rest of the paper is organized as follows.In Sect.2, we introduce the system model and describe the signal processing procedure.In Sect.3, we present the simulation results and evaluate the system's performance.Finally, in Sect.4, we provide a conclusion.

System model
In this section, a block diagram of a CP-free OFDM PDMA system is presented.

CP-free OFDM PDMA transmitter
Figure 1 presents the block diagram of CP-free OFDM PDMA transmitter.In the transmitter, the data of the j-th user undergoes a series of operations, including channel encoding, constellation mapping, pattern mapping, parallel to serial conversion (P/S), and CP-free OFDM modulation.Let S j denote the modulation symbol after constellation mapping, the PDMA pattern matrix Ŝj can be expressed as where Ŝj is a K × 1 vector.In Fig. 1, the combiner linearly superimposes the signals of all J users, which are then send into a serial to parallel conversion (S/P) unit, followed by IFFT, P/S unit, and digital-to-analog conversion (DAC) unit before wireless transmission.The IFFT is performed on Ŝj to generate an OFDM signal S , that is where F is IFFT matrix.The only difference from a typical OFDM system, which usually inserts CP between adjacent OFDM blocks, is that CP-insertion is absent between OFDM blocks in order to maintain high spectral efficiency.

CP-free OFDM receiver
Figure 2 illustrates the primary structure of the CP-free OFDM PDMA receiver, which comprises the DFE, cyclic prefix recovery (CPR), and PDMA multi-user detection modules.Due to the presence of multipath, the signal suffers from intersymbol interference (ISI) and inter-carrier interference (ICI).
Here, we name [R m (n)] K −1 n=0 as the Nyquist samples vector of the m-th OFDM symbol block in j-th user, it can be expressed as where ρ jl denotes the channel attenuation of the l-th path of the j-th user, 2πϕ j,l is a mul- tipath random phase that follows a uniform distribution over [0, 2π], Sm (n − l) is the (n-l)-th element of the m-th multi-user superposition time-domain vector transmitted by the base station, I m (n) is the tailing interference of the previous symbol when detect- ing the m-th symbol, and v(n) is the Gaussian noise on the n-th time-domain symbol.
Assuming that the channel state information is known, the previous samples are overlapped by the exceeding samples of the (m-1)-th symbol due to multipath propagation, with the length of the overlapping part being L-1.The reconstructed signal C m-1 is obtained by convolving the (m-10)-th symbol Sm−1 with the channel impulse response vector h j as depicted in [16], resulting in Eq. ( 6). Here, is a vector with M + L -1 elements.Additionally, we assume the latter L-1 elements of the exceeding samples to be denoted as , as expressed in Eq. (7).
Referring to Eq. ( 5), the acquisition of the interference component enables the elimination of intersymbol interference (ISI) by processing the interference term associated with the m-th symbol.
Furthermore, the symbol block is influenced by its inherent linear shift.To address this, the cyclic prefix recovery algorithm is employed to estimate and compensate for the trailing part of the m-th symbol, thereby bridging the initial gap.This process transforms the (5)

DEF Data
Fig. 2 The main structure of the CP-free OFDM PDMA receiver linearly shifted signal into a cyclically shifted one.Subsequently, the original signal is reinstated through frequency domain equalization following a fast Fourier transform (FFT).
A challenge arises from the fact that the signal received encompasses a multipath superposition.In this context, a step estimation approach can be employed to determine the sampling values across all paths.Disregarding noise and the removal of the trailing part of the preceding symbol, the initial sampling value for the m-th symbol block is as Eq. ( 8).
where ρ 0 and ϕ 0 represent the first channel attenuation and phase, respectively.The product of R m (0) and h j /h j (0) is used to estimate the first sampling values of each path, i.e., [ Rm,0 (0), Rm,1 (0), • • • , Rm,L−1 (0)] , where Rm,l (0) represents the first sampling value on the m-th symbol block in path l.Obviously, the second sampling value is represented by R m (1).The second sample value is interfered by the first sample value in the second path, where the latter has been estimated in the previous step.The estimated value of the second sampling value of the first path after eliminating the interference can be obtained by subtracting Rm,1 (0) from R m (1), we can get Then, Rm,0 (1) is multiplied by h j / h j (0) to estimate the second sample value of each path, i.e., [ Rm,0 (1), Rm,1 (1), • • • , Rm,L−1 (1)] .According to this inference, all K signal val- ues of all paths can be estimated to complete CP reconstruction.Then, performed FFT gets where v ′ (k) is the Gaussian noise on the k-th sub-carrier, ε ′ m (k) is the decision feedback equalization error on the k-th sub-carrier, and ε m (n) is the decision feedback error on the n-th time-domain symbol.

Multi-user detection
The receiver employs the SIC algorithm for multi-user detection, which is organized according to the signal-to-interference-noise ratio (SINR).Initially, after resolving the signal for user 1, this signal is subtracted from the aggregated multi-user signal to mitigate the interference user 1 causes to others.This process is repeated sequentially until the signal of the last user is detected.For illustration, consider a PDMA system equipped with four sub-carriers (K = 4) and six users (J = 6).This example will elucidate the SIC process further.
The pattern matrix is delineated in Eq. (2).To enhance user discrimination and achieve superior multi-user detection performance, PDMA typically introduces a power factor to the basic pattern matrix as follows The corresponding factor diagram is depicted in Fig. 3.Each row of G represents a carrier, and each column of G corresponds to a user.Consequently, the sum of the elements in a user's column in G indicates the transmit diversity order for that user.Similarly, the sum of the elements in a carrier's row in G indicates the overlap order for that carrier.The ratio of the number of columns to the number of rows represents the overload rate of the PDMA pattern.
The key of the proposed PDMA method lies in the pattern design, which can significantly affect both system performance and detection complexity.On the one hand, for a given K, different overload rates can support varying access connections.The larger the overload ratio, the better the system performance and the higher the detection complexity.On the other hand, for certain users, a higher transmit diversity order can achieve more reliable data transmission at the cost of more complex detection.Therefore, strategic pattern design is necessary to balance system performance and detection complexity.
As per the description above of the PDMA pattern, initially, the receiver performs detection priority sorting, which is based on the transmission diversity and average SINR of signals transmitted by each user.A higher transmission diversity yields more reliable transmission.If the transmission diversity is equal among users, their average SINR values are compared.For the 1st and 2nd users, the transmission diversities are 4 and 3, respectively, and are higher than those of subsequent users.So the 1st and 2nd users are detected preferentially.For the 3rd and 4th users, their transmission diversity values being equal, (11) G [4,6] =      g 1,1 g 1,2 g 1,3 g 1,4 00 g 2,1 g 2,2 0g 2,4 00 g 3,1 g 3,2 g 3,3 0 g 3,5 0 g 4,1 000 0 g 4,6 Fig. 3 The factor diagram of the PDMA pattern a comparison of their average SINR values is undertaken.With the use of SIC detection, when detecting the 3rd and 4th users, it is assumed that the interference from the 1st and 2nd users is entirely eliminated.The 3rd user faces interference form from the 4th, 5th, and 6th users, while the 4th user's interference is from the 3rd, 4th, and 6th users.Consequently, the average SINR of the 3rd user can be expressed as where γ 1,3 and γ 3,3 are the SINR of the 1st user on the 1st and 3rd sub-carriers, and σ 2 is the noise power.Similarly, the average SINR of the 4th user is calculated.By comparing their SINR, the one with the higher average SINR is detected first, using the maximum ratio combining method to amalgamate signals from different sub-carriers.Let y k, j denote the signals received by the j-th user on the k-th sub-carrier, the signal combined for decision-making is represented as follows where ψ j is the total SINR for the j-th user across all K sub-carriers, and if g k, j = 0, the SINR for the j-th user on the k sub-carrier equals zero and is not included in the total, λ k, j is the weight coefficient.λ k, j is actually the ratio of the SINR for the j-th user on the k sub-carrier to the total SINR across all K sub-carriers, as illustrated in Eqs.(13b, c).This definition confirms the optimality of the maximum ratio combination weight coefficient.
After applying the maximum ratio combination to the signal of the j-th user, the signal form any undetected user is treated as noise in the decision process.The equivalent noise power received by the j-th user can be expressed as (12a)

System capacity
The frequency domain response of the j-th user on the k-th sub-carrier is presented in Eq. ( 15).
For the j-th user, the signal transmitted across different sub-carriers following pattern mapping results in a total capacity C j , which equals the sum of the capacities of the signal on each sub-carrier as illustrated in [15], that can be expressed as where B denotes the system bandwidth, B/K represents the bandwidth of a single subcarrier, C j, k is the capacity of the j-th user on the k-th sub-carrier, P s is the signal power of the signal from the j-th user on the k-th sub-carrier, P i is the interference power from other users on the k-th sub-carrier, σ 2 is the noise power, and ||•|| is the modulus of the complex number.
The system's capacity, C, equals the sum of capacities of all users, as demonstrated in Eq. (17).
where ξj is the transmission efficiency.For CP-free OFDM systems, ξj = 1, and for CP OFDM systems, ξj = K/( K + Lj −1), where (Lj −1) is the maximum delay spread of the j-th user's channel, determined by hj.

Complexity analysis
Compared with CP OFDM PDMA, the scheme we proposed achieves an improvement in capacity and performance at the expense of increased receiver complexity.
In the detection of the m-th multi-user superimposed symbol block, it is necessary to use the detected (m − 1)-th symbol block that convolves with the channel response vector to reconstruct the tailing interference of the (m − 1)-th symbol block.In this (15) hj process, the (m − 1)-th symbol block also needs to undergo an IFFT to reconstruct the time-domain signal Sm−1 .The number of complex multiplications consumed in this pro- cess is K/2log 2 K.In the convolution operation, the length of the input signal sequence is numerically equal to the number of sub-carriers K, and the length of the channel response vector is L. Therefore, the computational overhead of the convolution operation is K × L. The total computational complexity generated by the DFE process is represented by Eq. ( 18).The computational complexity in the process of CP reconstruction is mainly reflected in using the sample values of the first path signal to estimate the sample values of other paths, that is , and then, the computational complexity generated by this process is represented by Eq. ( 19).
The additional computational complexity of CP-free OFDM PDMA is the sum of the complexities of the DFE and CP reconstruction processes, denoted as O DEF + O CPR .

Simulation results and discussions
This section presents simulated results.Unless otherwise specified in the simulation, the number of sub-carriers K is set to 16, the number of users per carrier is J, and the overload rate is defined as η = J/K.The modulation scheme employed is quadrature phase shift keying (QPSK), and the total bandwidth is set to 1 MHz.The CP length is set equal to the maximum channel delay spread for a traditional CP OFDM system.The channel models used in the simulation include the extended typical urban channel model (ETU) and the extended vehicle channel model (EVA) channel models.(18)  Fig. 4 BER performance of CP-free OFDM PDMA system and CP OFDM PDMA system in ETU channel model with overload rate η = 150% Figure 4 illustrates the simulation results of BER comparison between the CP-free OFDM PDMA system and the CP OFDM PDMA system for the 1st, 6th and 24th users when the overload rate in an ETU environment is 150%.From Fig. 4, it is evident that the BER performance improves as the SNR increases.However, the overall BER of the CP-free OFDM PDMA system is marginally higher than that of the CP OFDM PDMA system.For instance, considering the 1st user, to achieve a comparable BER performance of 10 −3 with the CP OFDM PDMA system, the CP-free OFDM PDMA system exhibits a 3.5 dB loss in signal-to-noise ratio (SNR) performance.This loss can be attributed to two main factors.Firstly, as analyzed in Sect.2, CP reconstruction introduces significant additional noise to the CP-free OFDM system, leading to decision feedback errors at the receiver.Secondly, the BER performance of users progressively worsens with the increase in the demodulation order.This deterioration is caused by the error propagation effect of SIC.
To investigate the impact of the overload rate η on the BER performance, simulations were conducted as illustrated in Fig. 5.These simulations compared the BER performance of the 6th user under CP-free OFDM PDMA and CP OFDM PDMA across different overload rates in ETU scenarios.Figure 5 demonstrates that the overload rate η significantly affects the system's BER performance.At an overload rate of η = 100%, where one user's date is transmitted on one sub-carrier, the multi-user interference is minimal, and the BER performance is optimal.As the overload rate η increases, so does the multi-user interference.When the BER reaches 10 −2 , the CP-free OFDM PDMA system with η = 150% experiences a 17 dB loss in SNR performance when compared to the CP-free OFDM PDMA system with η = 100%.At an overload rate of η = 200%, BER curve of the 6th user has shown a deterioration layer, and the main factor affecting the system's BER performance shifts from noise to the interference from other users.
Figure 6 provides a comparative assessment of the BER performance of a cyclic prefix CP-free OFDM PDMA system operating in an ETU channel environment, contrasted

BER SNR (dB)
CP-free OFDM, ŋ=200% CP OFDM, ŋ=200% CP-free OFDM, ŋ=150% CP OFDM, ŋ=150% CP-free OFDM, ŋ=100% CP OFDM, ŋ=100% Fig. 5 BER performance of the 6th user under CP-free OFDM PDMA system and CP OFDM PDMA system with different overload rates in ETU channel model against the time-domain precoding (TDP) OFDM technique described in [28] and the feature suppressed OFDM (FS-OFDM) algorithm presented in [29].FS-OFDM, recognized as a CP-free OFDM transmission strategy predicated on iterative interference suppression, involves iteratively reconstructing the CP until a stable BER performance level is achieved.It is important to note that, while the reconstructed CP serves as an approximation of the authentic CP, it is incapable of completely eliminating ICI.In Fig. 6, the symbol 'I' denotes the iteration count for FS-OFDM, demonstrating a progressive improvement in its BER performance as the number of iterations increases, this iterative reception strategy compromises the real-time processing capabilities of signal processing.In contrast, TDP-OFDM avoids the use of the conventional CP by dividing a lengthy OFDM symbol into numerous sub-symbols, interspersing these with protective intervals to substitute the conventional CP structure.This approach simultaneously preserves the intrinsic resilience of OFDM to frequency-selective fading while circumventing the bandwidth loss associated with CPs.In environments characterized by pronounced multipath effects, such as the ETU channel, TDP-OFDM exhibits a relatively strong BER performance.Nonetheless, it is inherently reliant on the accuracy and update frequency of the channel state information (CSI), inaccuracies in CSI may potentially compromise the BER.The CP-free OFDM PDMA system leverages pattern division to establish user orthogonality, thereby significantly enhancing both spectral efficiency and system capacity.Additionally, under multipath conditions, its BER performance closely resembles that of a CP OFDM system.
The SNR and overload factor fluctuation curves for the CP-free OFDM PDMA system and the CP OFDM PDMA system in the ETU channel model are depicted in Fig. 7.As observed in Fig. 7, for an equivalent overload factor, the CP-free OFDM PDMA system exhibits a higher system capacity than the CP OFDM PDMA system, attributable to the elimination of the CP cost.When the capacity is 1.6 × 107 and η = 250%, the CP OFDM PDMA system requires a 17 dB lower SNR compared to the CP OFDM PDMA system.Furthermore, at lower overload rates, the system capacity incrementally increases with rising overload rates.However, at higher overload rates, the escalation of multi-user interference becomes a significant hindrance, thus impeding further increases in system capacity.Figure 8 exhibits the fluctuation curve representing system capacity as a function of user count under varying SNR for CP-free OFDM PDMA and CP OFDM PDMA in channel model.It can be discerned from the diagram that, in the scenario of a limited user base, system capacity experiences an incremental elevation with the  progressive increase in users.Yet, as user numbers further proliferate, the gradient of the system capacity curve verges on becoming negligible.This saturation point arises due to the intensification of multi-user interference as the user population augments, leading to a leveling off in capacity.

Conclusions
In this manuscript, we introduce a novel CP-free OFDM scheme for PDMA transmissions.This innovative approach significantly augments the system's capacity by eradicating the CP overhead, thereby enhancing spectral efficiency.Nonetheless, it is imperative to recognize that the PDMA system's capacity enhancement is constrained by the inherent multi-user interference, which inherently limits the scalability of system capacity with increase in system overload factors.Our simulations elucidate that beyond a 250% overload rate, there is a negligible increment in system capacity, indicating a plateau in capacity enhancement under high overload conditions.Furthermore, the adoption of CP-free OFDM technology, while is beneficial for capacity improvement, introduces reconstruction errors due to the CP elimination process.These errors slightly compromise BER performance of the PDMA system utilizing CP-free OFDM in comparison with its CP OFDM PDMA counterpart.However, this degradation in BER performance is deemed tolerable when juxtaposed with the notable gains in system capacity.In essence, this research underscores the CP-free OFDM PDMA system's superior capacity and bandwidth efficiency, establishing it as a formidable alternative to the traditional CP OFDM PDMA system.Through a comprehensive examination of the trade-offs between system capacity, BER performance, and spectral efficiency, this work contributes significantly to the ongoing discourse on optimizing transmission schemes for PDMA systems, highlighting the CP-free OFDM PDMA system's potential to redefine performance benchmarks in multi-user communication environments.

Fig. 6
Fig.6BER performance of CP-free OFDM PDMA, TDP-OFDM, FS-OFDM, and CP OFDM system in ETU channel model

Fig. 7
Fig. 7 Capacity of CP-free OFDM PDMA and CP OFDM PDMA system with different overload factor in ETU channel model

Fig. 8
Fig. 8 Capacity of CP-free OFDM PDMA and CP OFDM PDMA system under different numbers of users in ETU channel model