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SNR gains vs. number of iterations, where both QPSK and 16-QAM signals are considered with P = 4 and P = 6.
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Recently proposed orthogonal time frequency space (OTFS) modulation has been considered as a promising candidate for accommodating various emerging communication and sensing applications in high-mobility environments. In this paper, we propose a novel cross domain iterative detection algorithm to enhance the error performance of OTFS modulation. Di...
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... Fig. 9, we provide the simulation results of the SNR gains of the proposed algorithm relative to the MMSE detection in the presence of fractional Doppler shifts, at BER ≈ 10 −3 . We set M = 32, N = 16, l max = 3, and k max = 6, where both QPSK and 16-QAM signals are considered. As shown in the figure, the SNR gains increase with iterations ...
Citations
... Many low-complexity detectors for OTFS and AFDM are currently being investigated. For OTFS, a cross-domain orthogonal approximate message passing (CD-OAMP) detector is developed in [3], which is iteratively performed between the time domain and the DD domain. A DD-OAMP has been proposed in the DD domain for OTFS [4]. ...
... Table I summarizes the comparison of OFDM, OTFS, AFDM, and IFDM. Note that IFDM has a modulation complexity of O(N logN ), which is the same as that of OFDM and AFDM [2] but less than that of OTFS [3]. Given the CD/DD-OAMP detector [3], [4] or the low-complexity CD-MAMP detector proposed later, IFDM is the best in terms of BER, while OFDM is the worst. ...
... Note that IFDM has a modulation complexity of O(N logN ), which is the same as that of OFDM and AFDM [2] but less than that of OTFS [3]. Given the CD/DD-OAMP detector [3], [4] or the low-complexity CD-MAMP detector proposed later, IFDM is the best in terms of BER, while OFDM is the worst. Fig. 3 shows that IFDM is integrated with MIMO. ...
In this letter, we study interleave frequency division multiplexing (IFDM) for multicarrier modulation in static multipath and mobile time-varying channels, which outperforms orthogonal frequency division multiplexing (OFDM), orthogonal time frequency space (OTFS), and affine frequency division multiplexing (AFDM) by considering practical advanced detectors. The fundamental principle underlying existing modulation techniques is to establish sparse equivalent channel matrices in order to facilitate the design of low-complexity detection algorithms for signal recovery, making a trade-off between performance and implementation complexity. In contrast, the proposed IFDM establishes an equivalent fully dense and right-unitarily invariant channel matrix with the goal of achieving channel capacity, ensuring that the signals undergo sufficient statistical channel fading. Meanwhile, a low-complexity and replica maximum a posteriori (MAP)-optimal cross-domain memory approximate message passing (CD-MAMP) detector is proposed for IFDM by exploiting the sparsity of the time-domain channel and the unitary invariance in interleave-frequency-domain channel. Numerical results show that IFDM with extremely low-complexity CD-MAMP outperforms OFDM, OTFS, and AFDM with state-of-the-art orthogonal approximate message passing detectors, particularly at low velocities.
... The channel fading coefficient is generated with a uniform power delay profile. Similar to [2], [18], the delay l k(i) and Doppler indices k k(i) are generated with equal probability within the range of [0, l max ] and [−k max , k max ], where the maximum delay index l max = 4 and the maximum Doppler index k max = 2 for lowmobility users, l max = 4 and k max = 4 for high-mobility users, respectively. Note that in the simulations, we consider nonfractional delay indices and fractional Doppler indices, which resemble practical channel conditions. ...
... The authors in [29] considered fractionally spaced sampling in the time-domain for enhanced pulse-shaped OTFS systems. Instead of performing detection in a single domain, a cross-domain OTFS detector was proposed in [30]. The idea of [30] is to carry out equalization in the time domain to exploit the natural sparsity of time-domain effective channels in OTFS systems, whilst performing the denoising in the DD domain. ...
... Instead of performing detection in a single domain, a cross-domain OTFS detector was proposed in [30]. The idea of [30] is to carry out equalization in the time domain to exploit the natural sparsity of time-domain effective channels in OTFS systems, whilst performing the denoising in the DD domain. Similar to the core idea of orthogonal approximate message passing (OAMP) [32], [33], the cross-domain detector passes extrinsic information (e.g. ...
... Inspired by [30], [32], [33], we propose to perform OTFS symbol estimation and SCMA decoding in different domains to fully exploit the advantages of OTFS and SCMA. Furthermore, we observe that the OTFS frames of all SCMA users carry the same information symbols, and different users may experience different physical channels. ...
This paper studies a downlink system that combines orthogonal-time-frequency-space (OTFS) modulation and sparse code multiple access (SCMA) to support massive connectivity in high-mobility environments. We propose a cross-domain receiver for the considered OTFS-SCMA system which efficiently carries out OTFS symbol estimation and SCMA decoding in a joint manner. This is done by iteratively passing the extrinsic information between the time domain and the delay-Doppler (DD) domain via the corresponding unitary transformation to ensure the principal orthogonality of errors from each domain. We show that the proposed OTFS-SCMA detection algorithm exists at a fixed point in the state evolution when it converges. To further enhance the error performance of the proposed OTFS-SCMA system, we investigate the cooperation between downlink users to exploit the diversity gains and develop a distributed cooperative detection (DCD) algorithm with the aid of belief consensus. Our numerical results demonstrate the effectiveness and convergence of the proposed algorithm and show an increased spectral efficiency compared to the conventional OTFS transmission.
... Moreover, the double-selective time-frequency (TF) channel is converted into the quasi-static DD domain channel based on the two-dimensional orthogonal basis functions, making OTFS capable of handling doubly-selective channels [6], [27]. However, the complexity of the OTFS system escalates, since the DD domain symbols are followed by the cascaded inverse symplectic finite Fourier transform (ISFFT)/SFFT and Heisenberg/Wigner transform pairs, especially for high numbers of time slots and subcarriers [7], [28]. ...
... To alleviate the ISI between OTSM frames, only a cyclic prefix (CP) length of L CP = l max + 1 is appended to the transmitted signal. The TD input-output relationship given by (7) can be further formulated as ...
... (32) 7 Therefore, the CPEP of (31) can be rewritten as ...
In orthogonal time sequency multiplexing (OTSM) modulation, the information symbols are conveyed in the delay-sequency domain upon exploiting the inverse Walsh Hadamard transform (IWHT). It has been shown that OTSM is capable of attaining a bit error ratio (BER) similar to that of orthogonal time-frequency space (OTFS) modulation at a lower complexity, since the saving of multiplication operations in the IWHT. Hence we provide its BER performance analysis and characterize its detection complexity. We commence by deriving its generalized input-output relationship and its unconditional pairwise error probability (UPEP). Then, its BER upper bound is derived in closed form under both ideal and imperfect channel estimation conditions, which is shown to be tight at moderate to high signal-to-noise ratios (SNRs). Moreover, a novel approximate message passing (AMP) aided OTSM detection framework is proposed. Specifically, to circumvent the high residual BER of the conventional AMP detector, we proposed a vector AMP-based expectation-maximization (VAMP-EM) detector for performing joint data detection and noise variance estimation. The variance auto-tuning algorithm based on the EM algorithm is designed for the VAMP-EM detector to further improve the convergence performance. The simulation results illustrate that the VAMP-EM detector is capable of striking an attractive BER vs. complexity trade-off than the state-of-the-art schemes as well as providing a better convergence. Finally, we propose AMP and VAMP-EM turbo receivers for low-density parity-check (LDPC)-coded OTSM systems. It is demonstrated that our proposed VAMP-EM turbo receiver is capable of providing both BER and convergence performance improvements over the conventional AMP solution.
... Moreover, the double-selective time-frequency (TF) channel is converted into the quasi-static DD domain channel based on the two-dimensional orthogonal basis functions, making OTFS capable of handling doubly-selective channels [6], [27]. However, the complexity of the OTFS system escalates, since the DD domain symbols are followed by the cascaded inverse symplectic finite Fourier transform (ISFFT)/SFFT and Heisenberg/Wigner transform pairs, especially for high numbers of time slots and subcarriers [7], [28]. ...
... To alleviate the ISI between OTSM frames, only a cyclic prefix (CP) length of L CP = l max + 1 is appended to the transmitted signal. The TD input-output relationship given by (7) can be further formulated as ...
... (32) 7 Therefore, the CPEP of (31) can be rewritten as ...
In orthogonal time sequency multiplexing (OTSM) modulation, the information symbols are conveyed in the delay-sequency domain upon exploiting the inverse Walsh Hadamard transform (IWHT). It has been shown that OTSM is capable of attaining a bit error ratio (BER) similar to that of orthogonal time-frequency space (OTFS) modulation at a lower complexity, since the saving of multiplication operations in the IWHT. Hence we provide its BER performance analysis and characterize its detection complexity. We commence by deriving its generalized input-output relationship and its unconditional pairwise error probability (UPEP). Then, its BER upper bound is derived in closed form under both ideal and imperfect channel estimation conditions, which is shown to be tight at moderate to high signal-to-noise ratios (SNRs). Moreover, a novel approximate message passing (AMP) aided OTSM detection framework is proposed. Specifically, to circumvent the high residual BER of the conventional AMP detector, we proposed a vector AMP-based expectation-maximization (VAMP-EM) detector for performing joint data detection and noise variance estimation. The variance auto-tuning algorithm based on the EM algorithm is designed for the VAMP-EM detector to further improve the convergence performance. The simulation results illustrate that the VAMP-EM detector is capable of striking an attractive BER vs. complexity trade-off than the state-of-the-art schemes as well as providing a better convergence. Finally, we propose AMP and VAMP-EM turbo receivers for low-density parity-check (LDPC)-coded OTSM systems. It is demonstrated that our proposed VAMP-EM turbo receiver is capable of providing both BER and convergence performance improvements over the conventional AMP solution.
... Moreover, the double-selective time-frequency (TF) channel is converted into the quasi-static DD domain channel based on the two-dimensional orthogonal basis functions, making OTFS capable of handling doublyselective channels [9], [10]. However, the complexity of the OTFS system escalates, since the DD domain symbols are followed by the cascaded inverse symplectic finite Fourier transform (ISFFT)/SFFT and Heisenberg/Wigner transform pairs, especially for high numbers of time slots and subcarriers [11], [12]. ...
In orthogonal time sequency multiplexing (OTSM) modulation, the information symbols are conveyed in the delay-sequency domain upon exploiting the inverse Walsh Hadamard transform (IWHT). It has been shown that OTSM is capable of attaining a bit error ratio (BER) similar to that of orthogonal time-frequency space (OTFS) modulation at a lower complexity, since the saving of multiplication operations in the IWHT. Hence we provide its BER performance analysis and characterize its detection complexity. We commence by deriving its generalized input-output relationship and its unconditional pairwise error probability (UPEP). Then, its BER upper bound is derived in closed form under both ideal and imperfect channel estimation conditions, which is shown to be tight at moderate to high signal-to-noise ratios (SNRs). Moreover, a novel approximate message passing (AMP) aided OTSM detection framework is proposed. Specifically, to circumvent the high residual BER of the conventional AMP detector, we proposed a vector AMP-based expectation-maximization (VAMP-EM) detector for performing joint data detection and noise variance estimation. The variance auto-tuning algorithm based on the EM algorithm is designed for the VAMP-EM detector to further improve the convergence performance. The simulation results illustrate that the VAMP-EM detector is capable of striking an attractive BER vs. complexity trade-off than the state-of-the-art schemes as well as providing a better convergence. Finally, we propose AMP and VAMP-EM turbo receivers for low-density parity-check (LDPC)-coded OTSM systems. It is demonstrated that our proposed VAMP-EM turbo receiver is capable of providing both BER and convergence performance improvements over the conventional AMP solution.
... However, when the time and frequency resources for OTFS are limited, the DD domain channel matrix could be dense due to insufficient resolution of delay and Doppler, and consequently, DD domain detection may suffer from high detection complexity. The cross-domain iterative detection proposed in [8] was a preliminary attempt to solve this issue by considering the detection in both time and DD domains via iterative processing, which can achieve almost the same error performance as ML detection even in the presence of fractional Doppler shifts. The cross-domain iterative detection is motivated by the unitary transformation between the time and DD domains, ensuring that the detection error in one domain is principally orthogonal to that in the other domain. ...
... Thus, it allows cross-domain iterations for signal detection without introducing error propagation. However, [8] applied a full-size linear minimum mean square error (LMMSE) filter in the time domain, which, as we will show later, does not fully exploit the advantages of cross-domain iteration. ...
... Based on (12), we propose to apply a reduced-size LMMSE estimator for eliminating the delay interference, which has a much lower complexity compared to the full-size LMMSE estimator adopted in [8]. However, such estimator cannot fully minimize the effect of Doppler. ...
In this paper, a reduced-complexity cross-domain iterative detection for orthogonal time frequency space (OTFS) modulation is proposed, which exploits channel properties in both time and delay-Doppler domains. Specifically, we first show that in the time domain effective channel, the path delay only introduces interference among samples in adjacent time slots, while the Doppler becomes a phase term that does not affect the channel sparsity. This ``band-limited'' matrix structure motivates us to apply a reduced-size linear minimum mean square error (LMMSE) filter to eliminate the effect of delay in the time domain, while exploiting the cross-domain iteration for minimizing the effect of Doppler by noticing that the time and Doppler are a pair of Fourier dual. The state (MSE) evolution was derived and compared with bounds to verify the effectiveness of the proposed scheme. Simulation results demonstrate that the proposed scheme achieves almost the same error performance as the optimal detection, but only requires a reduced complexity.
... The approximation of diag(η [ℓ] ) withη [ℓ] I N essentially treats the offdiagonal elements of the covariance matrix as zeros during the affine Fourier domain to time domain transformation, which effectively prevents bandwidth β spreading. Due to the absence of off-diagonal elements, this approximation may result in slight performance degradation [31]. Nevertheless, considering the significant reduction in complexity, this approximation is acceptable. ...
... and ∆B can be viewed as a matrix perturbation of the circulant channel matrix B. It is worth noting that the above formulas take into account the scenarios where multiple paths with different Doppler shifts share the same delay tap. Equation (31) implies that the magnitude matrix of T o with c 1 ̸ = 0 is equivalent to the magnitude matrix of T o with c 1 = 0 after undergoing a circular shift. Therefore, in order to simplify the analysis, we only analyze the case where c 1 = 0, and the analysis can be generalized to the case where c 1 ̸ = 0 using circular shift operations. ...
Affine Frequency Division Multiplexing (AFDM) is a chirp-transform modulation technique that has shown reliable performance in high-mobility scenarios, making it an attractive option for next generation communication systems. Recent literatures have suggested that AFDM can achieve optimal diversity performance with maximum likelihood (ML) detection and can achieve almost the same performance as Orthogonal Time Frequency Space (OTFS). However, the performance of AFDM with minimum mean square error equalization (MMSE-Eq) has not been extensively investigated in the existing literature. In this paper, we analyze the performance of AFDM with MMSE-Eq, derive a lower bound for the theoretical bit error rate (BER) of the AFDM system, and discuss the impact of chirp parameters on its performance. Then, aiming to optimize the transmission performance, we propose two different chirp parameter selection strategies for frequency selective and double-selective channels, respectively. Finally, we propose a low-complexity and high-performance iterative MMSE-Eq algorithm based on time-domain channel matrix operations. This algorithm resolves the issue encountered in traditional low-complexity methods, where different chirp parameter selections significantly impact the complexity. Simulation results are provided to demonstrate that the parameter selection strategies can enable optimal transmission of AFDM signals with MMSE-Eq. Additionally, the proposed iterative MMSE-Eq algorithm exhibits a significant BER performance improvement. Index Terms-Affine frequency division multiplexing, minimum mean square error (MMSE), channel equalization, doubly selective channel, frequency selective channel.
... As shown in Fig. 1, in this paper, we consider an OTFSenabled URLLC system [27], which consists of one singleantenna transmitter and one single-antenna receiver. In addition, we assume that there are M subcarriers and N time slots, where the subcarrier spacing and the time slot duration are 1/T and T , respectively. ...
... to minimize JDDCL(ω) in (26) based on BP algorithm it = it + 1 4: end while 5: Output: Well-trained gω * (·) as defined in (27) Online Precoder Design: 6: Input: Test dataH τ (i ) DD,t ...
To guarantee excellent reliability performance in ultra-reliable low-latency communications (URLLC), pragmatic precoder design is an effective approach. However, an efficient precoder design highly depends on the accurate instantaneous channel state information at the transmitter (ICSIT), which however, is not always available in practice. To overcome this problem, in this paper, we focus on the orthogonal time frequency space (OTFS)-based URLLC system and adopt a deep learning (DL) approach to directly predict the precoder for the next time frame to minimize the frame error rate (FER) via implicitly exploiting the features from estimated historical channels in the delay-Doppler domain. By doing this, we can guarantee the system reliability even without the knowledge of ICSIT. To this end, a general precoder design problem is formulated where a closed-form theoretical FER expression is specifically derived to characterize the system reliability. Then, a delay-Doppler domain channels-aware convolutional long short-term memory (CLSTM) network (DDCL-Net) is proposed for predictive precoder design. In particular, both the convolutional neural network and LSTM modules are adopted in the proposed neural network to exploit the spatial-temporal features of wireless channels for improving the learning performance. Finally, simulation results demonstrated that the FER performance of the proposed method approaches that of the perfect ICSI-aided scheme.
... The optimality of OAMP is studied under the minimum MSE (MMSE) measure in [16] and also under the mutual information measure in [21]. Various applications of OAMP have been reported for signal processing and communications problems [18], [22]- [26]. ...
Approximate message passing (AMP) algorithms break a (high-dimensional) statistical problem into parts then repeatedly solve each part in turn, akin to alternating projections. A distinguishing feature is their asymptotic behaviours can be accurately predicted via their associated state evolution equations. Orthogonal AMP (OAMP) was recently developed to avoid the need for computing the so-called Onsager term in traditional AMP algorithms, providing two clear benefits: the derivation of an OAMP algorithm is both straightforward and more broadly applicable. OAMP was originally demonstrated for statistical problems with a single measurement vector and single transform. This paper extends OAMP to statistical problems with multiple measurement vectors (MMVs) and multiple transforms (MTs). We name the resulting algorithms as OAMP-MMV and OAMP-MT respectively, and their combination as augmented OAMP (A-OAMP). Whereas the extension of traditional AMP algorithms to such problems would be challenging, the orthogonal principle underpinning OAMP makes these extensions straightforward. The MMV and MT models are widely applicable to signal processing and communications. We present an example of MIMO relay system with correlated source data and signal clipping, which can be modelled as a joint MMV-MT system. While existing methods meet with difficulties in this example, OAMP offers an efficient solution with excellent performance.
