Nonconstant Modulus Constellations Research Articles

Blind Channel Estimation for Downlink Massive MIMO Systems With Imperfect Channel Reciprocity

We consider the performance of time-division duplex (TDD) massive multiple-input multiple-output (MIMO) with imperfect calibration of the transmit and receive radio frequency chains. By deriving the achievable signal-to-interference-plus-noise ratio (SINR) and the per-user bit error rate (BER) for constant modulus constellations, we establish that, under linear precoding, reciprocity imperfections can result in substantial reduction of the array gain. To mitigate this loss, we propose an algorithm for blind estimation of the effective channel gain at each user. We show that, with sufficiently many downlink data symbols, our blind channel estimation algorithm restores the array gain. In addition, the proposed blind gain estimation algorithm can improve performance compared to standard hardening-based receivers even under perfect reciprocity. Following this, we derive the BERs for non-constant modulus constellations, viz. $M$ -PAM and $M$ -QAM. We corroborate all our derived results using numerical simulations.

Optimal Joint Channel Estimation and Data Detection for Massive SIMO Wireless Systems: A Polynomial Complexity Solution

By exploiting large antenna arrays, massive MIMO (multiple input multiple output) systems can greatly increase spectral and energy efficiency over traditional MIMO systems. However, increasing the number of antennas at the base station (BS) makes the uplink joint channel estimation and data detection (JED) challenging in massive MIMO systems. In this paper, we consider the JED problem for massive SIMO (single input multiple output) wireless systems, which is a special case of wireless systems with large antenna arrays. We propose exact Generalized Likelihood Ratio Test (GLRT) optimal JED algorithms with low expected complexity, for both constant-modulus and nonconstant-modulus constellations. We show that, despite the large number of unknown channel coefficients, the expected computational complexity of these algorithms is polynomial in channel coherence time (T) and the number of receive antennas (N), even when the number of receive antennas grows polynomially in the channel coherence time (N=O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sup> ) suffices to guarantee an expected computational complexity cubic in T and linear in N). Simulation results show that the GLRT-optimal JED algorithms achieve significant performance gains (up to 5 dB improvement in energy efficiency) with low computational complexity.

Enabling Sphere Decoding for SCMA

In this paper, we propose a reduced-complexity optimal modified sphere decoding (MSD) detection scheme for SCMA. As SCMA systems are characterized by a number of resource elements (REs) that are less than the number of the supported users, the channel matrix is rank-deficient, and sphere decoding (SD) cannot be directly applied. Inspired by the Tikhonov regularization, we formulate a new full-rank detection problem that it is equivalent to the original rank-deficient detection problem for constellation points with constant modulus and an important subset of non-constant modulus constellations. By exploiting the SCMA structure, the computational complexity of MSD is reduced compared with the conventional SD. We also employ list MSD to facilitate channel coding. Simulation results demonstrate that in uncoded SCMA systems the proposed MSD achieves the performance of the optimal maximum likelihood (ML) detection. Additionally, the proposed MSD benefits from a lower average complexity compared with MPA.

OFDM reference signal reconstruction exploiting subcarrier‐grouping‐based multi‐level Lloyd‐Max algorithm in passive radar systems

This study proposes a low-complexity subcarrier-grouping-based multi-level Lloyd-Max (SG-ML-LM) algorithm to reconstruct orthogonal frequency division multiplexing (OFDM) reference signal in passive radar systems. The authors’ previously proposed multi-level Lloyd-Max (ML-LM) algorithm is originally intended to blindly estimate the narrowband flat fading channel coefficients for non-constant modulus constellations. The authors extend it into broadband wireless environment to blindly estimate the frequency-selective fading channel coefficients by exploiting OFDM signal structure properties. Then a SG-ML-LM algorithm is further proposed to reduce the computational complexity of ML-LM by utilising OFDM subcarrier correlation. Numerical results validate that: (i) ML-LM and SG-ML-LM algorithms are both feasible schemes to reconstruct OFDM reference signals for passive radar applications; (ii) compared with ML-LM algorithm, SG-ML-LM algorithm significantly reduces the computational complexity at the cost of blind channel estimation error, but the OFDM reference signal reconstruction accuracy is nearly maintained, so that the cross-ambiguity function performance is nearly lossless despite of a negligibly slight increase of main lobe Doppler bandwidth.

Data-classification-based SNR estimation for linearly modulated signals

We present a new numerical approach to SNR estimation of linearly modulated signals after passing through a complex additive white Gaussian noise (AWGN) channel. This classified data (CD) based SNR estimator is particularly suitable for both constant and non-constant modulus constellations, including BPSK, M-PSK and M-QAM. In essence, the received data will be first classified into a number of classes, and then a look-up table (LUT) is searched to find an entry that closest matches with the classified data; this matched result in LUT corresponds to the SNR value of the received data. The performance of the proposed estimator in terms of accuracy and complexity is evaluated by numerical simulations and compared with a few well-known estimators such as moment and maximum likelihood based estimators.

Signal‐to‐noise ratio estimation for DVB‐S2 based on eigenvalue decomposition

Signal-to-noise ratio (SNR) estimation is required in many receivers and systems of communications. In this study, the authors propose a novel SNR estimation method for DVB-S2 over additive white Gaussian noise channel. This is a non-data-aided, eigenvalue decomposition-based method. First, they use the even and odd component vectors of received signal to construct a novel 2 × 2 matrix, and propose an estimation variable using the standard condition number of this matrix. And then, they derive the relationship between this variable and SNR, and give the detailed expressions for SNR estimation on different modulations used in DVB-S2, including quadrature phase shift keying, 8PSK, 16 amplitude PSK (APSK) and 32APSK. Monte Carlo simulation results show the good estimation performance for different modulations. The proposed method is asymptotic unbiased and can be used for both constant and non-constant modulus constellations such as M-ary PSK, MAPSK and M-ary quadrature amplitude modulation.

Blind Channel Estimation Based on Multilevel Lloyd-Max Iteration for Nonconstant Modulus Constellations

In wireless communications, knowledge of channel coefficients is required for coherent demodulation. Lloyd-Max iteration is an innovative blind channel estimation method for narrowband fading channels. In this paper, it is proved that blind channel estimation based on single-level Lloyd-Max (SL-LM) iteration is not reliable for nonconstant modulus constellations (NMC). Then, we introduce multilevel Lloyd-Max (ML-LM) iteration to solve this problem. Firstly, by dividing NMC into subsets, Lloyd-Max iteration is used in multilevel. Then, the estimation information is transmitted from one level to another. By doing this, accurate blind channel estimation for NMC is achieved. Moreover, when the number of received symbols is small, we propose the lacking constellations equalization algorithm to reduce the influence of lacking constellations. Finally, phase ambiguity of ML-LM iteration is also investigated in the paper. ML-LM iteration can be more robust to the phase of fading coefficient by dividing NMC into subsets properly. As the signal-to-noise ratio (SNR) increases, numerical results show that the proposed method’s mean-square error curve converges remarkably to the least squares (LS) bound with a small number of iterations.

Squared envelope-based SNR estimation

Two novel squared envelope (SQE)-based and nondata-aided (NDA) signal-to-noise ratio (SNR) estimators over slow fading channel are proposed, which are robust to carrier offsets and can be applied to both nonconstant modulus (NCM) and constant modulus (CM) constellations. First, the moment-based first-and-second moments (M1M2) estimator is proposed because of its simple computation. The NDA approach has some difficulties when it is applied to NCM constellations, mainly due to self-noise introduced by the variability in the amplitude of transmitted symbols. Therefore, an expectation-maximization (EM) estimator, to solve the problem, instead of using the maximum-likelihood estimation, which is too complex for practical implementation, is proposed. For comparison purposes, the normalized Cramér–Rao lower bound is numerically evaluated as the performance benchmark for those estimators. Simulation results have verified that the two novel SQE and NDA SNR estimators are better at estimating the corresponding SNR range for different constellations. Besides, two hybrid schemes combining M1M2 with EM estimators are suggested to reduce deficiencies caused by these methods when they are employed independently to estimate the SNR.

A Nondata-Aided SNR Estimation Technique for Multilevel Modulations Exploiting Signal Cyclostationarity

Signal-to-noise ratio (SNR) estimators of linear modulation schemes usually operate at one sample per symbol at the matched filter output. In this paper we propose a new method for estimating the SNR in the complex additive white Gaussian noise (AWGN) channel that operates directly on the oversampled cyclostationary signal at the matched filter input. Exploiting cyclostationarity proves to be advantageous due to the fact that a signal-free Euclidean noise subspace can be identified such that only second order moments of the received waveform need to be computed. The proposed method is nondata-aided (NDA), as well as constellation and phase independent, and only requires prior timing synchronization to fully exploit the cyclostationarity property. The estimator can also be applied to nonconstant modulus constellations without requiring any tuning, which is a feature not found in existing approaches. Implementation aspects and simpler suboptimal solutions are also provided.

Cramer-Rao lower bound and EM algorithm for envelope-based SNR estimation of nonconstant modulus constellations

Signal-to-noise ratio (SNR) estimation for linearly modulated signals is addressed in this letter, focusing on envelope-based estimators, which are robust to carrier offsets and phase jitter, and on the challenging case of nonconstant modulus constellations. For comparison purposes, the true Cramer-Rao lower bound is numerically evaluated, obtaining an analytical expression in closed form for the asymptotic case of high SNR values, which quantifies the performance loss with respect to coherent estimation. As the maximum-likelihood algorithm is too complex for practical implementation, an expectation-maximization (EM) approach is proposed, achieving a good tradeoff between complexity and performance for medium-to-high SNRs. Finally, a hybrid scheme based on EM and moments-based estimates is suggested, which performs close to the theoretical limit over a wide SNR range.

A DSFBC-OFDM for a Next Generation Broadcasting System With Multiple Antennas

In this paper, we propose a differential space-frequency block code-orthogonal frequency division multiplexing (DSFBC-OFDM) scheme as a multiple-input multiple-output (MIMO) transmission technique for next generation broadcasting system. A linear decoding method for DSFBC, which performs comparably to the ML decoding method, is derived for the cases of two or four transmit antennas. A simple table lookup method is proposed to improve the efficiency of the encoding/decoding process of DSFBC for the case of non-constant modulus constellations. This not only reduces the computational load, but also removes the necessity of channel estimation. Also, synchronization techniques with a DSFBC-encoded phase reference symbol (PRS) are discussed. Finally, an MIMO channel model for the next generation broadcasting system is developed by extending the 3GPP MIMO model to fit broadcasting environments. The MIMO channel model is then used to compare BER performances of differential space block code schemes for various channel environments. Simulation results show that the DSFBC-16QAM scheme using either four transmit antennas with one receive antenna or two transmit antennas with two receive antennas achieves a performance gain of 12 dB, with a data rate twice faster than that of the conventional DQPSK scheme

Sixth-Order Statistics-Based Non-Data-Aided SNR Estimation

Signal-to-noise ratio (SNR) estimation is an important task in many digital communication systems. With nonconstant modulus constellations, the performance of the classical second- and fourth-order moments estimate is known to degrade with increasing SNR. A new non-data-aided estimate is proposed, which makes use of the sixth-order moment of the received data, and which can be tuned for a particular constellation in order to extend the usable range of SNR values. The advantage of the new method is especially significant for constellations with two different amplitude levels, e.g. 16-amplitude-and-phase-shift keying (16-APSK)

An efficient generalized sphere decoder for rank-deficient MIMO systems

We derive a generalized sphere decoder (GSD) for rank-deficient multiple input multiple output (MIMO) systems using N transmit antennas and M receive antennas. This problem arises when N>M or when the channel gains are strongly correlated. The upper triangular factorization of the Grammian yields an under-determined system and the standard sphere decoding (SD) fails. For constant modulus constellations, we modify the maximum likelihood (ML) cost metric so that the equivalent Grammian is rank N. The resulting GSD algorithm has significantly lower complexity than previous algorithms. A method to handle nonconstant modulus constellations is also developed.

Differential Space Time Block Codes Using Nonconstant Modulus Constellations for Four Transmit Antennas

We extend the differential space time block code (STBC) using nonconstant modulus constellations of two transmit antennas to four transmit antennas case. The proposed method obtains larger minimum Euclidean distances than those of conventional differential STBC with PSK constellations. We derive the symbol error rate (SER) performance of the proposed method and demonstrate the SER performance using computer simulations for both static and fast fading channels. For transmission rates greater than 2 bits/channel use and 3 bits/channel use, the proposed method outperforms the conventional differential STBC.

SNR estimation for nonconstant modulus constellations

We propose a new technique to estimate the signal-to-noise ratio (SNR) over the flat-fading channel. This is a nondata-aided, envelope-based estimator that can be applied to nonconstant modulus constellations, which is a feature not found in existing approaches. We also analyze the performance of our estimators for both phase-shift keying (PSK) and non-PSK constellations by deriving their asymptotic variances and comparing with the Crame/spl acute/r-Rao Bounds (CRBs). Moreover, we discuss how the SNR estimates can be used to approximate the bit error rate (BER) and how the accuracy of the SNR estimate is related to that of the BER estimate, which justifies the necessity for the accurate estimation of SNR. The analytical performance is shown for both PSK and non-PSK constellations, such as 8 quadrature amplitude modulation (QAM) and 16 QAM. Monte Carlo simulation results corroborate our analysis.

Differential space time block codes using nonconstant modulus constellations

We propose differential space time block codes (STBC) using nonconstant modulus constellations, e.g., quadrature amplitude modulation (QAM), which cannot be utilized in the conventional differential STBC. Since QAM constellations have a larger minimum distance compared with the phase shift keying (PSK), the proposed method has the advantage of signal-to-noise ratio (SNR) gain compared with conventional differential STBC. The QAM signals are encoded in a manner similar to that of the conventional differential STBC. To decode nonconstant modulus signals, the received signals are normalized by the channel power estimated forgoing training symbols and then decoded with a conventional QAM decoder. Assuming the knowledge of the channel power at the receiver, the symbol error rate (SER) bound of the proposed method under independent Rayleigh fading assumption is derived, which shows better SER performance than the conventional differential STBC. When the transmission rate is more than 3 bits/channel use in time-varying channels, the simulation results demonstrate that the proposed method with the channel power estimation outperforms the conventional differential STBC. Specifically, the posed method using the channel power estimation obtains a 7.3 dB SNR gain at a transmission rate of 6 bits/channel use in slow fading channels. Although the performance gap between the proposed method and the conventional one decreases as the Doppler frequency increases, the proposed method still exhibits lower SER than the conventional one, provided the estimation interval L is chosen carefully.

Generalized differential encoding: a nonlinear signal processing perspective

Incoherent detection based on differential encoding has been successfully applied to phase-shift-keying (PSK) signals because it eliminates the need for carrier phase acquisition and tracking at the receiver. This paper generalizes the idea of differential encoding by using a nonlinear transformation called multilag high-order instantaneous moment (ml-HIM). The ml-HIM decoder is capable of removing the effects of phase ambiguity, Dopper frequency, Doppler rate, and even higher order phase distortions. The degrees of freedom offered by the different lags are exploited to optimize system performance. In addition to the classical M-ary PSK, the generalized differential encoding idea is also applied to nonconstant modulus constellations, such as M-ary QAM and AM-PM.

Least squares approach to fractionally spaced blind channel equalisation

A new approach to fractionally spaced blind equalisation of finite-duration impulse response (FIR) channels is proposed based on the method of least squares (LS) and the constant modulus algorithm (CMA). A nonlinear transformation of the equaliser parameters is formulated and solved for, so as to avoid the potential problems associated with fractionally spaced CMA (FS-CMA) when the channel input is correlated. The new algorithm has a fast convergence rate in comparison with FS-CMA and the resulting equaliser setting is invariant to channel input correlation so long as all finite-length subsequences of the channel input sequence occur with nonzero probability. Although the algorithm is based on the assumption of constant modulus channel inputs, its application to nonconstant modulus constellations such as M-ary quadrature amplitude modulation (QAM) is also illustrated. The issue of channel noise enhancement is studied in connection with subchannel zeros, and a method for reducing inflated equaliser norms is proposed by way of reduced rank matrix approximation. A modified recursive least squares implementation of the algorithm is simulated to demonstrate its superior performance vis-à-vis FS-CMA.