Constant Modulus Algorithm Research Articles

Constant modulus algorithms via low-rank approximation

Abstract We present a novel convex-optimization-based approach to the solutions of a family of problems involving constant modulus signals. The family of problems includes the constant modulus and the constrained constant modulus, as well as the modified constant modulus and the constrained modified constant modulus. These solutions are shown to constitute semidefinite programs (SDP), thus enabling efficient interior-point methods with polynomial time complexity. The performance of the proposed solutions, demonstrated in several simulated experiments for the task of blind beamforming, is shown to be superior to existing methods.

CMA adaptive equalization in subspace pre-whitened blind receivers

In blind channel equalization, the use of whitening transformation (WT) preceding the constant modulus algorithm (CMA), referred to as the pre-whitened CMA, may accelerate the rate of convergence of using the CMA alone. In this work, we develop a novel subspace pre-whitened CMA (SP-CMA) using the subspace WT. By allowing the channel matrix to be non-square, this approach for the subspace WT offers greater flexibility in the design of the WT than the conventional per-whitened CMA does. The similarities and differences between the SP-CMA and the conventional CMA were first examined and analyzed in terms of their convergence behaviors from a theoretical viewpoint. The convergence behavior of the SP-CMA was analyzed under the whiteness assumption in which the input of the blind equalizer has been whitened beforehand by the subspace WT. The upper bound of the improvement in convergence of the SP-CMA over its CMA counterpart was developed. This analysis can be readily extended to other existing pre-whitened CMA schemes. Computer simulations using two highly dispersive multipath channels confirmed the superior convergence of the SP-CMA.

Blind Fuzzy Adaptation Step Control for a Concurrent Neural Network Equalizer

Mobile communications, not infrequently, are disrupted by multipath propagation in the wireless channel. In this context, this paper proposes a new blind concurrent equalization approach that combines a Phase Transmittance Radial Basis Function Neural Network (PTRBFNN) and the classic Constant Modulus Algorithm (CMA) in a concurrent architecture, with a Fuzzy Controller (FC) responsible for adapting the PTRBFNN and CMA step sizes. Differently from the Neural Network (NN) based equalizers present in literature, the proposed Fuzzy Controller Concurrent Neural Network Equalizer (FC-CNNE) is a completely self-taught concurrent architecture that does not need any training. The Fuzzy Controller inputs are based on the estimated mean squared error of the equalization process and on its variation in time. The proposed solution has been evaluated over standard multipath VHF/UHF channels defined by the International Telecommunication Union. Results show that the FC-CNNE is able to achieve lower residual steady-state MSE value and/or faster convergence rate and consequently lower Bit Error Rate (BER) when compared to Constant Modulus Algorithm-Phase Transmittance Radial Basis Function Neural Network (CMA-PTRBFNN) equalizer.

Nonlinear Modified Concurrent Equalizer

Wireless and optical communication systems not infrequently are disrupted by intersymbol interference (ISI). Wireless communications are degraded by multipath propagation in the wireless channel, while optical communications are degraded due to chromatic and polarization mode dispersion. For both cases, equalization techniques are applied to circumvent the problem. In this context, this paper proposes a novel concurrent blind equalizer, based on the nonlinear modified constant modulus algorithm (NMCMA) and on the soft direct decision (SDD) algorithm. The proposed nonlinear modified concurrent equalizer (NMCE) combines the NMCMA sharp decision regions with the SDD fast convergence, resulting in improved performance. The NMCE is compared with the NMCMA and with the constant modulus algorithm CMA-SDD equalizers under static and dynamic multipath scenarios. Results show that the proposed solution presents fast convergence, lower steady-state mean squared error (MSE), and reduced symbol error rate (SER) when compared with the NMCMA and the CMA-SDD equalizers, even in dynamic propagation scenarios.

Uplink Interference Suppression by Semi-Blind Adaptive Array With Decision Feedback Channel Estimation on Multicell Massive MIMO Systems

This paper proposes a novel semi-blind uplink interference suppression scheme for multicell multiuser massive multiple-input multiple-output (MIMO) systems. The inter-cell interference (ICI) becomes intenser when a number of users are spatially multiplexed via massive MIMO. Here arises a pilot contamination problem due to the limitation of orthogonal sequence set and thus inter-user interference (IUI) also remains unsuppressed. Our proposal employs a constant modulus algorithm (CMA) known as a blind interference suppression scheme and exploits channel state information (CSI)-aided minimum mean square error (MMSE) postcoding weight as an initial weight of CMA. Although MMSE weight cannot suppress ICI and IUI due to the pilot contamination, it can help subsequent CMA to capture the desired signals which is required for CMA to effectively work. Since CMA can fully exploit degree of freedom of massive arrays by using uncorrelated data signal part, derived weight successfully suppresses ICI/IUI while resolving the pilot contamination problem. A decision feedback channel estimation (DFCE) is then introduced to further improve CSI estimation accuracy and subsequent interference suppression performance. A series of signal processing, MMSE–CMA–DFCE, can be iteratively performed. The computer simulation reveals that our proposal can improve outage probability and it satisfies below the fifth percentile of cumulative distribution with saved symbol amount.

The unsupervised optimum linear finite length filter for fourth order wide sense stationary single output systems

The finite length absolute minimum vectors of the Constant Modulus (CM) criterion with an arbitrary source and a single output linear regression model are shown to equal the Wiener filter or one of its phase shifted versions. The exact equations for these vectors are also provided and depend on the second and the fourth order statistical moments of the sample sequence only. These expressions describe the behavior of the Constant Modulus Algorithm (CMA) when it operates at the desired global minimum and expose key characteristics of the CM minimization approach. These characteristics include the existence conditions, the persistency of excitation, the computational complexity, the excess CM cost, the robustness to an additive noise, and the tolerance to the length variations. Moreover, several computer simulation experiments are included to validate our findings.

A Robust Maximum Likelihood Algorithm for Blind Equalization of Communication Systems Impaired by Impulsive Noise

To improve the performance of the blind equalizer (BE) in impulsive noise environments, a robust maximum likelihood algorithm (RMLA) is proposed for the communication systems using quadrature amplitude modulation signals. A novel robust maximum likelihood cost function based on the constant modulus algorithm is constructed to effectively suppress the influence of impulsive noise and ensure the computational stability. Theoretical analysis is presented to illustrate the robustness and good computational stability of the proposed algorithm under the impulsive noise ambient. Moreover, it is proved that the weight vector of the proposed BE can converge stably by LaSalle invariance principle. Simulation results are provided to further confirm the robustness and stability of the proposed RMLA.

Binary constant modulus blind equalization for multi-level QAM signals under Gaussian and non-Gaussian noise

A binary constant modulus algorithm (BCMA) is proposed in this work for the blind equalization of multi-level quadrature amplitude modulation (QAM) systems. A novel cost function and its corresponding iteration formula are developed, which removes both the excess error and the steady state error introduced in the design of the most popular constant modulus algorithm (CMA). Theoretical analysis indicates the proposed algorithm can solve the performance degradation of CMA applying to the non-constant QAM signals. Moreover, it can combat the impulsiveness of alpha stable noise. Simulation results illustrate the effectiveness and robustness of the new algorithm under both Gaussian and non-Gaussian environments.

Pilot decontamination for multi-cell massive MIMO systems using asynchronous pilot design and data-aided channel estimation

In this work, we investigate the pilot contamination problem in multi-cell massive multi-input multi-output (MIMO) systems. The conventional approaches to address pilot contamination for multi-cell MIMO systems impose orthogonal pilots constraint across multiple users and cells, which leads to bandwidth inefficiency under large number of active users. We propose an uplink (UL) transmission protocol in which the pilot segments of users partially overlap across cells and even overlap with data from other cells, which provides more freedom in system designing. We also apply semi-blind algorithms at base station (BS) in order to separate the data streams of each user. Our analysis shows that semi-blind receiver removes pilot contamination effectively under large UL data length. The simulation results demonstrate that the performance of the proposed protocol with semi-blind receiver employing Constant Modulus Algorithm (CMA) is superior as compared to the existing schemes.

A full-duplex optical fiber/wireless coherent communication system with digital signal processing at the receiver

In this paper, a full-duplex, 120 Gbps optical fiber/wireless system is presented for high-speed and multicasting communication link. Both the wired and wireless systems use Dual Polarization 16 Quadrature Amplitude Modulation (DP-16-QAM) with homodyne detection. Blind Phase Search (BPS) algorithm, Constant Modulus Algorithm (CMA) and Viterbi Phase Estimation (VPE) algorithm are used at the receiver end to compensate for signal distortions such as Chromatic Dispersion (CD), Polarization Mode Dispersion (PMD), nonlinear impairments, frequency and phase mismatch. The system performance is analyzed in terms of Error Vector Magnitude (EVM), Bit Error Rate (BER ≤ 10−12) and constellation diagram based on OptiSystem V.13 simulation platform. The maximum transmission distances achieved are 275 km and 1.1 km for Standard Single Mode Fiber (SSMF) and Free Space Optical Link (FSOL), respectively.

Null space method for tap value initialization in constant modulus algorithm for mode division multiplexing systems

In this paper, we propose a constant modulus algorithm (CMA) for mode division multiplexing (MDM) systems with improved convergence performance. In order to adapt to sparse channels with large differential mode group delay (DMGD) in MDM systems, the CMA adopts a variable step size similar to the improved proportionate normalized least-mean-square (IPNLMS) algorithm. In additional to that, when a singularity problem is encountered or the tap values fail to converge, it reinitializes the tap coefficients according to the tap vectors of the successfully de-multiplexed data tributaries. The proposed initialization approach is based on the fact that the channel matrix is unitary in the frequency domain in the absence of mode dependent loss (MDL), which means the channel coefficient vectors for each data tributary should be orthogonal to each other. By limiting the initial values of the taps within the null space of the complex conjugate vectors of the successfully de-multiplexed channels, singularity can be effectively avoided and the convergence of the taps is guaranteed. When the number of modes is two, the proposed algorithm becomes the constrained CMA, which has been commonly implemented in polarization division multiplexing (PDM) systems. Although the algorithm has been derived under zero MDL assumption, it is found that the proposed CMA can be quite resilient to MDL. No singularity/tap convergence failure problem occurs when the MDL is below 4 dB at both the input and the output ports.

An improved least squares constant modulus algorithm for collision messages separation in multichannel space‐based AIS

SummaryIn space‐based automatic identification system, message collision is inevitable. To separate collision messages in multichannel space‐based automatic identification system, an improved least squares constant modulus algorithm (LSCMA) is proposed in this paper. The algorithm initializes LSCMA by using the result of complex symmetric fast independent component analysis and reduces the dimension of the received signal by using the minimum description length algorithm. At the same time, a new metric named failure rate is proposed to measure the separation performance. The simulation shows that the performance loss caused by the delay, the frequency and phase offset, and the input signal‐to‐interference ratio is within an acceptable range. The proposed algorithm outperforms the original LSCMA and other algorithms in terms of red packet error rate and failure rate.

Pulse-Overlapping Super-Nyquist WDM System

In this paper, a pulse-overlapping super-Nyquist (Pol-SN) wavelength-division multiplexing (WDM) scheme is proposed to improve spectral efficiency of coherent optical transmission. In this scheme, two tributaries of polarization-division multiplexing quadrature phase-shift keying (PDM-QPSK) signals at the transmitter, carried by the same wavelength, are grouped together to allow for pulse overlapping in the time domain. Then, several wavelength channels are packed tightly in the frequency domain to form a super-Nyquist WDM system. At the receiver, three partial-response detection schemes are investigated and compared: (1) constant modulus algorithm (CMA) + duobinary shaping + maximum likelihood sequence estimation (MLSE); (2) duobinary shaping + multimodulus algorithm (MMA) + MLSE; and (3) channel shortening filter. Among these schemes, the easy-to-implement duobinary shaping +MMA + MLSE is selected by virtue of its best performance and lowest complexity. A 224-Gb/s Pol-SN PDM-QPSK system is investigated through numerical simulations. The simulation results show that the channel spacing of the 224 Gb/s Pol-SN PDM-QPSK system can be reduced to 20 GHz (11.2 bit/s/Hz SE) with 7% overhead hard-decision forward-error correction (HD-FEC, 3.8 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> ). At 26-GHz channel spacing, Pol-SN PDM-QPSK signals show 5.5-dB OSNR improvement, as compared with the signals of 224-Gb/s super-Nyquist PDM-16QAM. The simulation results also show that 224-Gb/s Pol-SN PDM-QPSK system, with 20-GHz spacing, can transmit up to 1000-km SSMF with 7% overhead HD-FEC.

Burst-Mode Digital Signal Processing That Pre-Calculates FIR Filter Coefficients for Digital Coherent PON Upstream

This paper proposes a burst-mode digital signal processing architecture for digital coherent time-divisionmultiplexed passive optical network (PON) upstream transmission, wherein two key advances are introduced to shorten the preamble length of burst signals required to complete adaptive equalization, as performed by a constant modulus algorithm: tap coefficients of a finite impulse response (FIR) filter are pre-calculated in the optical network unit (ONU) discovery process, and the feed-forward state of polarization compensation is employed before adaptive equalization. The bit error rate performance attained by the proposal is experimentally evaluated; successful burstmode coherent detection with a high sensitivity of -44.7 dBm is demonstrated for 20 Gbit/s single polarizationquadrature phase shift keying burst signals transmitted over a 40 km single mode fiber, even with the use of a short preamble of 1.3 μs, which meets the optical power specifications required for a 512-way split system. The proposal utilizes register request signals whose preamble is longer than those used in current PON systems to optimize the tap coefficients of the FIR filter, which may disrupt smooth ONU registration in high-splitting-ratio systems. To evaluate this impact, numerical calculations based on the Monte Carlo method are conducted, and the results show that the time required to complete the registration of 512 ONUs is just 40 s or so.

Modulation format identification enabled by the digital frequency-offset loading technique for hitless coherent transceiver.

We propose a blind and fast modulation format identification (MFI) enabled by the digital frequency-offset (FO) loading technique for hitless coherent transceiver. Since modulation format information is encoded to the FO distribution during digital signal processing (DSP) at the transmitter side (Tx), we can use the fast Fourier transformation based FO estimation (FFT-FOE) method to obtain the FO distribution of individual data block after constant modulus algorithm (CMA) pre-equalization at the receiver side, in order to realize non-data-aided (NDA) and fast MFI. The obtained FO can be also used for subsequent FO compensation (FOC), without additional complexity. We numerically investigate and experimentally verify the proposed MFI with high accuracy and fast format switching among 28 Gbaud dual-polarization (DP)-4/8/16/64QAM, time domain hybrid-4/16QAM, and set partitioning (SP)-128QAM. In particular, the proposed MFI brings no performance degradation, in term of tolerance of amplified spontaneous emission (ASE) noise, laser linewidth, and fiber nonlinearity. Finally, a hitless coherent transceiver enabled by the proposed MFI with switching-block of only 2048 symbols is demonstrated over 1500 km standard single mode fiber (SSMF) transmission.

Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system.

A window-split frequency domain Kalman scheme is proposed in this paper for the equalization of large polarization mode dispersion (PMD) and ultra-fast rotation of state-of-polarization (RSOP) which is an extreme environment due to the Kerr effect and the Faraday effect under the lightning strike near the fiber cables. In order to carry out the proposed Kalman scheme, we give a simplified and equivalent fiber channel model as a replacement for the general model of the polarization effect of the co-existence of PMD and RSOP. With this fiber channel model, we can conduct compensation for PMD in the frequency domain and tracking RSOP in time domain. A half analytical and half empirical theory for the initialization of the process and measurement noise covariance is also presented in theory and verified by the numerical simulation. The performance of the proposed Kalman scheme is checked in the 28Gbaud PDM-QPSK coherent system built on both simulation and experiment platforms. The simulation and experiment results confirm that compared with the generally used constant modulus algorithm (CMA), the proposed scheme provides excellent performance and stability to cope with large range DGD from 20ps to 200ps and RSOP from 200krad/s to 2Mrad/s, with less computational complexity.

A time and frequency synchronization method for CO-OFDM based on CMA equalizers

In this paper, an efficient time and frequency synchronization method based on a new training symbol structure is proposed for polarization division multiplexing (PDM) coherent optical orthogonal frequency division multiplexing (CO-OFDM) systems. The coarse timing synchronization is achieved by exploiting the correlation property of the first training symbol, and the fine timing synchronization is accomplished by using the time-domain symmetric conjugate of the second training symbol. Furthermore, based on these training symbols, a constant modulus algorithm (CMA) is proposed for carrier frequency offset (CFO) estimation. Theoretical analysis and simulation results indicate that the algorithm has the advantages of robustness to poor optical signal-to-noise ratio (OSNR) and chromatic dispersion (CD). The frequency offset estimation range can achieve [−Nsc2ΔfN,+Nsc2ΔfN] GHz with the mean normalized estimation error below 12×10−3 even under the condition of OSNR as low as 10 dB.

An Efficient Algorithm for Unit-Modulus Quadratic Programs With Application in Beamforming for Wireless Sensor Networks

In this letter, we consider a network of single-antenna sensors that aim at the estimation of an unknown deterministic parameter. The sensors collect the observations and forward them to a fusion center via applying a phase-only beamforming weight. The derivation of the optimal beamforming weights requires the solution of a unit-modulus quadratic program (UQP), which in the relevant literature is solved via the semidefinite relaxation (SDR) technique or via a variation of the analytic constant modulus algorithm. The former achieves better performance, though it exhibits high computational cost that increases drastically with the number of sensors. The latter requires much less complexity, though it achieves worse performance. To that end, we propose an efficient algorithm for the solution of UQPs based on the alternating direction method of multipliers. The new approach achieves almost identical performance to that of the SDR-based approach while exhibiting significantly reduced computational complexity. The convergence of the proposed algorithm to a Karush-Kuhn-Tucker point is theoretically studied, and its effectiveness is verified via numerical results.

Differential phase-shift keying and channel equalization in free space optical communication system

We present the performance benefits of differential phase-shift keying (DPSK) modulation in eliminating influence from atmospheric turbulence, especially for coherent free space optical (FSO) communication with a high communication rate. Analytic expression of detected signal is derived, based on which, homodyne detection efficiency is calculated to indicate the performance of wavefront compensation. Considered laser pulses always suffer from atmospheric scattering effect by clouds, intersymbol interference (ISI) in high-speed FSO communication link is analyzed. Correspondingly, the channel equalization method of a binormalized modified constant modulus algorithm based on set-membership filtering (SM-BNMCMA) is proposed to solve the ISI problem. Finally, through the comparison with existing channel equalization methods, its performance benefits of both ISI elimination and convergence speed are verified. The research findings have theoretical significance in a high-speed FSO communication system.

Laboratory Experiment of Blind Adaptive Array with Subcarrier Transmission Power Assignment in Spectrum Superposing Scenarios

This paper experimentally validates the basic feasibility of our proposed blind adaptive array (BAA) with subcarrier transmission power assignment (STPA) scheme using a prototype. The proposed STPA-BAA enables spectrum superposing without any channel state information between two different wireless communication systems: employing STPA-BAA to the secondary system, inter-system interference from/to the primary system can be suppressed. Exploiting the characteristics of constant modulus algorithm (CMA) and power inversion (PI), the secondary transmitter provides two levels of power density for each subcarrier: high and low levels. It enables the secondary receiver to suppress interference with almost the same level of the desired signal. It is also effective in reducing interference to the primary receiver that has no interference suppression function. This paper conducts laboratory experiments of spectrum superposition in two multicarrier systems by wired setup. Though the practical performance of STPA-BAA is limited due to the quantization level or dynamic range of the transceiver, effectiveness of the proposed scheme is confirmed.