Multimodulus algorithms for blind equalization

Assuming the intersymbol interference (ISI) has been removed by an adaptive equalizer which employs a suitable blind equalization algorithm, the output constellation may still be rotated by an arbitrary phase angle. These angles are the result of the existence of undesirable local minima in the phase parameter space. A simple technique to identify all such angles is presented. In addition, the stability of the rotated solutions and the range of phase angles that leads to such solutions can be determined too. Three blind algorithms, namely the multi-modulus algorithm (MMA), the reduced constellation algorithm (RCA), and the constant modulus algorithm (CMA) are analyzed using this approach. The immunity of the MMA towards such solutions is verified analytically while the susceptibility of the RCA towards them, especially the 45deg-solution, is verified through simulations. Our analysis also agrees with the well known fact that the CMA is indeed phase-blind.

This paper presents a new blind equalization algorithm called multimodulus algorithm (MMA). This algorithm combines the benefits of the well-known reduced constellation algorithm (RCA) and constant modulus algorithm (CMA). In addition, MMA provides more flexibility than RCA and CMA, and is better suited to take advantage of the symbol statistics of certain types of signal constellations, such as nonsquare constellations, very dense constellations, and some wrong solutions.

In this paper, we present Support Vector Machine (SVM)-based blind per tone equalization for OFDM systems. Blind per tone equalization using Constant Modulus Algorithm (CMA) and Multi-Modulus Algorithm (MMA) are used as the comparison benchmark. The SVM-based cost function utilizes a CMA-like error function and the solution is obtained by means of an Iterative Re-Weighted Least Squares Algorithm (IRWLS). Moreover, like CMA, the error function allows to extend the method to multilevel modulations. In this case, a dual mode algorithm is proposed. Dual mode equalization techniques are commonly used in communication systems working with multilevel signals. Practical blind algorithms for multilevel modulation are able to open the eye of the constellation, but they usually exhibit a high residual error. In a dual mode scheme, once the eye is opened by the blind algorithm, the system switches to another algorithm, which is able to obtain a lower residual error under a suitable initial ISI level. Simulation experiments show that the performance of blind per tone equalization using support vector machine has better than blind per tone equalization using CMA and MMA, from viewpoint of average Bit-Error Rate (BER).

In this paper we consider the blind equalization in the transform domain. For blind equalization, three blind algorithms, general Sato algorithm, constant modulus algorithm, multimodulus algorithm, which are known to be very efficient from implementation issues, are used. To improve the rate of convergence, we used the band-partitioning property of transform domain (TD). In fact, the TD causes the channel output samples to be uncorrelated and this increases the rate of convergence. Also, to further improve the rate of convergence, an optimal time variant step size is used. Simulation results are presented to show the fast convergence of the proposed blind equalizer.

This work investigates the effect of channel noise on the performances of the constant modulus algorithm (CMA), the multimodulus algorithm (MMA), and a phase tracker (PT), following the CMA for carrier phase recovery, when various symmetrical two-dimensional complex-valued source constellations are used. The contributions of this work are two-fold. First, a generalized formulation of the relocation of stationary points toward the origin owing to channel noise is derived for both the CMA and the MMA, allowing their sensitivity to channel noise to be explicitly compared when various constellations are used. Second, the curvatures of the error surface at the stationary points in both the radial and the angular directions are reduced in a way that is linearly proportional to the square of the distances between the stationary points and the origin as the stationary points of the CMA and the MMA relocate toward the origin in the presence of additive channel noise. The reduction of the error surface curvature in the radial direction reduces the rate of blind equalization convergence, whereas the reduction in the angular direction degrades the capacity for carrier phase recovery.

The constant modulus algorithm (CMA) for blind equalization requires a separate carrier-recovery system for phase recovery. A modified CMA, called the multimodulus algorithm (MMA), which may perform joint blind equalization and carrier recovery without the need for a separate carrier-recovery system for quadrature amplitude modulation (QAM) signal constellations. This letter mathematically analyzes the steady-state mean square error (MSE) of MMA. Analysis results indicate that MMA produces 50% fewer steady-state MSE than CMA.

Blind per tone equalization of multilevel signals using support vector machines for OFDM in wireless communication

Summary form only given. This paper presents a Variable Step Size (VSS) technique that is integrated with a Stop-And-Go (SAG) approach for blind adaptive equalization of Quadrature Amplitude Modulation (QAM) signals. The proposed technique is applied to the Constant-Modulus Algorithm (CMA) and the Multi-Modulus Algorithm (MMA). The SAG approach allows the VSS technique to either vary the step size or to hold the previous one depending on a specific error criterion. This leads to the reduction of the amount of computations involved Simulation results show that algorithms with the VSS technique result in less steady-state Mean-Squared Errors (MSEs) than those obtained for algorithms without the VSS technique for the same speed of convergence.

Experimental performances of blind adaptive training algorithms have been evaluated in real-time WiMax (3.5GHz) radio channels in this paper. Inter Symbol Interference (ISI) cancellation and Mean Square Error (MSE) performances of the most commonly used blind equalization techniques, Constant Modulus Algorithm (CMA), and CMA based modified CMA (M-CMA) and normalized CMA (N-CMA) algorithms are investigated in study.

Among the useful blind equalization algorithms, there are stochastic-gradient iterative equalization algorithms. Because these algorithms basically use a linear FIR filter except to adopt a memoryless nonlinear function of the equalizer output to generate the desired response, only linear channel distortion can be corrected well. To overcome nonlinear channel distortion two new blind equalization schemes using the structure of the complex multi-layer perceptron that can deal with signals of any constellation size are proposed. One is a modification of the constant modulus algorithm (CMA), the most widely used blind algorithm. The other is a modification of the radius-directed equalization (RDE) algorithm with multiple constant modulus that is more suitable for the node activation function having a multi-saturated output region. The experimental results show that the proposed schemes have very low MSE in the steady state and at the same time can recover the arbitrary phase rotation due to the channel when the major channel distortion is both linear and nonlinear.

This paper proposes a new computationally-efficient method for blind equalization and presents a comprehensive review of existing computationally-efficient methods. The proposed method selectively updates the equalizer taps based on the equalizer output radius for QAM signal constellations. Signed-error, dithered signed-error, quantized-error, block, and update-decimated methods are discussed for computationally-efficient blind equalization. The proposed and discussed methods are applied to the constant modulus algorithm (CMA) and multi-modulus algorithm (MMA), and comparisons are provided.

This paper expands our previously proposed semi-blind uplink interference suppression scheme on multicell multiuser massive MIMO systems to support multi modulus signals. The original proposal applies channel state information (CSI) aided blind adaptive array (BAA) interference suppression after the beamspace preprocessing and decision feedback channel estimation (DFCE). It employed the constant modulus algorithm (CMA) as the BAA scheme which can fully exploit the degree of freedom (DoF) of massive antenna arrays to suppress both inter-user interference (IUI) and inter- cell interference (ICI). Its effectiveness has been verified under the extensive pilot contamination constraint. Unfortunately, CMA basically works well only for the constant envelope signal such as QPSK and thus the proposed scheme should be expanded to cover QAM signals for more general use. This paper newly proposes to apply multi modulus algorithm (MMA) and data-aided sample matrix inversion based minimum mean square error (MMSE-SMI) weight derivation. It can successfully realize interference suppression under the use of multi-level envelope signals such as 16QAM with satisfactorily outage probability performance.

Adaptive beamforming plays an important role in smart antenna systems. Nonblind adaptive algorithm utilizes the known training sequence to update the weights of antenna array at the cost of low transmission efficiency. Without sending training sequences, blind adaptive algorithm is a solution to implement beamforming and increase the transmission efficiency. Two classic blind algorithms are studied, namely constant modulus (CM) algorithm and Least squares-constant modulus (LS-CM) algorithm. Considering the advantages of the nonblind and blind algorithms, a semi-blind adaptive beamforming based on CM algorithm is proposed in this paper. Simulation results indicate it has comparable performance compared with nonblind adaptive algorithm, while it only needs the training sequence in the first iteration.

In this paper, an efficient blind equalization with two‐stage single/multilevel modulus and decision directed (DD) least mean square (LMS)‐like algorithm is proposed for high‐order QAM (64/256/1024QAM) cable modem systems. In wire‐lined cable systems, it seems that the cable channel may display a nearly invariant multi‐path effect over a considerable time, and then the blind equalizer can have the capability to recover the received high‐order QAM symbols on wired cables. The proposed blind equalization algorithm applies a two‐stage convergence scheme, which results from the multi‐modulus algorithm (MMA), sometimes called the modified constant modulus algorithm (MCMA), and the DD LMS‐like method. In the first equalization stage, the joint MMA and DD LMS‐like methods are applied for the purpose of fast convergence. When the convergence process reaches an acceptably steady condition, the convergence detector changes the process into the second stage. In the second equalization stage, the enhanced DD LMS‐like method with the generalized multi‐modulus is applied to reduce the mean square error (MSE) of equalizations further. For high‐order QAM cable systems, the proposed method performs faster than the previous well‐known blind methods do. Simultaneously, the proposed algorithm also provides less MSE and lower symbol error rate (SER) than the other blind methods do at the same signal‐to‐noise ratio (SNR).

In this work, a new algorithm, based on the minimum-disturbance principle with relaxation, is presented for the blind equalization of complex signals. This algorithm combines the benefits of the well-known reduced constellation algorithm (RCA) and constant modulus algorithm (CMA). The convergence characteristics of the proposed algorithm are demonstrated by way of simulations. In addition, closed form expressions are obtained for the statistical (dispersion) constants used in these algorithms.