Constant Modulus Cost Function Research Articles

Blind Constant Modulus Multiuser Detection via Low-Rank Approximation

We present a novel convex-optimization-based solution to blind linear multiuser detection in direct-sequence code division multiple access systems. The solution is based on a convex low-rank approximation of the linearly constrained constant modulus cost function, thus guaranteeing its global minimization. Further, it can be cast as a semidefinite program, implying that it can be solved using interior-point techniques with polynomial time complexity. The solution is parameter free and is shown to be superior to existing solutions in terms of output signal-to-interference-plus-noise ratio and bit error rate, especially for a small number of samples.

Application of Constraint CMA Blind Equalization Algorithm in the Medical Image Restoration

A constraint constant module blind equalization algorithm for medical image based on dimension reduction was proposed. The constant modulus cost function applied to medical image was founded. In order to improve the effect of image restoration, a constraint item was introduced to restrict cost function, and it guarantees that the algorithm converge the optimal solution. Compared to the traditional methods, the novel algorithm improves peak signal to noise ratio and restoration effects. Computer simulations demonstrate the effectiveness of the algorithm.

Low-Complexity Variable Forgetting Factor Mechanism for Blind Adaptive Constrained Constant Modulus Algorithms

In this paper, we propose a novel low-complexity variable forgetting factor (VFF) mechanism for blind adaptive constrained constant modulus (CCM) recursive least squares (RLS) algorithms applied to linear interference suppression in direct-sequence code-division multiple access (DS-CDMA) systems. The proposed variable forgetting factor mechanism employs an updated component related to the time average of the constant modulus (CM) cost function to automatically adjust the forgetting factor in order to ensure good tracking of the interference and the channel. Convergence and tracking analyses are carried out. Analytical expressions for predicting the mean-squared error of the proposed adaptation technique are obtained. Simulation results are presented for nonstationary environments and show that the proposed variable forgetting factor mechanism achieves superior performance to previously reported methods at a reduced complexity.

Research of constraint constant modulus medical CT image blind equalization algorithm

In order to improve the Peak Signal-to-Noise Ratio(PSNR) of images,and to raise the reliability and restoration effects of the algorithm,the degradation and restoration process of image was transformed by a linear transform to be equivalent to one dimensional convolution.A constraint constant modulus cost function of blind equalization applied to medical CT images was founded,and its convexity was proved.A blind equalization algorithm based on dimension reduction was proposed.Computer simulations demonstrate the effectiveness of the algorithm.Matrix inversion is avoided to improve the efficiency of operations.

Blind source separation of convolutive mixtures of non-circular linearly modulated signals with unknown baud rates

This paper addresses the problem of blind separation of convolutive mixtures of BPSK and circular linearly modulated signals with unknown (and possibly different) baud rates and carrier frequencies. In previous works, we established that the Constant Modulus Algorithm (CMA) is able to extract a source from a convolutive mixture of circular linearly modulated signals. We extend the analysis of the extraction capabilities of the CMA when the mixing also contains BPSK signals. We prove that if the various source signals do not share any non-zero cyclic frequency nor any non-conjugate cyclic frequencies, the local minima of the constant modulus cost function are separating filters. Unfortunately, the minimization of the Godard cost function generally fails when considering BPSK signals that have the same rates and the same carrier frequencies. This failure is due to the existence of non-separating local minima of the Godard cost function. In order to achieve the separation, we propose a simple modification of the Godard cost function which only requires knowledge of the BPSK sources frequency offsets at the receiver side. We provide various simulations of realistic digital communications scenarios that support our theoretical statements.

Blind Reduced-Rank Adaptive Receivers for DS-UWB Systems Based on Joint Iterative Optimization and the Constrained Constant Modulus Criterion

A novel linear blind adaptive receiver based on joint iterative optimization (JIO) and the constrained-constant-modulus design criterion is proposed for interference suppression in direct-sequence ultrawideband systems. The proposed blind receiver consists of the following two parts: 1) a transformation matrix that performs dimensionality reduction and 2) a reduced-rank filter that produces the output. In the proposed receiver, the transformation matrix and the reduced-rank filter are jointly and iteratively updated to minimize the constant-modulus cost function subject to a constraint. Adaptive implementations for the JIO receiver are developed using the normalized stochastic gradient (NSG) and recursive least squares (RLS) algorithms. To obtain a low-complexity scheme, the columns of the transformation matrix with the RLS algorithm are individually updated. Blind channel estimation algorithms for both versions (i.e., NSG and RLS) are implemented. Assuming perfect timing, the JIO receiver only requires the spreading code of the desired user and the received data. Simulation results show that both versions of the proposed JIO receivers have excellent performance in terms of suppressing the intersymbol interference (ISI) and multiple-access interference (MAI) with low complexity.

Adaptive blind equalization for MIMO systems uadaptive blind equalization for MIMO systems under α-stable noise environmentnder α-stable noise environment

The presence of non-Gaussian impulsive noise in wireless system can degrade the performance of existing equalizers and signal detectors. In this paper, the problem of blind source separation and equalization for MIMO systems in heavy-tailed impulsive noise is studied. A generalized multiuser constant modulus cost function by employing the fractional lower-order statistic of the equalizer input signal is proposed. The associated adaptive blind equalization algorithm based on a stochastic gradient descent method is defined as fractional lower-order multi-user constant modulus algorithm (FLOS MU CMA). Computer simulations are presented to illustrate the performance of the new algorithm.

Space–time blind equalisation in impulsive noise

The presence of non-Gaussian impulsive noise in wireless system can degrade the performance of existing equalisers and signal detectors. The problem of blind source separation and equalisation for multiple-input/multiple-output (MIMO) channels under heavy-tailed impulsive noise environment is studied. A generalised multi-user constant modulus (CM) cost function is proposed by employing the fractional lower order CM property of the equaliser input signals as well as the fractional lower order cross-correlations between them. The associated adaptive blind equalisation algorithm based on a stochastic gradient descent method is defined as fractional lower order multi-user constant modulus algorithm (FMU_CMA), which is able to mitigate impulsive channel noise while recovering all input signals simultaneously. The steady-state mean-square error (MSE) performance of the FMU_CMA is studied in a noise-free environment; the approximate expression is derived based on the energy-preserving relation and the Taylor series expansion. Simulation studies are undertaken to support the analysis with CM and non-CM signal.

Blind Adaptive Constrained Reduced-Rank Parameter Estimation Based on Constant Modulus Design for CDMA Interference Suppression

This paper proposes a multistage decomposition for blind adaptive parameter estimation in the Krylov subspace with the code-constrained constant modulus (CCM) design criterion. Based on constrained optimization of the constant modulus cost function and utilizing the Lanczos algorithm and Arnoldi-like iterations, a multistage decomposition is developed for blind parameter estimation. A family of computationally efficient blind adaptive reduced-rank stochastic gradient (SG) and recursive least squares (RLS) type algorithms along with an automatic rank selection procedure are also devised and evaluated against existing methods. An analysis of the convergence properties of the method is carried out and convergence conditions for the reduced-rank adaptive algorithms are established. Simulation results consider the application of the proposed techniques to the suppression of multiaccess and intersymbol interference in DS-CDMA systems.

On the Misbehavior of Constant Modulus Equalizers for Improper Modulations

In this letter, the constant-modulus (CM) cost function is analyzed under the general assumptions that improper modulation schemes of practical interest are employed and the baseband equivalent of the channel impulse response is complex-valued. This study allows one to determine a broad family of undesired minima of the CM cost function, which do not lead to perfect symbol recovery in the absence of noise. The results developed herein generalize and subsume as a particular case existing studies of the CM cost function, which exclusively consider real-valued binary modulations.

Tracking Issues of Some Blind Equalization Algorithms

Due to the growing demand for mobile communications, blind adaptive algorithms have an important role in improving data transmission efficiency. In this context, the convergence and tracking analysis of such algorithms is a problem of interest. Recently, a tracking analysis of the Constant Modulus Algorithm was presented based on an energy conservation relation. In this letter we extend that analysis to blind quasi-Newton algorithms that minimize the Constant Modulus cost function. Under certain conditions, the considered algorithms can reach the same steady-state mean-square error. Close agreement between analytical and simulation results is shown.

Recursive Least Squares Constant Modulus Algorithm for Blind Adaptive Array

We consider the problem of blind adaptive signal separation with an antenna array, based on the constant modulus (CM) criterion. An approximation to the CM cost function is proposed, which allows the use of the recursive least squares (RLS) optimization technique. A novel RLS constant modulus algorithm (RLS-CMA) is derived, where the modulus power of the array output can take on arbitrary positive real values (i.e., fractional values allowed). Simulations are performed to compare the performance of the proposed RLS-CMA to other well-known algorithms for blind adaptive beamforming. Results indicate that the RLS-CMA has a significantly faster convergence rate and better tracking ability.

Blind Search for Optimal Wiener Equalizers Using an Artificial Immune Network Model

This work proposes a framework to determine the optimal Wiener equalizer by using an artificial immune network model together with the constant modulus (CM) cost function. This study was primarily motivated by recent theoretical results concerning the CM criterion and its relation to the Wiener approach. The proposed immune-based technique was tested under different channel models and filter orders, and benchmarked against a procedure using a genetic algorithm with niching. The results demonstrated that the proposed strategy has a clear superiority when compared with the more traditional technique. The proposed algorithm presents interesting features from the perspective of multimodal search, being capable of determining the optimal Wiener equalizer in most runs for all tested channels.

CM-based channel estimators for space-time block-coded M-PSK and M-QAM systems

A constant modulus (CM)-based channel estimation scheme for space-time block-coded M-PSK and M-QAM transmission systems is derived. Exploiting the orthogonal structure of the space-time block coded transmission model, the analytical solutions to minimize the CM cost function are given for 2/sup k/ PSK (k>2) and 2/sup 2k/-QAM schemes, and used to derive the expressions for the CM-based channel estimators. Illustrative results on the performance are presented.

An enhanced Godard blind equalizer based on the analysis of transient phase

This paper proposes an enhanced version of Godard blind equalizer, also known as the constant modulus algorithm, with faster convergence speed. To show that the cause of slow convergence is closely related to the behaviour of equalizer output power, the difference between the behaviour of equalizer output power and the reduction of intersymbol interference is studied by comparing the two types of minimum points of constant modulus cost function. For faster convergence, we insert a gain stage after the equalizer to control the behaviour of equalizer output power. We show the effects of gain stage on the convergence speed through simulations. We also show that the proposed method can be applied for other algorithms based on the modulus, such as the multimodulus algorithm.

Vector-CM stable equilibrium analysis

The vector-constant modulus (VCM) criterion is an extension of the constant modulus (CM) criterion introduced recently for equalization of channels involving Gaussian sources. In this letter, we analyze the behavior of VCM for arbitrary source distributions and combined channel-receiver impulse responses of finite dimension. We begin by pointing out the difference between the VCM and CM cost functions and showing that the VCM criterion can be expressed as a composite criterion combining the CM cost function and a penalty term involving cross-correlations of the equalizer output. We continue by providing conditions for noise-free channels, under which VCM admits stable minima corresponding to zero-forcing (ZF) channel receivers. We find that for sub-Gaussian sources, the VCM and CM criteria share the same global minima. For Gaussian and super-Gaussian sources, however, it appears that only ZF receivers corresponding to input/output transmission delays at the extremes of the range of possible delays are truly stable equilibria of VCM.

An analysis of constant modulus algorithm for array signal processing

The constant modulus (CM) cost function is analyzed for array signal processing. The analysis includes arbitrary source types. It is shown that CM receivers have the signal space property except in two special cases. Local minima of CM cost function are completely characterized for the noiseless case. In the presence of measurement noise, the existence of CM local minima in the neighborhood of Wiener receivers is established, and their mean squared error (MSE) performance, output power, estimation bias and residual interference are evaluated. Numerical examples are presented to demonstrate the effects of signal statistics on the locations of CM local minima.

Relationships between the constant modulus and Wiener receivers

The Godard (1980) or the constant modulus algorithm (CMA) is an effective technique for blind receiver design in communications. However, due to the complexity of the constant modulus (CM) cost function, the performance of the CM receivers has primarily been evaluated using simulations. Theoretical analysis is typically based on either the noiseless case or approximations of the cost function. The following question, while resolvable numerically for a specific example, remains unanswered in a generic manner. In the presence of channel noise, where are the CM local minima and what are their mean-squared errors (MSE)? In this paper, a geometrical approach is presented that relates the CM to Wiener (or minimum MSE) receivers. Given the MSE and the intersymbol/user interference of a Wiener receiver, a sufficient condition is given for the existence of a CM local minimum in the neighborhood of the Wiener receiver. The MSE bounds on CM receiver performance are derived and shown to be tight in simulations. The analysis shows that, while in some cases the CM receiver performs almost as well as the (nonblind) Wiener receiver, it is also possible that, due to its blind nature, the CM receiver may perform considerably worse than a (nonblind) Wiener receiver.

On blind beamforming for multiple non-Gaussian signals and the constant-modulus algorithm

A new method for blindly separating multiple cochannel non-Gaussian signals received by a sensor array is presented. The method is based on a cumulant-based least-squares criterion that, for identically distributed negative-kurtosis signals, is proven to be identical to the 2-2 constant-modulus (CM) cost function commonly used by CM algorithms. A computationally simple algorithm is proposed to minimize the criterion. The algorithm performs well even when the number of samples is small, thus allowing its application in dynamic environments (e.g., moving emitters). For the special case of two signals only, the minimization is obtained analytically. Simulation results are included.