Automatic phase recovery properties of adaptive blind equalization algorithms

Several algorithms were developed in the past for blind equalization. The two most common ones are the constant modulus algorithm (CMA) and the reduced constellation algorithm (RCA). RCA is simple to implement but is not very reliable. CMA is very reliable but has a high complexity. In this thesis, an improved blind algorithm, called multimodulus algorithm (MMA), is proposed. This new algorithm is more reliable than RCA and less complex than CMA. MMA makes better usage of the knowledge of the statistics of the data signal and employs multiple moduli to achieve initial convergence. The use of multiple moduli makes MMA very flexible and easily adaptable to applications using two-dimensional transmission schemes with nonsquare or very dense signal constellations. Most blind equalization algorithms minimize a cost function. Our investigation has shown that the minimum (residual) value of this cost function has a great influence on the reliability and speed of convergence of the corresponding blind algorithm. For example, for CMA, RCA and MMA, this residual value increases when the bandwidth efficiency of the transmission system increases, and the blind algorithms suffer a corresponding decrease in performance. Two generalized versions of MMA are proposed to circumvent the above problem. One is called generalized MMA (GMMA). The basic version of MMA uses multiple moduli primarily for non-square constellations when nonuniform symbol distributions are utilized. Multiple moduli are also used with GMMA but the additional purpose there is to decrease the residual value of the cost function. A similar effect is achieved by using another blind algorithm, called windowed MMA (WMMA). With WMMA, only one modulus is used, but the tap coefficients of the filter are only updated with some of the data, which leads to a reduction of the residual value of the cost function. The new MMA algorithm has been extensively tested with computer simulations. Most simulation results presented here were obtained with the 155 Mb/s 64-CAP transceiver specified in the ATM LAN standard for category 3 unshielded-twisted-pair office wiring. However, our results are equally applicable to other applications, such as the 52 Mb/s 16-CAP transceiver defined in DAVIC's specification for fiber-to-the-curb networks. Some preliminary experimental results obtained in the laboratory with such a 16-CAP transceiver are presented in this thesis.

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.

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.

The Concept of Adaptive Equalization is to counter Intersymbol Interference (ISI) which severely degrades the performance of a Communication System. Adaptive Equalization can be classified into Supervised/Trained Equalization and Unsupervised/Blind Equalization. Supervised Equalization is carried out with the aid of a training sequence and Blind Equalization is done without the aid of a training sequence. Godard's Constant Modulus Algorithm (CMA) was the very first blind equalization algorithm that could be used in two dimensional digital communication systems. It requires the knowledge of statistical properties of the transmitted data symbols to perform channel equalization. Picchi and Prati by modifying the standard Decision-Directed Algorithm proposed a new Blind Equalization Algorithm known as "Stop-and-Go" Decision-Directed Algorithm. This paper compares the performance of Blind Equalization algorithms: CMA and Stop-and-Go. Simulation results indicate faster convergence rate of Stop-and-Go than CMA for 16 and 64 Quadrature Amplitude Modulated (QAM) transmitted constellations. The results also indicate negligible difference between the MSE's of Stop-and-Go after convergence and supervized equalizer.

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).

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.

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.

In this paper, a new blind equalization algorithm is proposed based on guard interval (GI)-restoral property for quadrature amplitude modulation (QAM) signals. The proposed algorithm utilizes existing redundancy in the time domain to completely eliminates inter-symbol-interference. We prove that the usage of GI-restoral property is sufficient to obtain a zero-forcing equalizer of the channel. The main advantage of this approach resides in its simplicity as the equalizer coefficients are estimated using a second-order least square fitting (LSF) scheme. The equalizer uses the LSF in the initial phase and then switches to a proposed constant modulus algorithm (CMA). The proposed CMA is constructed through the exploitation of a fully data information. To achieve a dramatic improvement in equalization performance, a modified CMA (MCMA) is proposed for OFDM systems based on cyclic prefixing approach. The resulting scheme provides a state-of-the-art low complexity with superior performance blind equalization. Extensive simulation shows that LSF-MCMA has a better steady-state equalization performance than LSF-CMA.

We investigate the application of blind adaptive equalizers in GSM wireless communication systems. Based on a single-input two-output linear approximated model, a diversity based constant modulus algorithm (CMA) and super-exponential algorithm (SEA) adaptive equalizers are implemented to GSM systems without additional antennas. These diversity based CMA and SEA equalizers are robust to channel order mismatch that cannot be overcome by SOS blind channel algorithms. They are also applicable to time-variant mobile channels due to utilization of statistical information from the entire burst. The condition for the adaptive equalizers to converge is given. All simulation results are based on a more practical wireless channel model that is quasi-time-variant (piecewise time-invariant).

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.

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.

We propose a polarization dependent loss (PDL) and chromatic dispersion (CD) insensitive, low-complexity adaptive equalizer (AEQ) for short-reach coherent optical transmission systems. The AEQ contains a 1-tap butterfly finite impulse response (FIR) filter and two N-tap FIR filters. It first performs polarization demultiplexing using the 1-tap filter, of which the coefficients are obtained based on Stokes space. Then it mitigates the inter-symbol interference (ISI) using the two N-tap finite impulse response (FIR) filters and adjust the filter's coefficients by utilizing constant modulus algorithm (CMA). Through theoretical and experimental analysis, we verify that this proposed AEQ can perform robust polarization demultiplexing when PDL and CD exists. Besides, our proposed AEQ has faster convergence speed compared with recently proposed AEQs. In addition, it reduces the number of multipliers and thus reduce the computational complexity of conventional butterfly filter structure AEQ. And this proposed AEQ suffers little bit error ratio loss compared with the conventional AEQ. Due to the low-complexity and robustness to PDL and CD, this proposed AEQ is well-suited for future low-cost short-reach optical communication system.

The Godard (1980) or constant modulus algorithm (CMA) equalizer is perhaps the best known and the most popular scheme for blind adaptive channel equalization. Most published works on blind equalization convergence analysis are confined to T-spaced equalizers with real-valued inputs. The common belief is that analysis of fractionally spaced equalizers (FSEss) with complex inputs is a straightforward extension with similar results. This belief is, in fact, untrue. We present a convergence analysis of Godard/CMA FSEs that proves the important advantages provided by the FSE structure. We show that an FSE allows the exploitation of the channel diversity that supports two important conclusions of great practical significance: (1) a finite-length channel satisfying a length-and-zero condition allows Godard/CMA FSE to be globally convergent, and (2) the linear FSE filter length need not be longer than the channel delay spread. Computer simulation demonstrates the performance improvement provided by the adaptive Godard FSE.

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

Adaptive blind equalization has gained widespread use in communication receivers that operate without training signals. In particular, the constant modulus algorithm (CMA) has become a favourite of practitioners due to its LMS-like complexity and desirable robustness properties. The desire for further reduction in computational complexity has motivated signed-error versions of CMA (SE-CMA) which have been found to lack the robustness properties of CMA. Alternatively a dithered signed-error of CMA (DSE-CMA) based on the judicious use of dither, has been proposed which results in algorithm with robustness properties nearly identical to those of CMA with the cost of the degradation in steady-state mean square error (MSE) performance. This paper presents a modified dithered signed-error version of CMA (MDSE-CMA) which is motivated by a reduction of the steady-state MSE of DSE-CMA without increase in its computational complexity.