Modern communications systems employ bandwidth-efficient modulation techniques, such as quadrature amplitude modulated (QAM). With QAM, the amplitude and phase of a sinusoidal signal are both varied to transmit digital information. Most transmission mediums introduce distortion in the transmitted signal in amplitude as well as phase. This distortion is sometimes referred to as intersymbol interference (ISI) because weighted contributions of neighboring symbols are added to the current symbol.

To combat ISI, both wireless and wireline modems employ various equalization techniques. In principle, these techniques attempt to recover the original transmitted signal. The two most widely used techniques in practice are forward error correction (FEC) and channel equalization. For the latter, a training sequence is generally transmitted to help the adaptive equalizer extract channel information. The problem with this approach is that the training sequence consumes bandwidth. To remedy this problem, blind equalization algorithms have been proposed. In these algorithms, channel information is extracted from information data only.

The Constant Modulus Algorithm (CMA) is a member of this family. This algorithm has gained popularity in QAM-based wireless modems due to its insensitivity to carrier frequency offset. During acquisition phase, such insensitivity is required from an equalization algorithm. Yet the commercial use of CMA has been limited by its slow convergence rate. As it turns out, this slow convergence rate is an inherent problem in all blind equalization algorithms.

This article proposes a modified version of CMA in which the "Constant Modulus" is replaced by an "Adaptive Modulus." Simulation results show that this simple modification to CMA speeds up its convergence rate by an order of magnitude when channel distortion is moderate. For a heavily distorted received signal, a hybrid approach yields better results than those that are obtained when CMA is used alone. In this approach, CMA is used at startup and then it switches to AMA. Simulation results are now proving the validity of this new approach.

CMA ALGORITHM
The Constant Modulus Algorithm is a very popular adaptive-equalization algorithm. It is employed in modems to correct for the distortion introduced by the channel. CMA belongs to a special family of adaptive equalizers. In this family, the equalizer taps are updated "blindly" (i.e., without the transmission of a training sequence). This approach saves bandwidth.

CMA's popularity, especially in wireless applications, is due to two facts. First of all, it is able to extract channel information in the presence of carrier frequency offset. This feature is especially desirable at startup phase. Secondly, its LMS-like update equation makes CMA very easy to analyze and implement in power-limited applications. Unfortunately, these nice features do not come without a price. For example, CMA has a slow convergence rate. This issue has limited its use in some commercial applications, which need a rapid convergence rate.

To gain a better understanding of how CMA works, take a close look at the error-calculation and filter-taps update equations:

Here, x(k) = [x(k) x(k−1)...x(k − M + 1)]^T is a length-M vector of input samples to the equalizer-tapped delay line. The length-M equalizer taps is h(k). The adaptation step size is µ. The equalized output signal is y(k). Conjugation is denoted by (.)*. It should be mentioned that when equalization takes place in the baseband, all of the above-mentioned signals are complex. The value of the constant parameter β², called the dispersion factor, depends on the modulation scheme being used. It is equal to E{|a_i|⁴}/E{|a_i|²}, where E {.} denotes expectation. The a_i (i = 1,2...N) represents the QAM constellation points. For 16QAM (N = 16), where the in-phase and quadrature components are derived from the alphabet α = [± 1, ± 3], we have β² = 13.2.

Examining the 16QAM constellation in Figure 1, one can derive a rather useful interpretation of this parameter. CMA attempts to fit the output of the equalizer to a circle of radius β. This minimization criterion allows CMA to remove ISI. At the same time, it remains insensitive to carrier frequency offset. Unfortunately, this same criterion also slows down the convergence speed of the equalizer taps to their optimum values. In some cases, CMA will never be able to reduce the mean squared error (MSE) to an acceptable level in order for decision-directed equalization techniques to take over. Of course, they would take over after frequency-offset compensation. Better performance can be achieved by adapting the value of β² based on certain decisions.