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 β^{2}, 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 β^{2} = 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 β^{2} based on certain decisions.

AMA ALGORITHM

LMS-based adaptive equalization algorithms are derived by minimizing a certain cost function of the second order. For CMA, this cost function is E\{(β^{2}−|y(k)|^{2})^{2}\}. The choice of this cost function gives CMA its most important feature: insensitivity to phase rotations. Unfortunately, this same choice also degrades the algorithm's performance. To see why this happens, observe Figure 1 in more detail. This cost function attempts to minimize a squared-error expression. Yet this expression will never vanish, even if the filter taps miraculously converge to their optimum values. This is because the non-vanishing error term in Equation 2 keeps contributing to updating the filter taps. This is true even when an optimum solution is reached. In other words, as seen in the 16QAM example, the equalizer is not able to output a clean 16QAM constellation.

To remedy this problem without sacrificing CMA's insensitivity to phase rotations, it is proposed that an "adaptive" modulus should be employed instead of a constant one. Referring back to Figure 1, observe that the 16QAM constellation is composed of three sub-constellations: two QPSK and one 8PSK. Each of the inner and outer sub-constellations represents a QPSK constellation with a radius of √2 and √18, respectively. The middle sub-constellation represents 8PSK with a radius of √10. With this stated, one can now make the following statement using the Maximum Likelihood (ML) principle: The output of the equalizer will most likely belong to the closest ring. When using 16QAM modulation, the value of β ^{2} must therefore be adapted according to the following conditions:

- If |y(k)|
^{2}is less than or equal to 6, then β^{2}= 2. - If |y(k)|
^{2}is greater than 6 but less than or equal to 14, then β^{2}= 10. - If |y(k)|
^{2}is greater than 14, then β^{2}= 18.

For an example, refer again to Figure 1. The squared magnitude of y(k), which is located between the inner and middle rings, should be subtracted from 10—not from 13.2 as suggested in CMA. The above stated modifications to CMA clearly ensure that the magnitude of the proposed error signal is smaller than the CMA one. This will help speed up CMA's convergence rate. It also will guarantee the smallest achievable MSE. It should be mentioned that similar conditions could be derived for higher-order QAM constellations, as any QAM constellation can be decomposed into smaller MPSK sub-constellations.

The above modifications to CMA are inspired from decision-directed (DD) equalization algorithms. In general, these algorithms perform better than CMA in the absence of phase rotations. So the newly proposed algorithm combines the advantages of DD algorithms and CMA. Unfortunately, as with DD algorithms, AMA may fail to converge if channel distortion is severe or—more precisely—if p_{e }> 0.01. CMA rarely suffers from this problem, however. In this particular case, it is suggested to combine both CMA and AMA. CMA is used at startup and then followed by AMA when the effect of channel distortion is reduced to an acceptable level.

SIMULATION RESULTS

Simulink from The MathWorks, Inc. is used to compare the performance of AMA to that of CMA. This choice of a simulation tool is motivated by the following reasons:

- Simulink is a system-level tool that allows the user to study the interaction of various components in a straightforward manner. This is a must-have feature if one wants to perform tradeoff analysis on the system level rather than on the component level, as when programming languages are used.
- The DSP and communications libraries within Simulink contain most standard DSP and communications functions and algorithms. Some good examples are the adaptive filters and modulation techniques. They allow the user to build a simulation model very quickly without having to resort to manual coding.
- Simulink interacts seamlessly with MATLAB, a popular tool for scientific computing. As a result, the user can utilize MATLAB's powerful graphical and analytical tools, as well as its scripting capability for comparing different simulation results.

Figure 2 shows the baseband-to-baseband simulation setup using 16QAM as the modulation technique. The random-integer-generator block generates 16 equiprobable symbols at a rate of 6.25 Mbaud (equivalent to 25 Mbps). These symbols are then mapped to 16QAM complex numbers. For bandlimiting the transmitted signal, a square-root raised-cosine 96-tap filter (SRRCF) with 50% excess bandwidth and an 8X oversampling rate is used. Using the Digital Filter Design block in the DSP library, this pulse-shaping filter is designed interactively.

The carrier frequency offset is normally caused by the small mismatch that exists between the transmitter and receiver RF oscillators. To simulate this offset, the transmitted signal is multiplied by a complex sinusoid. The frequency of this sinusoid corresponds to the overall worst-case frequency offset. The magnitude of this frequency offset is directly proportional to the RF carrier and the accuracy of the RF oscillators.

In this case, the given was a 27.4-GHz RF carrier that has been allocated for Local Multipoint Distribution Service (LMDS). Assuming the use of Tx and Rx RF oscillators with 50-ppm accuracy, this offset is equal to 2.74 MHz (equivalent to 2 × 50 × 10^{−6 }× 27.4 × 10^{9}). To simulate the ISI introduced by the channel, a typical LMDS fixed-wireless non-line-of-sight (NLOS) FIR channel model was used (see table). For completeness, an AWGN channel block was added.

The receiver is composed of a complementary SRRCF, a downsampler, and an adaptive equalizer block. Underneath the latter is the LMS Adaptive Filter block found in the DSP library, as well as an error-generator block. This block generates the appropriate error signal depending on the algorithm used. Other graphical blocks are added to the model. They allow the visualization of the adaptive filters taps and the constellation diagram of the equalized signal.

To illustrate the validity of AMA, two computer simulations are performed under different channel conditions. In the first experiment, Channel 1 was used. It introduces relatively moderate channel distortion. The second simulation employed Channel 2, which introduces severe channel distortion. Both cases used 11-tap baud-rate channel equalizers. In addition, it was assumed that the pulse-shaping filters would not introduce any additional channel distortion. The incoming signal is sampled at the optimum sampling point. Furthermore, the MSE is approximated using the following time-based averaging window (see equation 3)

This MSE is further smoothed over 200 independent simulation runs. In both cases, the β used in computing the MSE is the adaptive one. This β better measures the extent to which the output constellation is clean. In both experiments, the equalizer taps are initialized to zero. The exception is the middle one, which is initialized to unity. Noise was not added in either case.

In the first experiment, different step sizes were chosen for CMA and AMA so that they both had the same normalized MSE upon convergence (in this case, −53 dB). This approach allowed a fair comparison of convergence rates. Looking at Figure 3, it can be concluded that in the presence of moderate channel distortion, AMA outperforms CMA with respect to convergence speed. While it takes CMA 5000 iterations to attain −18-dB MSE, for example, AMA reaches that level in less than 300 iterations. At that MSE level, this number represents a sixteenfold convergence speed improvement. Here is an idea of the simulation speed achieved by Simulink in this case: All 200 independent runs take 163s to complete on a PIII 600-MHz computer with 328 M of RAM.

In the second experiment, it was determined that AMA fails to converge to the right solution (results not shown here). The severe distortion introduced by Channel 2 is to blame. This is expected, as AMA makes boundary decisions at the output of the equalizer. In this scenario, most of those decisions are wrong. This issue causes the equalizer taps to diverge from the right solution.

On the contrary, CMA has no convergence problems at all. Compared to the first case, however, it does take longer for the equalizer taps to converge to the right solution. Under severe channel conditions like this, it is suggested that a hybrid approach be used. In this approach, the equalizer uses the CMA error at startup. It can then be switched to the AMA error when MSE reaches an acceptable level.

Figure 4 shows the MSE curves for both CMA and hybrid CMA-AMA algorithms. By switching to AMA (in this case at 5000 iterations), faster convergence speed can clearly be obtained. As an example, the hybrid CMA-AMA MSE curve reaches −37 dB at 5200 iterations. The CMA curve reaches that level after only 10,000 iterations.

In this article, a simple and elegant solution has been proposed to solve the slow convergence problem of CMA. By employing an Adaptive Modulus instead of a Constant one, designers can quicken the convergence speed of CMA. This Adaptive Modulus has been inspired by decision-directed equalization methods. In the absence of phase errors, these equalization methods generally outperform CMA.

Simulation results using 16QAM modulation prove that in the presence of light-to-moderate channel distortions, the new algorithm—known as baptized AMA—converges faster than CMA. Under severe channel conditions, however, AMA might fail to converge to the correct solution. In this case, a hybrid CMA-AMA approach is proposed. CMA can be used at startup until the MSE reaches an acceptable level. Then, the designer can switch to AMA.