P. Bert Schniter - Cornell University Blind Equalization Research Group

The behavior of the Constant Modulus Algorithm (CMA) as used to adapt a Fractionally Spaced Equalizer (FSE) is illustrated below using simple examples. Some important conclusions are then drawn from these examples: (1) in the presence of noise or channel undermodelling, proper initialization may be necessary to sucessfully equalize the channel, and (2) channels containing nearly reflected roots exhibit high levels of noise gain.


In digital communications systems, the path from source to receiver (i.e. the channel) generally introduces dispersion and multipath, making the received pulses difficult or impossible to detect without a high probability of error. The use of equalization in the receiver attempts to combat these linear distortions imposed by the channel. As a linear equalizer filtering the received signal, usually a transversal FIR implementation is chosen. When sampling the received signal at the system baud rate, we refer to the equalizer as baud spaced. When operating at a sample rate higher than baud (a typical choice being twice the baud rate), we refer to the equalizer as fractionally spaced. The use of FSEs is nearly ubiquitous in recent designs due to advantages in performance over baud spaced equalizers. It can be shown [2] that perfect equalization is possible with FSEs longer than the channel delay spread for channels not exhibiting reflected roots.

In addition to dispersion and multipath, the channel can introduce noise. In general, the equalizer must make a compromise between compensating for linear channel distortions and attenuating noise. In the presence of noise, the ability to perfectly equalize is lost.

In many cases, the receiver does not know the channel characteristics (possibly because of a newly initiated connection or because of a rapidly varying channel). To remedy this, it may be possible to transmit a known training signal from which the receiver can extract the current channel characteristics. In some situations, however, the amount of channel capacity used by this scheme cannot be tolerated, and the receiver must instead determine the channel characteristics blindly. CMA is the most widely established algorithm for accomplishing blind equalization.


Similar to LMS, CMA can be considered a stochastic gradient algorithm. This means that the convergence of the equalizer parameter vector under CMA can be viewed as a (stochastic) traversal of the CMA cost surface with average movement in the direction of steepest descent. Thus, great insights into the dynamical behavior of CMA can be gained from an understanding of the topology of the CMA cost surface.

What follows are simple two-dimensional examples that help to illustrate the salient characteristics of the CMA cost surface in the case of fractionally spaced equalization. Restricting the scope to two real-valued equalizer taps makes the surface possible to visualize as a 3D object and allows us to identify important features which also exist in the general (i.e. higher dimensional) case.

Illustrations of the CMA-FSE Cost Surface

In one regard, the goal of CMA is to transfer a set of initial equalizer parameters to the lowest point on the CMA cost surface. As evident below, there may be multiple locations on the cost surface attaining the minimum cost. As such, they represent equally desirable solutions.

All of the contour plots exhibit the following properties: at the origin there exists a maximum, surrounded by a ring of up to four minima. Minima reflected across the origin correspond to solutions differing only in sign, which is inconsequential when using differential coding. Finally, as we travel further from the origin, the surface rises steeply in all directions.

It should be noted that, though the following surfaces were created from particular channels, they are intended to be prototypical of their respective channel classes. For simplicity, BPSK was chosen as the model for the source statistics, though the thrust of the conclusions below applies to higher-order constellations (e.g. 64-QAM) as well.

Perfect Equalizability with Well-Behaved Channel

In the case of perfect equalizability, all four minima correspond to the same (optimal) CMA cost of zero. Each minima reflects a particular combination of system delay and system polarity. The asterisk indicates the location of a particular Wiener solution. Since there is no noise, it appears coincident with the CMA cost minimum of corresponding system delay and polarity.

Perfect Equalizability with Nearly-Reflected Channel Roots

As channel roots become more nearly reflected, the surface becomes more elongated. Note the scaling on the vertical axis is 100 times that on the horizontal axis! This characteristic will be manifested as an extremely slow convergence of the 'f1' equalizer tap due to the shallowness of the gradient along that dimension. Theoretically, however, perfect equalizability is still possible in the noiseless case. In other words, all minima correspond to zero cost and are aligned with the respective Wiener solutions.

Noisy but Well-Behaved Channel

When the well-behaved channel (i.e. the first plot above) is operated in a 7dB SNR environment, note the appearance of local minima distinct from the global minima. For particular initializations, local minima are capable of capturing the parameter trajectory and preventing it from attaining an optimum setting. Note that all minima correspond to costs greater than zero since perfect equalization is not possible in the presence of noise.

Noisy Channel with Nearly-Reflected Roots

When the channel with nearly reflected roots (above) is operated in a 7dB SNR environment, we see a dramatic change in the cost surface. Note the equivalent scaling of horizontal and vertical axes; the cost surface has become much less elongated than in the noiseless case (see the second picture in the series). Also note the merging of two pairs of minima into one pair. For this channel, the optimal solution in the presence of noise is very different from the solution which perfectly equalizes the channel.

Undermodelled but Well-Behaved Channel, No Noise

When the delay spread of the channel is longer than the delay spread of the FSE, we lose the ability to perfectly equalize. Thus, none of the minima correspond to zero cost. More importantly, note the existence of local minima with greater CMA cost than the global minima. This implies that the equalizer initialization will determine the optimality of the resulting CMA solution.


The examples above emphasize a few facts. First, the FSE-CMA cost surface is multimodal, by virtue of the existence of multiple minima. In the case where an FSE is able to perfectly equalize the channel, all minima correspond to the same optimal CMA cost. Furthermore, it can be shown that these multiple minima occur very near to the Wiener solutions corresponding to particular combinations of system polarity and system delay [3]. By system delay, we mean the delay through the cascaded channel and equalizer. By sytem polarity, we mean {+,-} in the case of real signals or {+,-,+j,-j} in the case of complex signals.

Next, channels with nearly-reflected roots will most likely exhibit very slow FSE parameter convergence rates in the absence of noise. Adding the effects of noise, the FSE may not even come close to equalizing the channel. Well behaved channels are not as sensitive to noise as those with nearly reflected roots.

Any time noise is present, though, we can expect local minima. This implies the potential for convergence to non-optimal CMA solutions and places importance on the choice of initialization. The same holds true for FSEs that undermodel the channel, even for well-behaved channels in the absence of noise. Initialization can be extremely important if, in these cases, there exist bad local minima which do not correspond to open eye solutions. By open eye, we mean that the channel has been equalized enough to successfully transfer adaptation to a decision- directed equalization scheme. If CMA converges to one of these bad local minima, there is little chance of successfully demodulating the incoming data. In such cases, the system must be restarted, possibly with a different initialization, in the hope that CMA will converge to an open-eye solution.

Questions for Practitioners

The examples presented in this demonstration help to motivate the search for techniques that avoid the problems associated with CMA's sensitivity to poor choices of initialization. In practice, it is typical to use a center-spike initialization, whereby the center equalizer tap is made unity and all other taps are made zero. Though originally suggested in [1], we know of no theoretical work that has rigorously defended this technique.

The question remains as to how well the center-spike initialization really works. We have received reports of the center-spike method failing on occasion, but have not received any specific examples. Can anyone provide us with some?


[1] D.N. Godard, "Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems," IEEE Trans. Commun., Vol. COM-28, No.11, Nov. 1980.

[2] J.R. Treichler, I. Fijalkow, and C.R. Johnson, Jr., "Fractionally Spaced Equalizers - How long should they really be?," Signal Processing, pp.65-81, May 1996.

[3] H.H. Zeng, L. Tong, and C.R. Johnson, Jr., "Relationships Between CMA and Wiener Receivers," Submitted to IEEE Trans. on Inform. Theory.

Further Reading

The examples and discussion above are drawn from a technical report written by members of Cornell's Blind Equalization Research Group (BERG). The report includes a derivation of the equations respresenting CMA-FSE cost with arbitrary source kurtosis (i.e. for any constellation) and in the presence of additive white gaussian channel noise.

Matlab Code

The BERGulator can be used to generate these surfaces and others.