A rectangular-constellation-based blind-equalization (RECBEQ) technique implemented by a real-time, recursive algorithm has been developed to improve the performances of radio receivers in recovering unknown signals that are modulated with digital information and that have been distorted in propagation by multipath channels and carrier offsets. The technique is so named because it is intended specifically to enable the equalization of large-order rectangular signal constellations; for example, that of quadrature amplitude modulation (QAM).

Blind equalization provides for the recovery of unknown signals via a finite-dimensional linear projection of a channel output data vector; namely,

where *z _{n}* is the complex output sample from the blind equalizer at the

*n*th sampling interval,

*w*is the

_{i}*i*th of

*L*blind-equalizer coefficients, and

*y*is the

_{j}*j*th complex sample from the unknown channel. The latter sample can be expressed as a convolution of the sampled channel impulse response

*f*with an unknown sequence of independent and identically distributed source symbols

_{k}*a*; that is,

_{l}This equalization process is said to be "blind" because the *w _{i}*s are derived from available channel output data only, without knowledge of either the transmitted signal waveform or the linear channel.

The present blind-equalization technique belongs to a class of such techniques in which the *w _{i}*s are chosen to maximize or minimize objective functions (e.g., cost functions). The objective function for this technique is derived from a uniformly most powerful (UMP) scale-invariant hypothesis test between factored (rectangular) generalized Gaussian distributions. The net result of the derivation is the following time-recursive equation for updating the blind-equalizer coefficients:

where β_{rect} is a positive "step size" which controls the rate of adaptation (smaller values of _{βrect }result in lower adaptation rates); the subscripts *x *and *y* denote the real and imaginary parts, respectively, of the affected quantities; *s* is a positive constant, greater than 2, which helps determine the steady-state performance of the equalizer (larger values of *s* yield lower steady-state adaptation noise but result in greater implementation complexity and greater sensitivity to additive receiver noise - it has been observed that *s* = 8 provides a good tradeoff between algorithm adaptation noise in steady state and sensitivity to additive receiver noise); *R*_{Orect }is a positive constant which controls the scale of the equalized constellation at convergence and is given by:

and *E*( )is the expectation operator. The equation for *w _{k}*(

*n*+ 1) converges rapidly for input rectangular constellations distorted by multipath.

When residual carrier offsets are present, the receiver must also include a data-directed phase-locked loop (PLL). Part of the figure depicts a conventional receiver architecture that incorporates a data PLL along with an older blind equalizer of a type called "CMA" (constant-modulus algorithm). The equalizer output is phase-corrected ("derotated") by the PLL output, which is driven by symbol decisions based on the phase-corrected equalizer output. This architecture is viable because the CMA is not affected by phase rotations of the input signal constellation, and therefore phase correction can occur downstream from the CMA equalizer.

Unlike the CMA, the RECBEQ algorithm is sensitive to the phase orientation. Extensive tests have revealed that the RECBEQ algorithm can acquire a rotating rectangular constellation, but not at the same level of precision that would be achieved if the constellation were static. This finding led to the development of the modified architecture, also shown in the figure. Here, the input to the RECBEQ equalizer is phase-corrected by the PLL output, which again, is driven by symbol decisions based directly on the equalizer output. In this architecture, the RECBEQ initially equalizes the rotating constellation to such an extent that the PLL can lock up and finalize the joint equalization/carrier-recovery process.

Results of computational tests show that for rectangular constellations, the RECBEQ algorithm converges much more quickly, to lower-noise solutions, than does the CMA. For other, more-rounded constellations [e.g., those of *M*-ary-phase-shift keying (MPSK)], the CMA performs better. Computationally, both algorithms are comparable.

*This work was done by Edgar Satorius of Caltech and James Mulligan of TASC for *NASA's Jet Propulsion Laboratory*. **NPO-20324 *

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This article first appeared in the March, 1999 issue of *NASA Tech Briefs* Magazine.

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