Blind Equalization of Cross-QAM Signals - Semantic Scholar
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006
745
Blind Equalization of Cross-QAM Signals Shafayat Abrar, Student Member, IEEE, and Ijaz Mansoor Qureshi
Abstract—This letter presents two new blind equalization algorithms that are specifically designed for cross-QAM signals. Proposed algorithms minimize the dispersion of the equalizer output with respect to single or multiple cross-shaped zero error contour(s). Simulations and analysis demonstrate the good performance of the proposed algorithms. Index Terms—Adaptive equalizers, blind equalization.
I. INTRODUCTION
Fig. 1. Illustration of the cross.
E ADDRESS the problem of blind equalization of linear channels in digital communication systems that employ (odd-bit) cross-quadrature amplitude modulation (QAM). Assuming a time-invariant channel, the channel and the equalizer outputs at time instant are given by and , respectively, where and are the channel and equalizer imand are the channel pulse response vectors, respectively, input and additive noise samples, respectively, and subscripts and denote the in-phase and the quadrature components of the complex entity, respectively. There exist many multimodulus blind equalization algorithms (MMA) [1]–[3] that separately minimize the dispersion of the in-phase and the quadrature levels of received QAM signals. For square-QAM signals, where the in-phase and the quadrature levels are independent of each other, these MMA work reasonably well. However, for cross-QAM signals, where the in-phase and the quadrature levels are not independent of each other, the performances of different MMA are not very impressive. Note that, in a (distortion-free) cross-QAM, if the in-phase level belongs to the set , then the quadrature level must belong to the set and vice versa (where denotes the number of different symbols on QAM constellation). To develop a blind equalization algorithm for cross-QAM, the authors in [4] presented an idea to separately minimize the dispersion of the disjoint sets and . This idea has also been the subject of a U.S. patent [5]. However, the authors did not provide any general expression for the evaluation of dispersion constants and the threshold that was required to determine which dispersion constant to be used. In another attempt, the author of [6] proposed to use a separate dispersion constant for each symbol such that the resulting zero-error contour was -point
cross-shaped. However, the convergence capability of that algorithm was found to be poor for high-order QAM signals.
W
II. PROPOSED ALGORITHMS Recently, three separate groups of researchers independently presented a novel idea for the blind equalization of square-QAM signals. They proposed to minimize the dispersion of the in-phase or the quadrature level, only one at a time, depending on their relative absolute amplitudes [7]–[9, Chap. 15]. This scheme resulted in the minimization of the dispersion w.r.t. a square-shaped zero-error contour, and for this reason, it was termed as the constant square algorithm (CQA) [7]. The idea was realized by minimizing the following cost function:1 CQA
Notice that is an expression of square with radius . If we replace this expression with the expression of cross, then the resulting cost function will be able to yield a cross-shaped zero-error contour over the signal-space. As a result, the equalizer will be able to minimize the dispersion w.r.t. a cross-shaped contour without determining the set or to which symbols belong. Consider the expression of the cross on -plane as follows:
where
, , and (refer to Fig. 1). Using the expression of cross, we can develop the following criterion for the blind equalization of cross-QAM signal: (2) where
Manuscript received March 2, 2006; revised May 17, 2006. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Philip Schniter. S. Abrar is with the Electrical Engineering Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan (e-mail: shafayat@ieee. org; [email protected]). I. M. Qureshi is with the Electrical Engineering Department, Mohammad Ali Jinnah University, Islamabad 44000, Pakistan. Digital Object Identifier 10.1109/LSP.2006.879828
(1)
is given by (3)
The zero-error cross-shaped contour, exhibited by (2) is depicted in Fig. 2(a) for . To obtain a stochastic gradient 1Readers
are referred to [10] for a detail analysis of (1).
1070-9908/$20.00 © 2006 IEEE
746
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006
adaptive algorithm, we drop the expectation in (2) and minimize to obtain the following: it w.r.t (4) where
, and
and
are obtained as follows:
Fig. 2. J and J surfaces over signal-space for 32-QAM.
The algorithm (4) is named constant cross algorithm (CXA). and in (4) are described in The optimum values of Appendices A and B, respectively. Note that CXA minimizes the dispersion of the equalized symbols w.r.t. a cross-shaped contour. However, due to the presence of single contour, CXA results in a significant misadjustment in steady state for large QAM constellations, even if the equalizer has converged successfully with correct orientation. The problem of misadjustment in Bussgang-type algorithms due to the single contour can be minimized by introducing multiple contours. This idea has recently been discussed in the scenario of a multimodulus algorithm [11]. According to this idea, increased number of contours can be obtained by the joint use of dispersion constant and the sliced symbol (where is the decision made on ). The idea was realized by replacing the dispersion constant with , where is some suitable positive-valued even-symmetrical function. As a result, the values of the resulting contours become proportional to the magnitude of . So the dispersion in the small and large values of are, respectively, minimized w.r.t. small- and large-valued contours. To introduce multiple contours in CXA, we use in the CXA cost function (2) as follows:2 (5) where
is given by
The adaptive algorithm for (6) is given by (7) The algorithm (7) is named sliced constant cross algorithm (SCXA). The mesh and the contour plots of (5) are depicted in
j 0 j jj j 0j
2If J = (1=pq ) [ f (y ) R ] is a cost function for some Bussgang-type blind equalization algorithm, then its (heuristic) sliced version can be = (1=pq ) [ f (y ) f (^ a) R ], where r s = p r . obtained as J If p is an even number, then r and s can be selected as r = s = p=2 to ease the computational cost. Moreover, based on an idea which is recently presented in a U.S. patent [12], it is deduced that the slicing can also be obtained = (1=pq ) [ f (^ a) f (y ) R ]. as J
j
Fig. 2(b). The optimum values of and in Appendices A and B, respectively.
in (7) are described
III. SIMULATION RESULTS (6)
jj
Fig. 3. ISI traces for 32-QAM signaling.
j 1 jj
j
j 0 j
j
0
The CXA and SCXA were applied in a simulation of four outdoor wireless channel models. The microwave channel models “chan1,” “chan2,” “chan4,” and “chan5” in the Signal Processing Information Base [13] were used with a -spaced 40-tap FIR equalizer. The residual ISI traces (as defined in [14]) were measured and compared as a performance parameter for CXA, SCXA, and the traditional constant modulus algorithm (CMA) on 32- and 128-QAM signaling for SNR dB. Each of the ISI traces was obtained by averaging 50 Monte Carlo experiments. Single-spike initialization was considered in all cases such that the tap-20 was initialized as one and the rest were set to zero. The simulation results are summarized in Figs. 3 and 4. Notice that, irrespective of channels and QAM sizes, both CXA and SCXA are outperforming CMA by offering much
ABRAR AND QURESHI: BLIND EQUALIZATION OF CROSS-QAM SIGNALS
747
Fig. 6. Theoretical and experimental EMSE for CXA and SCXA.
TABLE I OPTIMUM VALUES OF k AND R FOR SOME RROSS-QAM
Fig. 4. ISI traces for 128-QAM signaling.
APPENDIX A OPTIMUM VALUES OF FOR CXA AND SCXA We borrow from Goupil et al. [7] that the dispersion constant must be selected in such a way that the perfect equalizer (in the sense of the zero-forcing criterion) is the minimum of the in a noiseless environment. The condition is cost criterion expressed by the relation [7]
(8)
Fig. 5. Cost for CXA and SCXA under phase-offset. The jagged appearance of the cost for SCXA is the consequence of the use of sliced symbols within its computation.
faster convergence speed. In addition to fast ISI mitigation capability, CXA and SCXA are also found capable of recovering the correct orientation of QAM constellation. The effect of phaseoffset on CXA and SCXA cost functions is depicted in Fig. 5 for a cross-QAM, where SCXA can be noticed to be more sensitive to the phase-offset than CXA. We also compared the experimental excess mean square error (EMSE) and the theoretical EMSE for the CXA and SCXA as the function of step-size in a noise-free environment and ideally equalized condition (refer to Appendix B for EMSE expressions). The experimental values were generated as averages over ten trials for 32-QAM using a 20-tap -spaced equalizer (with single-spike initialization). As depicted in Fig. 6, the result demonstrates that the theoretical values predicted by expression (14) and (16) match the experimental results reasonably well.
where is the coefficient of the one-tap equalizer. It is described in [7] that for a perfect channel, the coefficient should converge to 1. In other words, the equalizer must at least be able to recover the power of the signal [7]. For CXA, we formulate an optimization problem: find such that . For , we have , where . We find . Solving , we get
(9) For SCXA, we formulate: find
such that . Dif-
ferentiating w.r.t. , we have . Solving find
and substituting , leading to
, we (10)
The values of
(9), (10) are listed in Table I.
748
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006
leading to (14) where
,
, optimum value of
,
, and for CXA is obtained as follows:
. The
(15)
CXA
Fig. 7. Area of regions II and IV are 0:5R (1 while the total area is R .
0
For SCXA, we find and Next, we obtain k)
and 0:5R
k
, respectively,
APPENDIX B OPTIMUM VALUES OF FOR CXA AND SCXA We assume that the value of is optimum if it minimizes EMSE. The EMSE for a Bussgang-type algorithm with cost function can be expressed as follows [15]:
(11)
. . We substitute the values of
and
in (11) to get
(16) where
,
, optimum value of
, and for SCXA is obtained as follows:
. The
(17)
SCXA The values of
,
(15), (17) are listed in Table I.
,
where , and For CXA, with
. ,
and . After
some steps, we obtain . Taking a further derivative w.r.t.
, we obtain
the values of
and
. Now, substituting in (11), we obtain
(12) where (refer to Fig. 7) (Reg. ) (Reg. ) (Reg. ) (Reg. ).
(13)
Now we make use of the QAM orientation symmetries, which for any one of the in-phase or quadraenable us to evaluate ture components in one of the quadrants. We select the regions II and IV in the first quadrant to evaluate . The value of is equal to 4 and in regions II and IV, respectively. Using the relative area of different regions, we evaluate the expectation, as given by
Similarly, we compute the expectation in the numerator of (12),
REFERENCES [1] K. Wesolowski, “Self-recovering adaptive equalization algorithms for digital radio and voiceband data modems,” in Proc. Eur. Conf. Circuit Theory Design, 1987, pp. 19–24. [2] K. N. Oh and Y. O. Chin, “Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm,” in Proc. IEEE ICC, Jun. 1995, vol. 1, pp. 498–502. [3] S. Abrar, A. Zerguine, and M. Deriche, “Soft constraint satisfaction multimodulus blind equalization algorithms,” IEEE Signal Process. Lett., vol. 12, no. 9, pp. 637–640, Sep. 2005. [4] J. Yang, J.-J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 997–1015, Jun. 2002. [5] J.-J. Werner and J. Yang, “Technique for Improving the Blind Convergence of an Adaptive Equalizer Using a Transition Algorithm,” U.S. Patent 5809074, Sep. 1998. [6] S. Abrar, “Compact constellation algorithm for blind equalization of QAM signals,” in Proc. IEEE INCC, Jun. 2004, pp. 170–174. [7] A. Goupil and J. Palicot, “Constant norm algorithms class,” in Proc. EUSIPCO, 2002. [8] T. Thaiupathump, “New algorithms for blind equalization and blind source separation/phase recovery,” Ph.D. dissertation, Univ. Pennsylvania, Philadelphia, PA, 2002. [9] N. Benvenuto and U. Cherubini, Algorithms for Communications Systems and Their Applications. New York: Wiley, 2002. [10] T. Thaiupathump, L. He, and S. A. Kassam, “Square contour algorithm for blind equalization of QAM signals,” Signal Process. (Elsevier), February 16, 2006, accepted (galley proof is available online). [11] S. Abrar and R. A. Axford, Jr., “Sliced multi-modulus blind equalization algorithm,” ETRI J., vol. 27, no. 3, pp. 257–266, Jun. 2005. [12] J.-J. Werner and J. Yang, “Blind Equalization Algorithm With the Joint Use of the Constant R and the Sliced Symbols,” U.S. Patent 6 493 381, Dec. 2002. [13] [Online]. Available: http://spib.rice.edu/spib/microwave.html. Signal Processing Information Base (SPIB), Rice Univ. [14] O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of nonminimum phase systems (channels),” IEEE Trans. Inf. Theory, vol. 36, no. 2, pp. 312–321, Mar. 1990. [15] A. Goupil and J. Palicot, “A geometrical derivation of the excess mean square error for Bussgang algorithms in a noiseless environment,” Signal Process., vol. 84, pp. 311–315, 2004.
745
Blind Equalization of Cross-QAM Signals Shafayat Abrar, Student Member, IEEE, and Ijaz Mansoor Qureshi
Abstract—This letter presents two new blind equalization algorithms that are specifically designed for cross-QAM signals. Proposed algorithms minimize the dispersion of the equalizer output with respect to single or multiple cross-shaped zero error contour(s). Simulations and analysis demonstrate the good performance of the proposed algorithms. Index Terms—Adaptive equalizers, blind equalization.
I. INTRODUCTION
Fig. 1. Illustration of the cross.
E ADDRESS the problem of blind equalization of linear channels in digital communication systems that employ (odd-bit) cross-quadrature amplitude modulation (QAM). Assuming a time-invariant channel, the channel and the equalizer outputs at time instant are given by and , respectively, where and are the channel and equalizer imand are the channel pulse response vectors, respectively, input and additive noise samples, respectively, and subscripts and denote the in-phase and the quadrature components of the complex entity, respectively. There exist many multimodulus blind equalization algorithms (MMA) [1]–[3] that separately minimize the dispersion of the in-phase and the quadrature levels of received QAM signals. For square-QAM signals, where the in-phase and the quadrature levels are independent of each other, these MMA work reasonably well. However, for cross-QAM signals, where the in-phase and the quadrature levels are not independent of each other, the performances of different MMA are not very impressive. Note that, in a (distortion-free) cross-QAM, if the in-phase level belongs to the set , then the quadrature level must belong to the set and vice versa (where denotes the number of different symbols on QAM constellation). To develop a blind equalization algorithm for cross-QAM, the authors in [4] presented an idea to separately minimize the dispersion of the disjoint sets and . This idea has also been the subject of a U.S. patent [5]. However, the authors did not provide any general expression for the evaluation of dispersion constants and the threshold that was required to determine which dispersion constant to be used. In another attempt, the author of [6] proposed to use a separate dispersion constant for each symbol such that the resulting zero-error contour was -point
cross-shaped. However, the convergence capability of that algorithm was found to be poor for high-order QAM signals.
W
II. PROPOSED ALGORITHMS Recently, three separate groups of researchers independently presented a novel idea for the blind equalization of square-QAM signals. They proposed to minimize the dispersion of the in-phase or the quadrature level, only one at a time, depending on their relative absolute amplitudes [7]–[9, Chap. 15]. This scheme resulted in the minimization of the dispersion w.r.t. a square-shaped zero-error contour, and for this reason, it was termed as the constant square algorithm (CQA) [7]. The idea was realized by minimizing the following cost function:1 CQA
Notice that is an expression of square with radius . If we replace this expression with the expression of cross, then the resulting cost function will be able to yield a cross-shaped zero-error contour over the signal-space. As a result, the equalizer will be able to minimize the dispersion w.r.t. a cross-shaped contour without determining the set or to which symbols belong. Consider the expression of the cross on -plane as follows:
where
, , and (refer to Fig. 1). Using the expression of cross, we can develop the following criterion for the blind equalization of cross-QAM signal: (2) where
Manuscript received March 2, 2006; revised May 17, 2006. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Philip Schniter. S. Abrar is with the Electrical Engineering Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan (e-mail: shafayat@ieee. org; [email protected]). I. M. Qureshi is with the Electrical Engineering Department, Mohammad Ali Jinnah University, Islamabad 44000, Pakistan. Digital Object Identifier 10.1109/LSP.2006.879828
(1)
is given by (3)
The zero-error cross-shaped contour, exhibited by (2) is depicted in Fig. 2(a) for . To obtain a stochastic gradient 1Readers
are referred to [10] for a detail analysis of (1).
1070-9908/$20.00 © 2006 IEEE
746
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006
adaptive algorithm, we drop the expectation in (2) and minimize to obtain the following: it w.r.t (4) where
, and
and
are obtained as follows:
Fig. 2. J and J surfaces over signal-space for 32-QAM.
The algorithm (4) is named constant cross algorithm (CXA). and in (4) are described in The optimum values of Appendices A and B, respectively. Note that CXA minimizes the dispersion of the equalized symbols w.r.t. a cross-shaped contour. However, due to the presence of single contour, CXA results in a significant misadjustment in steady state for large QAM constellations, even if the equalizer has converged successfully with correct orientation. The problem of misadjustment in Bussgang-type algorithms due to the single contour can be minimized by introducing multiple contours. This idea has recently been discussed in the scenario of a multimodulus algorithm [11]. According to this idea, increased number of contours can be obtained by the joint use of dispersion constant and the sliced symbol (where is the decision made on ). The idea was realized by replacing the dispersion constant with , where is some suitable positive-valued even-symmetrical function. As a result, the values of the resulting contours become proportional to the magnitude of . So the dispersion in the small and large values of are, respectively, minimized w.r.t. small- and large-valued contours. To introduce multiple contours in CXA, we use in the CXA cost function (2) as follows:2 (5) where
is given by
The adaptive algorithm for (6) is given by (7) The algorithm (7) is named sliced constant cross algorithm (SCXA). The mesh and the contour plots of (5) are depicted in
j 0 j jj j 0j
2If J = (1=pq ) [ f (y ) R ] is a cost function for some Bussgang-type blind equalization algorithm, then its (heuristic) sliced version can be = (1=pq ) [ f (y ) f (^ a) R ], where r s = p r . obtained as J If p is an even number, then r and s can be selected as r = s = p=2 to ease the computational cost. Moreover, based on an idea which is recently presented in a U.S. patent [12], it is deduced that the slicing can also be obtained = (1=pq ) [ f (^ a) f (y ) R ]. as J
j
Fig. 2(b). The optimum values of and in Appendices A and B, respectively.
in (7) are described
III. SIMULATION RESULTS (6)
jj
Fig. 3. ISI traces for 32-QAM signaling.
j 1 jj
j
j 0 j
j
0
The CXA and SCXA were applied in a simulation of four outdoor wireless channel models. The microwave channel models “chan1,” “chan2,” “chan4,” and “chan5” in the Signal Processing Information Base [13] were used with a -spaced 40-tap FIR equalizer. The residual ISI traces (as defined in [14]) were measured and compared as a performance parameter for CXA, SCXA, and the traditional constant modulus algorithm (CMA) on 32- and 128-QAM signaling for SNR dB. Each of the ISI traces was obtained by averaging 50 Monte Carlo experiments. Single-spike initialization was considered in all cases such that the tap-20 was initialized as one and the rest were set to zero. The simulation results are summarized in Figs. 3 and 4. Notice that, irrespective of channels and QAM sizes, both CXA and SCXA are outperforming CMA by offering much
ABRAR AND QURESHI: BLIND EQUALIZATION OF CROSS-QAM SIGNALS
747
Fig. 6. Theoretical and experimental EMSE for CXA and SCXA.
TABLE I OPTIMUM VALUES OF k AND R FOR SOME RROSS-QAM
Fig. 4. ISI traces for 128-QAM signaling.
APPENDIX A OPTIMUM VALUES OF FOR CXA AND SCXA We borrow from Goupil et al. [7] that the dispersion constant must be selected in such a way that the perfect equalizer (in the sense of the zero-forcing criterion) is the minimum of the in a noiseless environment. The condition is cost criterion expressed by the relation [7]
(8)
Fig. 5. Cost for CXA and SCXA under phase-offset. The jagged appearance of the cost for SCXA is the consequence of the use of sliced symbols within its computation.
faster convergence speed. In addition to fast ISI mitigation capability, CXA and SCXA are also found capable of recovering the correct orientation of QAM constellation. The effect of phaseoffset on CXA and SCXA cost functions is depicted in Fig. 5 for a cross-QAM, where SCXA can be noticed to be more sensitive to the phase-offset than CXA. We also compared the experimental excess mean square error (EMSE) and the theoretical EMSE for the CXA and SCXA as the function of step-size in a noise-free environment and ideally equalized condition (refer to Appendix B for EMSE expressions). The experimental values were generated as averages over ten trials for 32-QAM using a 20-tap -spaced equalizer (with single-spike initialization). As depicted in Fig. 6, the result demonstrates that the theoretical values predicted by expression (14) and (16) match the experimental results reasonably well.
where is the coefficient of the one-tap equalizer. It is described in [7] that for a perfect channel, the coefficient should converge to 1. In other words, the equalizer must at least be able to recover the power of the signal [7]. For CXA, we formulate an optimization problem: find such that . For , we have , where . We find . Solving , we get
(9) For SCXA, we formulate: find
such that . Dif-
ferentiating w.r.t. , we have . Solving find
and substituting , leading to
, we (10)
The values of
(9), (10) are listed in Table I.
748
IEEE SIGNAL PROCESSING LETTERS, VOL. 13, NO. 12, DECEMBER 2006
leading to (14) where
,
, optimum value of
,
, and for CXA is obtained as follows:
. The
(15)
CXA
Fig. 7. Area of regions II and IV are 0:5R (1 while the total area is R .
0
For SCXA, we find and Next, we obtain k)
and 0:5R
k
, respectively,
APPENDIX B OPTIMUM VALUES OF FOR CXA AND SCXA We assume that the value of is optimum if it minimizes EMSE. The EMSE for a Bussgang-type algorithm with cost function can be expressed as follows [15]:
(11)
. . We substitute the values of
and
in (11) to get
(16) where
,
, optimum value of
, and for SCXA is obtained as follows:
. The
(17)
SCXA The values of
,
(15), (17) are listed in Table I.
,
where , and For CXA, with
. ,
and . After
some steps, we obtain . Taking a further derivative w.r.t.
, we obtain
the values of
and
. Now, substituting in (11), we obtain
(12) where (refer to Fig. 7) (Reg. ) (Reg. ) (Reg. ) (Reg. ).
(13)
Now we make use of the QAM orientation symmetries, which for any one of the in-phase or quadraenable us to evaluate ture components in one of the quadrants. We select the regions II and IV in the first quadrant to evaluate . The value of is equal to 4 and in regions II and IV, respectively. Using the relative area of different regions, we evaluate the expectation, as given by
Similarly, we compute the expectation in the numerator of (12),
REFERENCES [1] K. Wesolowski, “Self-recovering adaptive equalization algorithms for digital radio and voiceband data modems,” in Proc. Eur. Conf. Circuit Theory Design, 1987, pp. 19–24. [2] K. N. Oh and Y. O. Chin, “Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm,” in Proc. IEEE ICC, Jun. 1995, vol. 1, pp. 498–502. [3] S. Abrar, A. Zerguine, and M. Deriche, “Soft constraint satisfaction multimodulus blind equalization algorithms,” IEEE Signal Process. Lett., vol. 12, no. 9, pp. 637–640, Sep. 2005. [4] J. Yang, J.-J. Werner, and G. A. Dumont, “The multimodulus blind equalization and its generalized algorithms,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 997–1015, Jun. 2002. [5] J.-J. Werner and J. Yang, “Technique for Improving the Blind Convergence of an Adaptive Equalizer Using a Transition Algorithm,” U.S. Patent 5809074, Sep. 1998. [6] S. Abrar, “Compact constellation algorithm for blind equalization of QAM signals,” in Proc. IEEE INCC, Jun. 2004, pp. 170–174. [7] A. Goupil and J. Palicot, “Constant norm algorithms class,” in Proc. EUSIPCO, 2002. [8] T. Thaiupathump, “New algorithms for blind equalization and blind source separation/phase recovery,” Ph.D. dissertation, Univ. Pennsylvania, Philadelphia, PA, 2002. [9] N. Benvenuto and U. Cherubini, Algorithms for Communications Systems and Their Applications. New York: Wiley, 2002. [10] T. Thaiupathump, L. He, and S. A. Kassam, “Square contour algorithm for blind equalization of QAM signals,” Signal Process. (Elsevier), February 16, 2006, accepted (galley proof is available online). [11] S. Abrar and R. A. Axford, Jr., “Sliced multi-modulus blind equalization algorithm,” ETRI J., vol. 27, no. 3, pp. 257–266, Jun. 2005. [12] J.-J. Werner and J. Yang, “Blind Equalization Algorithm With the Joint Use of the Constant R and the Sliced Symbols,” U.S. Patent 6 493 381, Dec. 2002. [13] [Online]. Available: http://spib.rice.edu/spib/microwave.html. Signal Processing Information Base (SPIB), Rice Univ. [14] O. Shalvi and E. Weinstein, “New criteria for blind deconvolution of nonminimum phase systems (channels),” IEEE Trans. Inf. Theory, vol. 36, no. 2, pp. 312–321, Mar. 1990. [15] A. Goupil and J. Palicot, “A geometrical derivation of the excess mean square error for Bussgang algorithms in a noiseless environment,” Signal Process., vol. 84, pp. 311–315, 2004.
