Adaptive Solution for Blind Equalization and Carrier-Phase Recovery ...
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010
791
Adaptive Solution for Blind Equalization and Carrier-Phase Recovery of Square-QAM Shafayat Abrar, Graduate Student Member, IEEE, and Asoke Kumar Nandi, Senior Member, IEEE
Abstract—In this letter, we adaptively optimize the equalizer output energy to obtain a joint blind equalization and carrier-phase recovery solution. The resulting (multimodulus) update algorithm possesses a particular zero-memory Bussgang-type nonlinearity. We provide evidence of good performance, in comparison to existing adaptive methods, like RCA, MMA and CMA, through computer simulations for higher-order quadrature amplitude modulation signalling on symbol- and fractionally-spaced channels. Index Terms—Adaptive equalizers, blind equalization.
I. INTRODUCTION BLIND EQUALIZER optimizes, through the choice of its coefficients , a certain cost , such that its output provides an estimate of the source up to some indeterminacies [1]. Historically, the first-ever cost for this purpose was developed by Allen and Mazo [2] in 1974. They analyt, while ically showed that the optimization of energy anchoring the leading tap, is capable of inverting the channel. The potential of this idea remained unexplored until 1990, when Feyh and Klemt [3] studied the problem
A
(1)
and reported better results than those obtained from Shalvi–Weinstein algorithm [5]. Recently, Meng et al. [6] formulated
(3) as a quadratic programming problem for blind equalization of square-QAM and reported several impressive results. The parameter denoted the largest quadrature component of . subject to difNote that 1) costs (1)–(3) optimize ferent constraints, 2) costs (2) and (3) are capable of recovering carrier-phase, 3) cost (3) is able to restore true signal-energy, 4) costs (1)–(3) have been optimized originally in block-processing manner, and 5) due to nondifferentiable constraints in (2) and (3), it is not trivially possible to use gradient-based adaptive optimization. Motivated by these considerations, in this letter, we present a pertinent deterministic and differentiable cost, and obtain an adaptive blind equalization and carrier-phase recovery solution for square-QAM. II. PROPOSED COST AND ADAPTIVE ALGORITHM 2Suppose is a cost function to be optimized for blind equalization purpose and that depends solely on the statistics of . The stochastic gradient-based adaptive optimization of is obtained as [7, Ch. 6],
using eigenvalue analysis and discussed its inadmissibility. In 1993, using order statistics, Satorius and Mulligan [4] solved1 (2)
Manuscript received March 09, 2010; revised June 16, 2010; accepted June 24, 2010. Date of publication July 01, 2010; date of current version July 15, 2010. The work of S. Abrar work was supported by the ORSAS (UK), the University of Liverpool (U.K.) and the COMSATS Institute of Information Technology (Pakistan). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ricardo Merched. S. Abrar is with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K., on leave from the Department of Electrical Engineering, COMSATS Institute of Information Technology, Islamabad, Pakistan (e-mail: [email protected]; [email protected]). A. K. Nandi is with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2010.2055853 1Notations z and z denote respectively the real and the imaginary parts : of z 2 at time index n. For a sequence fg g, we define range(fg g) = max(fg g) 0 min(fg g), where max(fg g) and min(fg g), respectively, denote the maximum and the minimum value in fg g.
(4) where the sign or is required, respectively, for maximization or minimization of the cost with respect , the asterisk denotes the complex conjugate of to is a small positive step-size. Note the base entity and , and that , where . to be an error-function and Defining assuming a maximization scenario, we obtain (5) satisfies Bussgang condition upon sucIn principle, the cessful convergence [1], i.e., . Based on the discussion in Section I, we present a deterministic version of (3) involving instantaneous constraints, viz 2The baseband transmission of square-QAM symbols fa g is considered in the presence of additive white Gaussian noise through a moving-average channel fh g. An N -tap adaptive blind equalizer is employed to combat the intersymbol interference (ISI) caused by the channel. The received and equalized signals are x = h a + and y = w x , respectively, where w = [w ; 1 1 1 ; w ] and x = [x ; 1 1 1 ; x ] . Superscripts T and H denote, respectively, transpose and conjugate-transpose.
1070-9908/$26.00 © 2010 IEEE
792
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010
(6) where, for some
,
is defined as if if
. Due to the four-quadrant symmetry of QAM growth of and , note that a single signal (i.e., and . value of is required to be computed for both Denoting as one of the components of , we write . The is considered to be zero-mean Gaussian with and pdf , where is the variance of . variance Now combining (12) and (13), we get
(7)
.
The constellation of an -alphabet square-QAM (free from distortion, noise and rotation) is contained in a square region , . If the equalcentered at origin with perimeter falls inside region , then both constraints in ized sample cost (6) are satisfied; we simply need to maximize . Howlies outside region , then depending on where ever, when is residing at, either one or both of the constraints is/are viwould ensure that olated. In such a case, an ideal update the resulting a posteriori output comes inside region and the energy stays close to . is differentiable. For some , we can show The
(14) and
Further, are evaluated as
(8) where is the standard the Lagrangian multipliers,
and
function. Next employing , we obtain
(9) Differentiating (9) with respect to
, we obtain (10a) (10b)
where
denotes either
where and .3 Note that the has an asymptotic value for small noise. Considering a square-QAM, , and and small , we evaluate (14) to get assuming
or . Solving (5) and (10), we get (11)
If , then and the constraint is satisfied. On the other hand, the condition yields ; such that the Bussgang condihere, we suggest to compute tion is satisfied. This consideration leads to
(15)
where
. Assuming a diminishing noise, we get (16)
(12) The evaluation of (12) can be simplified by assuming that the update (11) is in the vicinity of an open-eye solution [1, Sect. is the sum of delayed source 2.8]. As a result, the output and convolutional noise [1]. For , (12) is signal satisfied due to the identical-and-independent distribution propand ; for , however, the constraints erty exhibited by may be satisfied by assuming
when In Fig. 1, we demonstrate that We summarize our algorithm as follows: if if
.
(17)
(13)
Note that the polarity of variable determines the direction is minimized away of adaptation such that the dispersion in and this property has a major from four corner points role in carrier-phase recovery. In [10], a term multimodulus was
where the negative sign is used to update in the opposite direction to bring the equalized symbol inside or close to the corner-points of region , and the is introduced to limit the
3For the evaluation of , we have considered 0:5 , where is the variance of additive noise and > 1 is an empirically determined constant. In future, we aim to explore the feasibility of established adaptive methods for the estimation of convolutional noise , like those addressed in [8] and [9].
ABRAR AND NANDI: ADAPTIVE SOLUTION FOR BLIND EQUALIZATION
Fig. 1. Parameter versus SNR
793
for some square-QAM.
Fig. 2. Carrier-phase recovery capability of MMA.
coined for the algorithm which can jointly solve blind equalization and carrier-phase recovery; we use this terminology to . denote (17) as -multimodulus algorithm Interestingly, the update (17) can be obtained by minimizing the following cost:4
Fig. 3. Plots of ISI convergence. Each of the traces has been obtained by taking average of 300 Monte-Carlo realizations with independent generation of noise and data samples. All traces in a given subplot exhibit same point of convergence as marked by a vertical dashed line.
III. SIMULATION RESULTS AND CONCLUSIONS (19) where the flags and are as defined in (17). Assuming and , we depict the sensitivity of cost exhibits (19) to phase-offset in Fig. 2. Clearly, the carrier-phase recovery capability. Finally, note that another adaptive realization is possible if . constraints are imposed on a posteriori output Taking conjugate-transpose of (11) and post-multiplying it with , we get . If , then the requirement yields , where . Note that is , and is similar to (13). favorably negative for 4Cost
(19) is similar to the following cost of traditional MMA [10]:
min where R =
y
0R
+
y
0R
[a ]= [a ] is termed as dispersion constant.
(18)
We simulate adaptive equalizers implementing reduced-constellation algorithm (RCA) [11], constant modulus algorithm (CMA) [12], multimodulus algorithm (MMA) [10], and the pro, while considering square-QAM transmisposed one sion over complex-valued baud and fractionally spaced (normalized) channels, evaluating (transient) ISI [5, Eq.(50)] and (steady-state) symbol-error rate (SER) performances. Experiment A: Firstly, we consider symbol-spaced equalization (TSE) of a voiceband channel [13]. A seven-tap equalizer is employed with central spike initialization. The converging ISI traces are summarized in Fig. 3(a)–(b) for 16- and 64-QAM, reis providing much lower ISI spectively. Note that the floor than all others while the traditional MMA is performing better than RCA and CMA. Secondly, we consider fractionally-spaced equalization -spaced microwave radio channel (channel-1, (FSE) of a SPIB [14]). We follow the multichannel equalizer architecture
794
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010
MMA is again performing better than RCA and CMA. We have and 2.0, respectively, for TSE and FSE. used Experiment B: The evaluation of SER can provide the performance comparison over a range of SNR values and it can incorporate any degradation due to imperfect restoration of carrier-phase and/or signal-energy. Here we simulate MMA and over the same two channels as we used in Experiment A. In Fig. 4(a), we depict SER performances for 16/64/256-QAM successover the voiceband channel. Both MMA and fully restored the 45 phase-offset, introduced by the channel, in all independent simulation runs. Observe that at lower SNR values, both MMA and performed almost identical; but, for higher SNR values, outperformed MMA for all QAM sizes. In Fig. 4(b), we depict SER results over the microwave radio channel. Again, we obis yielding much lower SER than MMA. serve that We have proposed a deterministic cost for joint blind equalization, carrier-phase recovery and energy restoration of square-QAM signals. We have experimentally showed that the can give better solution in resulting new algorithm terms of removing ISI and low SER, under the presence of noise, than some existing adaptive algorithms, like RCA, CMA and MMA. REFERENCES
Fig. 4. Plots of SER versus SNR . Step-sizes have been selected such that (a) for TSE, both MMA and MMA acquired steady-state around 3000th, 6000th and 20,000th iteration for 16-, 64- and 256-QAM, respectively, and (b) for FSE, both MMA and MMA acquired steady-state around 1500th, 3000th and 10,000th iteration for 16-, 64- and 256-QAM, respectively. The product of and is kept constant for MMA to maintain its convergence rate for all SNR values.
as described in [12] and implement a 42-tap equalizer, where we have 21 taps each in even and odd sets of coefficients with central spike initialization in even-set of coefficients. The converging ISI traces are summarized in Fig. 3(c)–(d) for 16is providing and 64-QAM, respectively. Note that the remarkably much lower ISI floor than all others while the
[1] S. Bellini, “Bussgang techniques for blind deconvolution and equalization,” in Blind Deconvolution (Edited). Upper Saddle River, NJ: Prentice-Hall, 1994, pp. 8–59. [2] J. Allen and J. Mazo, “A decision-free equalization scheme for minimum-phase channels,” IEEE Trans. Commun., vol. 22, no. 10, pp. 1732–1733, Oct. 1974. [3] G. Feyh and R. Klemt, “Blind equalizer based on autocorrelation lags,” in Proc. IEEE ACSSC, 1990, pp. 268–272. [4] E. H. Satorius and J. J. Mulligan, “An alternative methodology for blind equalization,” Digital Signal Process.: Rev. J., vol. 3, no. 3, pp. 199–209, 1993. [5] O. Shalvi and E. Weinstein, “New criteria for blind equalization of nonminimum phase systems,” IEEE Trans. Inf. Theory, vol. 36, no. 2, pp. 312–321, 1990. [6] C. Meng, J. Tuqan, and Z. Ding, “A quadratic programming approach to blind equalization and signal separation,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2232–2244, Jun. 2009. [7] R. Battiti, M. Brunato, and F. Mascia, Reactive Search and Intelligent Optimization. Berlin, Germany: Springer, 2008. [8] S. J. Nowlan and G. E. Hinton, “A soft decision-directed LMS algorithm for blind equalization,” IEEE Trans. Commun., vol. 41, no. 2, pp. 275–279, Feb. 1993. [9] M. Lazaro, I. Santamaria, D. Erdogmus, K. E. Hild, C. Pantaleon, and J. C. Principe, “Stochastic blind equalization based on PDF fitting using Parzen estimator,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 696–704, Feb. 2005. [10] 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, 2002. [11] A. Benveniste and M. Goursat, “Blind equalizers,” IEEE Trans. Commun., vol. COM-32, no. 8, pp. 871–883, Aug. 1984. [12] C. R. Johnson, Jr., P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE, vol. 86, no. 10, pp. 1927–1950, Oct. 1998. [13] G. Picchi and G. Prati, “Blind equalization and carrier recovery using a ‘stop-and-go’ decision-directed algorithm,” IEEE Trans. Commun., vol. COM-35, no. 9, pp. 877–887, Sep. 1987. [14] [Online]. Available: http://spib.rice.edu/spib/microwave.html. SPIB, Rice Univ. Houston, TX
791
Adaptive Solution for Blind Equalization and Carrier-Phase Recovery of Square-QAM Shafayat Abrar, Graduate Student Member, IEEE, and Asoke Kumar Nandi, Senior Member, IEEE
Abstract—In this letter, we adaptively optimize the equalizer output energy to obtain a joint blind equalization and carrier-phase recovery solution. The resulting (multimodulus) update algorithm possesses a particular zero-memory Bussgang-type nonlinearity. We provide evidence of good performance, in comparison to existing adaptive methods, like RCA, MMA and CMA, through computer simulations for higher-order quadrature amplitude modulation signalling on symbol- and fractionally-spaced channels. Index Terms—Adaptive equalizers, blind equalization.
I. INTRODUCTION BLIND EQUALIZER optimizes, through the choice of its coefficients , a certain cost , such that its output provides an estimate of the source up to some indeterminacies [1]. Historically, the first-ever cost for this purpose was developed by Allen and Mazo [2] in 1974. They analyt, while ically showed that the optimization of energy anchoring the leading tap, is capable of inverting the channel. The potential of this idea remained unexplored until 1990, when Feyh and Klemt [3] studied the problem
A
(1)
and reported better results than those obtained from Shalvi–Weinstein algorithm [5]. Recently, Meng et al. [6] formulated
(3) as a quadratic programming problem for blind equalization of square-QAM and reported several impressive results. The parameter denoted the largest quadrature component of . subject to difNote that 1) costs (1)–(3) optimize ferent constraints, 2) costs (2) and (3) are capable of recovering carrier-phase, 3) cost (3) is able to restore true signal-energy, 4) costs (1)–(3) have been optimized originally in block-processing manner, and 5) due to nondifferentiable constraints in (2) and (3), it is not trivially possible to use gradient-based adaptive optimization. Motivated by these considerations, in this letter, we present a pertinent deterministic and differentiable cost, and obtain an adaptive blind equalization and carrier-phase recovery solution for square-QAM. II. PROPOSED COST AND ADAPTIVE ALGORITHM 2Suppose is a cost function to be optimized for blind equalization purpose and that depends solely on the statistics of . The stochastic gradient-based adaptive optimization of is obtained as [7, Ch. 6],
using eigenvalue analysis and discussed its inadmissibility. In 1993, using order statistics, Satorius and Mulligan [4] solved1 (2)
Manuscript received March 09, 2010; revised June 16, 2010; accepted June 24, 2010. Date of publication July 01, 2010; date of current version July 15, 2010. The work of S. Abrar work was supported by the ORSAS (UK), the University of Liverpool (U.K.) and the COMSATS Institute of Information Technology (Pakistan). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ricardo Merched. S. Abrar is with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K., on leave from the Department of Electrical Engineering, COMSATS Institute of Information Technology, Islamabad, Pakistan (e-mail: [email protected]; [email protected]). A. K. Nandi is with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2010.2055853 1Notations z and z denote respectively the real and the imaginary parts : of z 2 at time index n. For a sequence fg g, we define range(fg g) = max(fg g) 0 min(fg g), where max(fg g) and min(fg g), respectively, denote the maximum and the minimum value in fg g.
(4) where the sign or is required, respectively, for maximization or minimization of the cost with respect , the asterisk denotes the complex conjugate of to is a small positive step-size. Note the base entity and , and that , where . to be an error-function and Defining assuming a maximization scenario, we obtain (5) satisfies Bussgang condition upon sucIn principle, the cessful convergence [1], i.e., . Based on the discussion in Section I, we present a deterministic version of (3) involving instantaneous constraints, viz 2The baseband transmission of square-QAM symbols fa g is considered in the presence of additive white Gaussian noise through a moving-average channel fh g. An N -tap adaptive blind equalizer is employed to combat the intersymbol interference (ISI) caused by the channel. The received and equalized signals are x = h a + and y = w x , respectively, where w = [w ; 1 1 1 ; w ] and x = [x ; 1 1 1 ; x ] . Superscripts T and H denote, respectively, transpose and conjugate-transpose.
1070-9908/$26.00 © 2010 IEEE
792
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010
(6) where, for some
,
is defined as if if
. Due to the four-quadrant symmetry of QAM growth of and , note that a single signal (i.e., and . value of is required to be computed for both Denoting as one of the components of , we write . The is considered to be zero-mean Gaussian with and pdf , where is the variance of . variance Now combining (12) and (13), we get
(7)
.
The constellation of an -alphabet square-QAM (free from distortion, noise and rotation) is contained in a square region , . If the equalcentered at origin with perimeter falls inside region , then both constraints in ized sample cost (6) are satisfied; we simply need to maximize . Howlies outside region , then depending on where ever, when is residing at, either one or both of the constraints is/are viwould ensure that olated. In such a case, an ideal update the resulting a posteriori output comes inside region and the energy stays close to . is differentiable. For some , we can show The
(14) and
Further, are evaluated as
(8) where is the standard the Lagrangian multipliers,
and
function. Next employing , we obtain
(9) Differentiating (9) with respect to
, we obtain (10a) (10b)
where
denotes either
where and .3 Note that the has an asymptotic value for small noise. Considering a square-QAM, , and and small , we evaluate (14) to get assuming
or . Solving (5) and (10), we get (11)
If , then and the constraint is satisfied. On the other hand, the condition yields ; such that the Bussgang condihere, we suggest to compute tion is satisfied. This consideration leads to
(15)
where
. Assuming a diminishing noise, we get (16)
(12) The evaluation of (12) can be simplified by assuming that the update (11) is in the vicinity of an open-eye solution [1, Sect. is the sum of delayed source 2.8]. As a result, the output and convolutional noise [1]. For , (12) is signal satisfied due to the identical-and-independent distribution propand ; for , however, the constraints erty exhibited by may be satisfied by assuming
when In Fig. 1, we demonstrate that We summarize our algorithm as follows: if if
.
(17)
(13)
Note that the polarity of variable determines the direction is minimized away of adaptation such that the dispersion in and this property has a major from four corner points role in carrier-phase recovery. In [10], a term multimodulus was
where the negative sign is used to update in the opposite direction to bring the equalized symbol inside or close to the corner-points of region , and the is introduced to limit the
3For the evaluation of , we have considered 0:5 , where is the variance of additive noise and > 1 is an empirically determined constant. In future, we aim to explore the feasibility of established adaptive methods for the estimation of convolutional noise , like those addressed in [8] and [9].
ABRAR AND NANDI: ADAPTIVE SOLUTION FOR BLIND EQUALIZATION
Fig. 1. Parameter versus SNR
793
for some square-QAM.
Fig. 2. Carrier-phase recovery capability of MMA.
coined for the algorithm which can jointly solve blind equalization and carrier-phase recovery; we use this terminology to . denote (17) as -multimodulus algorithm Interestingly, the update (17) can be obtained by minimizing the following cost:4
Fig. 3. Plots of ISI convergence. Each of the traces has been obtained by taking average of 300 Monte-Carlo realizations with independent generation of noise and data samples. All traces in a given subplot exhibit same point of convergence as marked by a vertical dashed line.
III. SIMULATION RESULTS AND CONCLUSIONS (19) where the flags and are as defined in (17). Assuming and , we depict the sensitivity of cost exhibits (19) to phase-offset in Fig. 2. Clearly, the carrier-phase recovery capability. Finally, note that another adaptive realization is possible if . constraints are imposed on a posteriori output Taking conjugate-transpose of (11) and post-multiplying it with , we get . If , then the requirement yields , where . Note that is , and is similar to (13). favorably negative for 4Cost
(19) is similar to the following cost of traditional MMA [10]:
min where R =
y
0R
+
y
0R
[a ]= [a ] is termed as dispersion constant.
(18)
We simulate adaptive equalizers implementing reduced-constellation algorithm (RCA) [11], constant modulus algorithm (CMA) [12], multimodulus algorithm (MMA) [10], and the pro, while considering square-QAM transmisposed one sion over complex-valued baud and fractionally spaced (normalized) channels, evaluating (transient) ISI [5, Eq.(50)] and (steady-state) symbol-error rate (SER) performances. Experiment A: Firstly, we consider symbol-spaced equalization (TSE) of a voiceband channel [13]. A seven-tap equalizer is employed with central spike initialization. The converging ISI traces are summarized in Fig. 3(a)–(b) for 16- and 64-QAM, reis providing much lower ISI spectively. Note that the floor than all others while the traditional MMA is performing better than RCA and CMA. Secondly, we consider fractionally-spaced equalization -spaced microwave radio channel (channel-1, (FSE) of a SPIB [14]). We follow the multichannel equalizer architecture
794
IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 9, SEPTEMBER 2010
MMA is again performing better than RCA and CMA. We have and 2.0, respectively, for TSE and FSE. used Experiment B: The evaluation of SER can provide the performance comparison over a range of SNR values and it can incorporate any degradation due to imperfect restoration of carrier-phase and/or signal-energy. Here we simulate MMA and over the same two channels as we used in Experiment A. In Fig. 4(a), we depict SER performances for 16/64/256-QAM successover the voiceband channel. Both MMA and fully restored the 45 phase-offset, introduced by the channel, in all independent simulation runs. Observe that at lower SNR values, both MMA and performed almost identical; but, for higher SNR values, outperformed MMA for all QAM sizes. In Fig. 4(b), we depict SER results over the microwave radio channel. Again, we obis yielding much lower SER than MMA. serve that We have proposed a deterministic cost for joint blind equalization, carrier-phase recovery and energy restoration of square-QAM signals. We have experimentally showed that the can give better solution in resulting new algorithm terms of removing ISI and low SER, under the presence of noise, than some existing adaptive algorithms, like RCA, CMA and MMA. REFERENCES
Fig. 4. Plots of SER versus SNR . Step-sizes have been selected such that (a) for TSE, both MMA and MMA acquired steady-state around 3000th, 6000th and 20,000th iteration for 16-, 64- and 256-QAM, respectively, and (b) for FSE, both MMA and MMA acquired steady-state around 1500th, 3000th and 10,000th iteration for 16-, 64- and 256-QAM, respectively. The product of and is kept constant for MMA to maintain its convergence rate for all SNR values.
as described in [12] and implement a 42-tap equalizer, where we have 21 taps each in even and odd sets of coefficients with central spike initialization in even-set of coefficients. The converging ISI traces are summarized in Fig. 3(c)–(d) for 16is providing and 64-QAM, respectively. Note that the remarkably much lower ISI floor than all others while the
[1] S. Bellini, “Bussgang techniques for blind deconvolution and equalization,” in Blind Deconvolution (Edited). Upper Saddle River, NJ: Prentice-Hall, 1994, pp. 8–59. [2] J. Allen and J. Mazo, “A decision-free equalization scheme for minimum-phase channels,” IEEE Trans. Commun., vol. 22, no. 10, pp. 1732–1733, Oct. 1974. [3] G. Feyh and R. Klemt, “Blind equalizer based on autocorrelation lags,” in Proc. IEEE ACSSC, 1990, pp. 268–272. [4] E. H. Satorius and J. J. Mulligan, “An alternative methodology for blind equalization,” Digital Signal Process.: Rev. J., vol. 3, no. 3, pp. 199–209, 1993. [5] O. Shalvi and E. Weinstein, “New criteria for blind equalization of nonminimum phase systems,” IEEE Trans. Inf. Theory, vol. 36, no. 2, pp. 312–321, 1990. [6] C. Meng, J. Tuqan, and Z. Ding, “A quadratic programming approach to blind equalization and signal separation,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2232–2244, Jun. 2009. [7] R. Battiti, M. Brunato, and F. Mascia, Reactive Search and Intelligent Optimization. Berlin, Germany: Springer, 2008. [8] S. J. Nowlan and G. E. Hinton, “A soft decision-directed LMS algorithm for blind equalization,” IEEE Trans. Commun., vol. 41, no. 2, pp. 275–279, Feb. 1993. [9] M. Lazaro, I. Santamaria, D. Erdogmus, K. E. Hild, C. Pantaleon, and J. C. Principe, “Stochastic blind equalization based on PDF fitting using Parzen estimator,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 696–704, Feb. 2005. [10] 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, 2002. [11] A. Benveniste and M. Goursat, “Blind equalizers,” IEEE Trans. Commun., vol. COM-32, no. 8, pp. 871–883, Aug. 1984. [12] C. R. Johnson, Jr., P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE, vol. 86, no. 10, pp. 1927–1950, Oct. 1998. [13] G. Picchi and G. Prati, “Blind equalization and carrier recovery using a ‘stop-and-go’ decision-directed algorithm,” IEEE Trans. Commun., vol. COM-35, no. 9, pp. 877–887, Sep. 1987. [14] [Online]. Available: http://spib.rice.edu/spib/microwave.html. SPIB, Rice Univ. Houston, TX
