Adaptive Minimum Entropy Equalization Algorithm - Semantic Scholar
966
IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 10, OCTOBER 2010
Adaptive Minimum Entropy Equalization Algorithm Shafayat Abrar, Graduate Student Member, IEEE, and Asoke K. Nandi, Senior Member, IEEE
Abstract—We employ minimum entropy deconvolution principle for blind equalization of complex digital signals. In essence, we maximize the output energy, constrain the equalizer output not to exceed certain level and obtain an adaptive solution capable of opening a closed-eye with the recovery of true energy of signal. We provide evidence of good performance in comparison to existing adaptive methods, like CMA and its variants, through simulations. Index Terms—Adaptive equalizers, blind equalization, minimum entropy deconvolution, constant modulus algorithm.
I. M INIMUM E NTROPY E QUALIZATION
B
LIND equalization of an unknown, non-minimum phase channel is an important problem in data communication systems. One of the popular methods in blind equalization is minimum entropy deconvolution (MED). This principle was introduced by Wiggins [1] in seismic data analysis, who sought to determine the inverse channel 𝒘 † that maximizes the kurtosis of the deconvolved data 𝑦𝑛 . Later, Gray [2] generalized the MED principle with two degrees of freedom as follows: 1 ∑𝐵 𝑝 𝑏=1 ∣𝑦𝑛−𝑏+1 ∣ 𝐵 † 𝒘 = arg max J𝑝,𝑞 , J𝑝,𝑞 (𝑦) ≡ ( ) 𝑝𝑞 ∑𝐵 𝒘 1 𝑞 ∣𝑦 ∣ 𝑛−𝑏+1 𝑏=1 𝐵
(1) The ratio (1) is formulated with an assumption that the sequence 𝑦𝑛 is characterized by the generalized Gaussian family of probability distribution. This measure is proportional to the probability that 𝑦𝑛 is a member of the distribution characterized by the shape parameter 𝑞 and not 𝑝. Several cases of MED, in the context of blind deconvolution of seismic data, have appeared in literature, like J2,1 [3], J4,2 [1], lim𝜀→0 J𝑝−𝜀,𝑝 [4], J𝑝,2 [5], J2𝑝,𝑝 [6]. The feasibility of MED for blind equalization of digital signals has been studied in [7], [8] and [9]. In this work, we consider a specific case of MED for adaptive blind equalization of complex digital signals. Consider a baseband transmission of complex symbols 𝑎𝑛 in the presence of additive white Gaussian noise 𝜁𝑛 through a moving-average channel ℎ𝑘 . An 𝑁 -tap blind adaptive equalizer 𝒘𝑛 is employed at receiver to combat the intersymbol interference (ISI) caused by ∑ the channel. The received and equalized signals are 𝑥𝑛 = 𝑘 ℎ𝑘 𝑎𝑛−𝑘 +𝜁𝑛 and 𝑦𝑛 = 𝒘𝐻 𝑛 𝒙𝑛 , Manuscript received July 3, 2010. The associate editor coordinating the review of this letter and approving it for publication was Y.-C. Wu. The authors are with the Department of Electrical Engineering and Electronics, Liverpool L69 3GJ, U.K. (e-mail: {shafayat, a.nandi}@liverpool.ac.uk). The work of S. Abrar is supported by the ORSAS (U.K.), the University of Liverpool, and the COMSATS Institute of Information Technology (Pakistan). Digital Object Identifier 10.1109/LCOMM.2010.083110.101168
respectively, where1 𝒘 𝑛 = [𝑤𝑛,0 , 𝑤𝑛,1 , ⋅ ⋅ ⋅ , 𝑤𝑛,𝑁 −1 ]𝑇 and 𝒙𝑛 = [𝑥𝑛 , 𝑥𝑛−1 , ⋅ ⋅ ⋅ , 𝑥𝑛−𝑁 +1 ]𝑇 . Now considering (1) with 𝑞 → ∞, 𝑝 = 2, and large 𝐵, we obtain ] [ E ∣𝑦𝑛 ∣2 𝒘† = arg max J2,∞ (𝑦𝑛 ), J2,∞ (𝑦𝑛 ) = 2 (2) 𝒘 (max {∣𝑦𝑛 ∣}) As deduced from [10], the cost (2) serves as a discrimination measure for uniform distribution against normal one. Effectively, the maximization of (2) enhances the non-Gaussianity in 𝑦𝑛 and tends to make the distribution of ∣𝑦𝑛 ∣ as uniform as possible. The scale-invariant cost (2) is incapable of recovering the true energy of signal 𝑎𝑛 ; we modify it to incorporate local knowledge of signal 𝑎𝑛 , for true energy recovery, as follows: ] [ 𝒘† = arg max E ∣𝑦𝑛 ∣2 s.t. max {∣𝑦𝑛 ∣} ≤ 𝑅𝑎 (3) 𝒘
where 𝑅𝑎 is the largest amplitude of 𝑎𝑛 . II. A DAPTIVE A LGORITHM AND S TABILITY A NALYSIS For a stochastic gradient-based implementation of (3), we need to further modify it to involve a differentiable constraint; one of the possibilities is2 [ ] 𝒘 † = max E ∣𝑦𝑛 ∣2 s.t. fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) = 𝑅𝑎 𝒘
(4)
where (for 𝑎, 𝑏 ∈ ℂ), we define ∣𝑎∣ + ∣𝑏∣ + ∣𝑎∣ − ∣𝑏∣ { ∣𝑎∣, if ∣𝑎∣ ≥ ∣𝑏∣ fmax(∣𝑎∣, ∣𝑏∣) ≡ = ∣𝑏∣, otherwise. 2 If ∣𝑦𝑛 ∣ < 𝑅𝑎 , then the cost (4) simply maximizes output energy. However, if ∣𝑦𝑛 ∣ > 𝑅𝑎 , then the constraint is violated and the new update 𝒘𝑛+1 is required to be computed such that the magnitude of a posteriori output 𝒘 𝐻 𝑛+1 𝒙𝑛 becomes smaller than or equal to 𝑅𝑎 . Next, employing Lagrangian multiplier, we get 𝒘 † = arg max E[∣𝑦𝑛 ∣2 ] + 𝜆(fmax (𝑅𝑎 , ∣𝑦𝑛 ∣) − 𝑅𝑎 ). 𝒘
(5)
The stochastic approximate gradient-based optimization of 𝒘 † = arg max𝒘 E[𝒥 ] is realized as 𝒘𝑛+1 = 𝒘𝑛 + 𝜇 ∂𝒥 /∂𝒘∗𝑛 , where asterisk ∗ denotes the complex conjugate 1𝑇
and 𝐻 denote transpose and conjugate-transpose, respectively. fmax is differentiable (below sgn denotes the signum function) ⎧ ( ( )) ⎨ 𝑎/(2∣𝑎∣), if ∣𝑎∣ > ∣𝑏∣ 𝑎 1 + sgn ∣𝑎∣ − ∣𝑏∣ ∂ fmax(∣𝑎∣, ∣𝑏∣) = = 𝑎/(4∣𝑎∣), if ∣𝑎∣ = ∣𝑏∣ ∗ ⎩ 0, ∂𝑎 4∣𝑎∣ if ∣𝑎∣ < ∣𝑏∣ 2 The
Differentiable functions, other than fmax, are possible, like those in [11].
c 2010 IEEE 1089-7798/10$25.00 ⃝
ABRAR and NANDI: ADAPTIVE MINIMUM ENTROPY EQUALIZATION ALGORITHM
and 𝜇 > 0 is a small step-size. Differentiating (5) with respect to 𝒘∗𝑛 gives ∂∣𝑦𝑛 ∣2 ∂∣𝑦𝑛 ∣2 ∂𝑦𝑛 = = 𝑦𝑛∗ 𝒙𝑛 (6a) ∗ ∂𝒘𝑛 ∂𝑦𝑛 ∂𝒘∗𝑛 ∂fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) ∂fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) ∂𝑦𝑛 𝑔𝑛 𝑦𝑛∗ 𝒙𝑛 = = ∂𝒘∗𝑛 ∂𝑦𝑛 ∂𝒘∗𝑛 4∣𝑦𝑛 ∣ (6b) where 𝑔𝑛 ≡ 1 + sgn (∣𝑦𝑛 ∣ − 𝑅𝑎 ). From (6), we obtain ( ( )) 𝒘𝑛+1 = 𝒘𝑛 + 𝜇 1 + 𝜆 𝑔𝑛 / 4∣𝑦𝑛 ∣ 𝑦𝑛∗ 𝒙𝑛
∣𝑦𝑛 ∣>𝑅𝑎
′
In steady-state, we assume 𝑦𝑛 = 𝑎𝑛′ + 𝑢𝑛 , where 𝑛 is time index to indicate delay, and 𝑢𝑛 is convolutional noise. For 𝑖 ∕= 0, (8) is satisfied due to uncorrelated 𝑎𝑛 and independent and identically distributed samples of 𝑢𝑛 . Let 𝑎𝑛 comprises 𝑀 distinct symbols on 𝐿 moduli {𝑅1 , ⋅ ⋅ ⋅ , 𝑅𝐿 }, where 𝑅𝐿 = 𝑅𝑎 is the largest modulus. Let 𝑀𝑖 denote number of unique ∑the 𝐿 symbols on the 𝑖th modulus, i.e., 𝑙=1 𝑀𝑙 = 𝑀 . With negligible 𝑢𝑛 , we solve (8) for 𝑖 = 0 to get 1 𝛽 2 2 2 + 𝑀𝐿 𝑅𝐿 − 𝑀𝐿 𝑅𝐿 = 0. (9) 𝑀1 𝑅12 + ⋅ ⋅ ⋅ + 𝑀𝐿−1 𝑅𝐿−1 2 2 The last two terms indicate that, when ∣𝑦𝑛 ∣ is close to 𝑅𝐿 , it would be equally likely to update in either direction. Noting ∑𝐿 that 𝑙=1 𝑀𝑙 𝑅𝑙2 = 𝑀 𝑃𝑎 , where 𝑃𝑎 = E[∣𝑎𝑛 ∣2 ] is the average energy of signal 𝑎𝑛 , the simplification of (9) gives 𝛽=2
𝑀 𝑃𝑎 −1 𝑀𝐿 𝑅𝑎2
(10)
The use of (10) ensures recovery of true signal energy upon successful convergence. The proposed algorithm is given by 𝒘𝑛+1 = 𝒘 𝑛 + 𝜇 f(𝑦𝑛 ) 𝑦𝑛∗ 𝒙𝑛 , ⎧ ⎨ 1, if ∣𝑦𝑛 ∣ < 𝑅𝑎 f(𝑦𝑛 ) = −𝛽, if ∣𝑦𝑛 ∣ > 𝑅𝑎 ⎩ 0, otherwise.
To study the stability of (11), we subtract zero-forcing solution 𝒘 [∗] from it and obtain 𝒘 𝑛+1 = 𝒘𝑛 − 𝜇 f(𝑦𝑛 )𝑦𝑛∗ 𝒙𝑛 , where 𝒘 𝑛 ≡ 𝒘[∗] − 𝒘𝑛 . If error 𝑒𝑎 = 𝑦𝑛 − 𝑎𝑛′ = −𝒘𝐻 𝑛 𝒙𝑛 is small, we can replace f(𝑦𝑛 )𝑦𝑛∗ with f(𝑎𝑛′ )(𝑎∗𝑛′ + 𝑒∗𝑛 ) to get ∗ 𝒘𝑛+1 = 𝒘𝑛 − 𝜇 𝜛 𝒙𝑛 𝒙𝐻 𝑛 𝒘 𝑛 + 𝜇 𝜛 𝑎𝑛′ 𝒙𝑛 ,
(11)
Note that the error-function f(𝑦𝑛 )𝑦𝑛∗ has 1) finite derivative at the origin, 2) becomes zero solely at 𝑅𝑎 , 3) is increasing for ∣𝑦𝑛 ∣ < 𝑅𝑎 and 4) decreasing for ∣𝑦𝑛 ∣ > 𝑅𝑎 . In [13], these properties 1)-4) have been regarded as essential features of a constant modulus algorithm (CMA); this motivates us to denote (11) as 𝛽CMA.
(12)
where 𝜛 ≡ −f(𝑎𝑛′ ). From [14], we know that the update (12) is stable stochastically in mean-square sense, if 0 0, which leads to 𝒘𝑛+1 = 𝒘 𝑛 + 𝜇(−𝛽)𝑦𝑛∗ 𝒙𝑛 . The Bussgang condition requires [ ] ] [ ∗ ∗ E 𝑦𝑛 𝑦𝑛−𝑖 + (−𝛽) E 𝑦𝑛 𝑦𝑛−𝑖 = 0, ∀𝑖 ∈ ℤ (8) ∣𝑦𝑛 ∣ 𝑅𝑎 ] − Pr [∣𝑎𝑛 ∣ < 𝑅𝑎 ] (14a) ≈ ((𝛽 + 1) E [∣𝑎∣] − 𝑅𝑎 ) 𝑅𝑎−1 .
(14b)
Similarly we obtain [ ] [ ] E 𝜛2 = E f 2 = 𝛽 2 Pr [∣𝑎𝑛 ∣ > 𝑅𝑎 ] + Pr [∣𝑎𝑛 ∣ < 𝑅𝑎 ] (15a) ) −1 ) (( 2 (15b) ≈ 𝛽 − 1 E [∣𝑎∣] + 𝑅𝑎 𝑅𝑎 . The stochastic stability bound for 𝛽CMA is thus given by 0
IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 10, OCTOBER 2010
Adaptive Minimum Entropy Equalization Algorithm Shafayat Abrar, Graduate Student Member, IEEE, and Asoke K. Nandi, Senior Member, IEEE
Abstract—We employ minimum entropy deconvolution principle for blind equalization of complex digital signals. In essence, we maximize the output energy, constrain the equalizer output not to exceed certain level and obtain an adaptive solution capable of opening a closed-eye with the recovery of true energy of signal. We provide evidence of good performance in comparison to existing adaptive methods, like CMA and its variants, through simulations. Index Terms—Adaptive equalizers, blind equalization, minimum entropy deconvolution, constant modulus algorithm.
I. M INIMUM E NTROPY E QUALIZATION
B
LIND equalization of an unknown, non-minimum phase channel is an important problem in data communication systems. One of the popular methods in blind equalization is minimum entropy deconvolution (MED). This principle was introduced by Wiggins [1] in seismic data analysis, who sought to determine the inverse channel 𝒘 † that maximizes the kurtosis of the deconvolved data 𝑦𝑛 . Later, Gray [2] generalized the MED principle with two degrees of freedom as follows: 1 ∑𝐵 𝑝 𝑏=1 ∣𝑦𝑛−𝑏+1 ∣ 𝐵 † 𝒘 = arg max J𝑝,𝑞 , J𝑝,𝑞 (𝑦) ≡ ( ) 𝑝𝑞 ∑𝐵 𝒘 1 𝑞 ∣𝑦 ∣ 𝑛−𝑏+1 𝑏=1 𝐵
(1) The ratio (1) is formulated with an assumption that the sequence 𝑦𝑛 is characterized by the generalized Gaussian family of probability distribution. This measure is proportional to the probability that 𝑦𝑛 is a member of the distribution characterized by the shape parameter 𝑞 and not 𝑝. Several cases of MED, in the context of blind deconvolution of seismic data, have appeared in literature, like J2,1 [3], J4,2 [1], lim𝜀→0 J𝑝−𝜀,𝑝 [4], J𝑝,2 [5], J2𝑝,𝑝 [6]. The feasibility of MED for blind equalization of digital signals has been studied in [7], [8] and [9]. In this work, we consider a specific case of MED for adaptive blind equalization of complex digital signals. Consider a baseband transmission of complex symbols 𝑎𝑛 in the presence of additive white Gaussian noise 𝜁𝑛 through a moving-average channel ℎ𝑘 . An 𝑁 -tap blind adaptive equalizer 𝒘𝑛 is employed at receiver to combat the intersymbol interference (ISI) caused by ∑ the channel. The received and equalized signals are 𝑥𝑛 = 𝑘 ℎ𝑘 𝑎𝑛−𝑘 +𝜁𝑛 and 𝑦𝑛 = 𝒘𝐻 𝑛 𝒙𝑛 , Manuscript received July 3, 2010. The associate editor coordinating the review of this letter and approving it for publication was Y.-C. Wu. The authors are with the Department of Electrical Engineering and Electronics, Liverpool L69 3GJ, U.K. (e-mail: {shafayat, a.nandi}@liverpool.ac.uk). The work of S. Abrar is supported by the ORSAS (U.K.), the University of Liverpool, and the COMSATS Institute of Information Technology (Pakistan). Digital Object Identifier 10.1109/LCOMM.2010.083110.101168
respectively, where1 𝒘 𝑛 = [𝑤𝑛,0 , 𝑤𝑛,1 , ⋅ ⋅ ⋅ , 𝑤𝑛,𝑁 −1 ]𝑇 and 𝒙𝑛 = [𝑥𝑛 , 𝑥𝑛−1 , ⋅ ⋅ ⋅ , 𝑥𝑛−𝑁 +1 ]𝑇 . Now considering (1) with 𝑞 → ∞, 𝑝 = 2, and large 𝐵, we obtain ] [ E ∣𝑦𝑛 ∣2 𝒘† = arg max J2,∞ (𝑦𝑛 ), J2,∞ (𝑦𝑛 ) = 2 (2) 𝒘 (max {∣𝑦𝑛 ∣}) As deduced from [10], the cost (2) serves as a discrimination measure for uniform distribution against normal one. Effectively, the maximization of (2) enhances the non-Gaussianity in 𝑦𝑛 and tends to make the distribution of ∣𝑦𝑛 ∣ as uniform as possible. The scale-invariant cost (2) is incapable of recovering the true energy of signal 𝑎𝑛 ; we modify it to incorporate local knowledge of signal 𝑎𝑛 , for true energy recovery, as follows: ] [ 𝒘† = arg max E ∣𝑦𝑛 ∣2 s.t. max {∣𝑦𝑛 ∣} ≤ 𝑅𝑎 (3) 𝒘
where 𝑅𝑎 is the largest amplitude of 𝑎𝑛 . II. A DAPTIVE A LGORITHM AND S TABILITY A NALYSIS For a stochastic gradient-based implementation of (3), we need to further modify it to involve a differentiable constraint; one of the possibilities is2 [ ] 𝒘 † = max E ∣𝑦𝑛 ∣2 s.t. fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) = 𝑅𝑎 𝒘
(4)
where (for 𝑎, 𝑏 ∈ ℂ), we define ∣𝑎∣ + ∣𝑏∣ + ∣𝑎∣ − ∣𝑏∣ { ∣𝑎∣, if ∣𝑎∣ ≥ ∣𝑏∣ fmax(∣𝑎∣, ∣𝑏∣) ≡ = ∣𝑏∣, otherwise. 2 If ∣𝑦𝑛 ∣ < 𝑅𝑎 , then the cost (4) simply maximizes output energy. However, if ∣𝑦𝑛 ∣ > 𝑅𝑎 , then the constraint is violated and the new update 𝒘𝑛+1 is required to be computed such that the magnitude of a posteriori output 𝒘 𝐻 𝑛+1 𝒙𝑛 becomes smaller than or equal to 𝑅𝑎 . Next, employing Lagrangian multiplier, we get 𝒘 † = arg max E[∣𝑦𝑛 ∣2 ] + 𝜆(fmax (𝑅𝑎 , ∣𝑦𝑛 ∣) − 𝑅𝑎 ). 𝒘
(5)
The stochastic approximate gradient-based optimization of 𝒘 † = arg max𝒘 E[𝒥 ] is realized as 𝒘𝑛+1 = 𝒘𝑛 + 𝜇 ∂𝒥 /∂𝒘∗𝑛 , where asterisk ∗ denotes the complex conjugate 1𝑇
and 𝐻 denote transpose and conjugate-transpose, respectively. fmax is differentiable (below sgn denotes the signum function) ⎧ ( ( )) ⎨ 𝑎/(2∣𝑎∣), if ∣𝑎∣ > ∣𝑏∣ 𝑎 1 + sgn ∣𝑎∣ − ∣𝑏∣ ∂ fmax(∣𝑎∣, ∣𝑏∣) = = 𝑎/(4∣𝑎∣), if ∣𝑎∣ = ∣𝑏∣ ∗ ⎩ 0, ∂𝑎 4∣𝑎∣ if ∣𝑎∣ < ∣𝑏∣ 2 The
Differentiable functions, other than fmax, are possible, like those in [11].
c 2010 IEEE 1089-7798/10$25.00 ⃝
ABRAR and NANDI: ADAPTIVE MINIMUM ENTROPY EQUALIZATION ALGORITHM
and 𝜇 > 0 is a small step-size. Differentiating (5) with respect to 𝒘∗𝑛 gives ∂∣𝑦𝑛 ∣2 ∂∣𝑦𝑛 ∣2 ∂𝑦𝑛 = = 𝑦𝑛∗ 𝒙𝑛 (6a) ∗ ∂𝒘𝑛 ∂𝑦𝑛 ∂𝒘∗𝑛 ∂fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) ∂fmax(𝑅𝑎 , ∣𝑦𝑛 ∣) ∂𝑦𝑛 𝑔𝑛 𝑦𝑛∗ 𝒙𝑛 = = ∂𝒘∗𝑛 ∂𝑦𝑛 ∂𝒘∗𝑛 4∣𝑦𝑛 ∣ (6b) where 𝑔𝑛 ≡ 1 + sgn (∣𝑦𝑛 ∣ − 𝑅𝑎 ). From (6), we obtain ( ( )) 𝒘𝑛+1 = 𝒘𝑛 + 𝜇 1 + 𝜆 𝑔𝑛 / 4∣𝑦𝑛 ∣ 𝑦𝑛∗ 𝒙𝑛
∣𝑦𝑛 ∣>𝑅𝑎
′
In steady-state, we assume 𝑦𝑛 = 𝑎𝑛′ + 𝑢𝑛 , where 𝑛 is time index to indicate delay, and 𝑢𝑛 is convolutional noise. For 𝑖 ∕= 0, (8) is satisfied due to uncorrelated 𝑎𝑛 and independent and identically distributed samples of 𝑢𝑛 . Let 𝑎𝑛 comprises 𝑀 distinct symbols on 𝐿 moduli {𝑅1 , ⋅ ⋅ ⋅ , 𝑅𝐿 }, where 𝑅𝐿 = 𝑅𝑎 is the largest modulus. Let 𝑀𝑖 denote number of unique ∑the 𝐿 symbols on the 𝑖th modulus, i.e., 𝑙=1 𝑀𝑙 = 𝑀 . With negligible 𝑢𝑛 , we solve (8) for 𝑖 = 0 to get 1 𝛽 2 2 2 + 𝑀𝐿 𝑅𝐿 − 𝑀𝐿 𝑅𝐿 = 0. (9) 𝑀1 𝑅12 + ⋅ ⋅ ⋅ + 𝑀𝐿−1 𝑅𝐿−1 2 2 The last two terms indicate that, when ∣𝑦𝑛 ∣ is close to 𝑅𝐿 , it would be equally likely to update in either direction. Noting ∑𝐿 that 𝑙=1 𝑀𝑙 𝑅𝑙2 = 𝑀 𝑃𝑎 , where 𝑃𝑎 = E[∣𝑎𝑛 ∣2 ] is the average energy of signal 𝑎𝑛 , the simplification of (9) gives 𝛽=2
𝑀 𝑃𝑎 −1 𝑀𝐿 𝑅𝑎2
(10)
The use of (10) ensures recovery of true signal energy upon successful convergence. The proposed algorithm is given by 𝒘𝑛+1 = 𝒘 𝑛 + 𝜇 f(𝑦𝑛 ) 𝑦𝑛∗ 𝒙𝑛 , ⎧ ⎨ 1, if ∣𝑦𝑛 ∣ < 𝑅𝑎 f(𝑦𝑛 ) = −𝛽, if ∣𝑦𝑛 ∣ > 𝑅𝑎 ⎩ 0, otherwise.
To study the stability of (11), we subtract zero-forcing solution 𝒘 [∗] from it and obtain 𝒘 𝑛+1 = 𝒘𝑛 − 𝜇 f(𝑦𝑛 )𝑦𝑛∗ 𝒙𝑛 , where 𝒘 𝑛 ≡ 𝒘[∗] − 𝒘𝑛 . If error 𝑒𝑎 = 𝑦𝑛 − 𝑎𝑛′ = −𝒘𝐻 𝑛 𝒙𝑛 is small, we can replace f(𝑦𝑛 )𝑦𝑛∗ with f(𝑎𝑛′ )(𝑎∗𝑛′ + 𝑒∗𝑛 ) to get ∗ 𝒘𝑛+1 = 𝒘𝑛 − 𝜇 𝜛 𝒙𝑛 𝒙𝐻 𝑛 𝒘 𝑛 + 𝜇 𝜛 𝑎𝑛′ 𝒙𝑛 ,
(11)
Note that the error-function f(𝑦𝑛 )𝑦𝑛∗ has 1) finite derivative at the origin, 2) becomes zero solely at 𝑅𝑎 , 3) is increasing for ∣𝑦𝑛 ∣ < 𝑅𝑎 and 4) decreasing for ∣𝑦𝑛 ∣ > 𝑅𝑎 . In [13], these properties 1)-4) have been regarded as essential features of a constant modulus algorithm (CMA); this motivates us to denote (11) as 𝛽CMA.
(12)
where 𝜛 ≡ −f(𝑎𝑛′ ). From [14], we know that the update (12) is stable stochastically in mean-square sense, if 0 0, which leads to 𝒘𝑛+1 = 𝒘 𝑛 + 𝜇(−𝛽)𝑦𝑛∗ 𝒙𝑛 . The Bussgang condition requires [ ] ] [ ∗ ∗ E 𝑦𝑛 𝑦𝑛−𝑖 + (−𝛽) E 𝑦𝑛 𝑦𝑛−𝑖 = 0, ∀𝑖 ∈ ℤ (8) ∣𝑦𝑛 ∣ 𝑅𝑎 ] − Pr [∣𝑎𝑛 ∣ < 𝑅𝑎 ] (14a) ≈ ((𝛽 + 1) E [∣𝑎∣] − 𝑅𝑎 ) 𝑅𝑎−1 .
(14b)
Similarly we obtain [ ] [ ] E 𝜛2 = E f 2 = 𝛽 2 Pr [∣𝑎𝑛 ∣ > 𝑅𝑎 ] + Pr [∣𝑎𝑛 ∣ < 𝑅𝑎 ] (15a) ) −1 ) (( 2 (15b) ≈ 𝛽 − 1 E [∣𝑎∣] + 𝑅𝑎 𝑅𝑎 . The stochastic stability bound for 𝛽CMA is thus given by 0
