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Stop-and-Go Algorithms for Blind Channel Equalization in QAM Data Communication Systems Shafayat Abrar Electrical Engineering Department, COMSATS-IIT (CIIT), Islamabad, Pakistan. sabrar@comsats.edu.pk, shafayat1972@yahoo.com Abstract: This paper presents a framework of stop-and-go input data {an }, each element of which belongs to a complex algorithms for the blind equalization of QAM data commu- alphabet A of QAM symbols. The data sequence {a n } is nication systems. Based on the proposed framework, three sent through a complex LTI channel whose output x n is new stop-and-go algorithms are presented. It is discussed observed by the receiver. The function of the blind equalizer that each of the existing and proposed stop-and-go algorithm at the receiver is to estimate the original data {a n } from forces the equalizer output to match some statistical contour the received signal x n . The input/output relationship of the on QAM constellation space. Using computer simulations, it QAM system can be written as: is shown that the Picchi and Prati’s “stop-and-go decision- K−1 directed algorithm”, which forces the equalizer output to xn = an−iT ci + νn , (1) lie on point contours, is the best among other stop-and-go i=0 algorithms. where T is the symbol (or baud) period and K is the Keywords: Blind equalization, adaptive equalizers, stop- length of channel impulse-response. The channel noise ν n is and-go, decision-directed, QAM. assumed to be stationary, Gaussian, and independent of the channel input a n . Denote the equalizer parameter vector with I. INTRODUCTION N + 1 elements as wn = [w0,n , w1,n , · · · , wN,n ]T , where the In most digital communication systems, inter-symbol in- superscript T represents transpose. In addition, deﬁne the terference (ISI) occurs due to bandwidth limited channel or received signal vector as x n = [xn , xn−1 , · · · , xn−N +1 ]T . T multipath propagation. Channel equalization is one of the The output signal of the equalizer is thus y n = wn xn = techniques to mitigate the effect of ISI. Adaptive algorithms yR,n + yI,n , where R and I represent the real and imaginary are used to initialize and adjust equalizer coefﬁcients when a parts of yn , respectively. If {h} = {c} ∗ {w} represents channel is unknown and possibly time-varying. Convention- the overall channel-equalizer impulse response. The channel ally, initial setting of the equalizer tap weights is achieved output xn can be expressed as: by a training sequence before data transmission. xn = an−iT hi + νn = h0 an + ai hn−iT + νn . However, when sending a training sequence is impractical i i=0 or impossible, it is desirable to equalize a channel without the aid of a training sequence. Equalizing a channel without signal+ISI+noise training mode is known as blind equalization. A typical In blind equalization, the channel input a n−D is unavailable, and thus different minimization criteria are explored. The crudest blind equalization algorithm is the decision-directed v(n) scheme that updates the adaptive equalizer coefﬁcients ac- x(n) a(n) Channel Equalizer y(n) a(n) cording to C(z) + W(z) Decision Device wn+1 = wn − µ(yn − D[yn ])x∗ n (2) where D[yn ] is the closest symbol to yn . Under high ISI, Blind Algorithm the convergence behavior of decision-directed equalizer is very poor. Better blind adaptive equalization algorithms are Fig. 1. Baseband model of communication system. designed to minimizing special non-MSE cost functions that do not directly involve the input a n while still reﬂect the blind equalization setup is depicted in Fig. 1 using a simple current level of ISI in the equalizer output. Deﬁne the mean system diagram. The complex baseband model for a typical cost function as: QAM (quadrature amplitude modulated) data communication J(w) = E[Ψ[yn ]] (3) system consists of an unknown linear time-invariant (LTI) channel cn which represents the physical inter-connection where Ψ[·] is a scalar function of the equalizer output. J(w) between the transmitter and the receiver at baseband. The should be speciﬁed such that at its minimum, the correspond- transmitter generates a sequence of complex-valued random ing wn results in a minimum ISI or MSE equalizer. Using (3), the stochastic gradient descent minimization algorithm symbols. Let us consider the following error function (say is easily derived as global), which is used in adaptation process to minimize ∂Ψ[yn ] the difference between the statistics of y n and some pre- wn+1 = wn − µ · = wn − µ · Ψ [yn ]x∗ . n calculated statistics (RR and RI ) of the transmitted QAM ∂wn signal: Let ψ be the ﬁrst derivative of Ψ. ψ is often called the error function since it replaces the prediction error in the LMS ψG [yn ] = (g1 (yR,n , yI,n ) − RR ) · g2 (yR,n , yI,n ) adaptation. The resulting blind equalization algorithm can + (g3 (yR,n , yI,n ) − RI ) · g4 (yR,n , yI,n ). (6) be written as: where g1 (·), g2 (·), g3 (·) and g4 (·) are zero-memory (and wn+1 = wn − µ · ψ[yn ]x∗ . n (4) preferably continuous) functions. The resulting weight adap- Thus the design of the blind equalizer thus translates into the tation process is wn+1 = wn − µ · ψG [yn ]x∗ . RR and RI n selection of a suitable function Ψ (or ψ) such that the local are positive (dispersion constants) and are computed so that: minima of J(w) correspond to a signiﬁcant removal of ISI from the equalizer output y n . E[ψG [an ]a∗ ] = 0. n (7) II. STOP-AND-GO BLIND EQUALIZATION The error functions of almost all existing stochastic gradient ALGORITHMS based blind equalization algorithms can be mapped onto (6). Given the standard form of the blind equalization algo- Now we want to design an error function ψ L [yn ] that rithm in (4), it is apparent that the convergence characteristics ˆ incorporates a n (the local information). There are many ways of blind algorithms are largely determined by the sign of to do it. One way is to allow dual mode; that is based on the error signal ψ[y n ]. In order, for the coefﬁcients of a the error level between y n and an , the equalizer will switch ˆ blind equalizer, to converge to the vicinity of the optimum between certain blind and decision-directed adaptations, as minimum-MSE (MMSE) solution as achieved by LMS adap- reported in [2]. So the algorithm switches between a suitable tation (under supervision), the sign of its error signal should ˆ ψG [yn ] and ψL [yn ] = yn − an . Another method exploits agree with the sign of the LMS prediction error y n − an−D ˆ an directly in the weight adaptation process without going most of the time. Slow or ill convergence can occur if the ˆ into decision-directed mode [3]. It embeds a n and dispersion sign of the two errors differ sufﬁciently often. constants (R) together, such that ψ G [yn ] becomes function The idea behind the stop-and-go algorithms is to allow the ˆ of both dispersion constant(s) and a n . adaptation “to go” only when the error function is more likely The third approach, which is the subject of this paper, is to have the correct sign for the gradient descent direction. “stop-and-go” scheme. In this approach, the adaptation pro- Given several criteria for blind equalization, one can expect cess apparently uses only a n and yn from startup to ﬁnal con- ˆ a more accurate descent direction when more than one of vergence. It is achieved by replacing the dispersion constants the existing algorithms agree on the sign (direction) of the ˆ RR and RI with some suitable nonlinear functions of a n . For error functions. When the error signs differ for a particular example, replacing R R and RI with g1 (|ˆR,n |, |ˆI,n |) and a a output sample, parameter adaptation is “stopped”. Consider g3 (|ˆR,n |, |ˆI,n |) in (6) will incorporate local information a a two algorithms with error functions ψ 1 and ψ2 . The following in weight adaptation process. Note that, it will result in an stop-and-go algorithm can be devised [1]: increase in the number of contours and as a result, the QAM symbols in alphabet A lie on (at least) one or more contours. wn − µψ1 [yn ]x∗ , if sgn[ψ1 [yn ]] = sgn[ψ2 [yn ]] n wn+1 = (5) The resulting error function (say local) ψ L [yn ] can be given wn , if sgn[ψ1 [yn ]] = sgn[ψ2 [yn ]] as Error functions ψ 1 and ψ2 should be selected such that they maximize reliable regions and make most of the local and the a a ψL [yn ] = (g1 (yR,n , yI,n ) − g1 (|ˆR,n |, |ˆI,n |)) · g2 (yR,n , yI,n ) global knowledge of the constellation. Given the equalizer + (g3 (yR,n , yI,n ) − g3 (|ˆR,n |, |ˆI,n |)) · g4 (yR,n , yI,n ) a a (8) ˆ output yn , the closest symbol an = D[yn ] can be considered as a local information; while the size (number of alphabets), Note that ψL [yn ] forces yn to lie on the contour which also shape (square or cross) and energy (mean distance between contains the closest symbol a n = D(yn ) on it. It can easily ˆ the symbols) of the constellation can be considered as global be understood that due to multiple contours exhibited by ˆ information. a n is termed local as it may change from one ψL [yn ], the steady-state misadjustment offered by ψ L [yn ] output to another; while the size, shape and energy are ﬁxed is much lower compared to that of ψ G [yn ]. If ψG [yn ] is and don’t depend on any speciﬁc value of y n . Most of the capable of removing ISI, then the use of ψ L [yn ] will be stochastic gradient descent algorithms employ error functions beneﬁcial, only when the sign of the two error functions, which exhibit global information. They compute an estimate ψG [yn ] and ψL [yn ], match. If ψL [yn ] is incorporated in the of an , by doing some nonlinear operation on the current weight adaptation process then the real and the imaginary equalizer output y n such that the certain statistics of y n are parts of ψL [yn ] should be weighted with binary ﬂags f R forced to match with global statistics of the transmitted data and fI , respectively, to indicate the sign match. Flags, f R and fI , are obtained as follows: wn+1 = wn − µ [ψL [yn ]R · fR + ψL [yn ]I · fI ] x∗ , where n ψL [yn ] is computed as decision-directed error: 1 + sgn[ψL [yn ]R ] · sgn[ψG [yn ]R ] fR = (9) 2 ˆ ˆ ψL [yn ] = yR,n − aR,n + (yI,n − aI,n ) (16) 1 + sgn[ψL [yn ]I ] · sgn[ψG [yn ]I ] fI = (10) Note that (16) can be expressed in an equivalent form as 2 follows: The resulting weight adaptation rule is, ψL [yn ] = yR,n − |ˆR,n | sgn[yR,n ] + (yI,n − |ˆI,n |sgn[yI,n ]) (17) a a wn+1 = wn − µ · (ψL [yn ]R · fR + ψL [yn ]I · fI )x∗ . (11) n Comparing (15) and (17); it can be observed that Picchi and where subscripts R and I denote the real and imaginary components, respectively. Since ψ L [yn ] is being used in Benveniste−Rudget’s Reduced Constellation Algorithm (1983) Picchi−Prati’s Point−Contour Stop−and−Go Algorithm (1987) Reliable regions for yR . Reliable regions for y I weight adaptation process, the values of R R and RI need 4 4 4 not to satisfy (7); instead, they can be selected such that the 2 2 2 sign of the two error functions agree most of the time. It is 0 0 0 observed that if R R and RI are selected as the outermost −2 −2 −2 contour, then the reliable regions can be maximized. −4 −4 −4 −4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4 RR = max[g1 ({|aR,n |}, {|aI,n |})] (12) (a) (b) (c) RI = max[g3 ({|aR,n |}, {|aI,n|})] (13) Fig. 2. Contours in (a) RCA (b)-(c) PC-SAGA The above mentioned scheme (Equations (6), (8), (9), (10), Prati algorithm is a stop-and-go version of RCA obtained by (11), (12) and (13)) can be applied to any stochastic gradient replacing RR and RI with |ˆR,n | and |ˆ I,n |, respectively. a a descent based blind equalization scheme to develop its stop- Similar to RCA, which forces y n to belong to one of the and-go version. In subsequent sections, we will review two four possible point contours, the Picchi and Prati algorithm existing stop-and-go blind equalization algorithms based on forces yn to belong to the nearest constellation symbol. It can this framework and three new stop-and-go algorithms will be said that it modiﬁes the RCA’s 4-point contours into an be presented. M-point contours. Picchi and Prati algorithm is thus named Point-Contour Stop-and-Go Algorithm (PC-SAGA). Based on III. POINT-CONTOUR STOP-AND-GO ALGORITHM the proposed framework in (12) or (13), we compute R R = The ﬁrst blind equalizer for multilevel PAM signals was RI = β for QAM as introduced by Sato [4]. In essence, it is identical to the β max[{|a |}] = max[{|aI |}] = decision-directed algorithm when the PAM input is binary √ R (±1). For M -level PAM signals, it is deﬁned by the error M − 1, for square QAM = 3 M − 1, for cross QAM. function (18) E[a2 ] 2 2 n ψG [yn ] = yn − R1 sgn[yn ], where R1 = . (14) IV. DIAMOND-CONTOUR STOP-AND-GO E[|an |] ALGORITHM The Sato algorithm was extended to complex signals (QAM) The ﬁrst complexity-efﬁcient (signed-error) blind equal- by Benveniste et al. [5] by separating signals into their real izer was proposed by Weerackody et al. [7]. The and imaginary parts as Weerackody-Kassam hard-limited algorithm (WK-HLA) is ψG [yn ] = yn,R − RR sgn[yn,R ] + (yn,I − RI sgn[yn,I ]) (15) speciﬁed by wn+1 = wn − µψG [yn ]x∗ , where ψG [yn ] is n computed as: where RR and RI are computed as E[a 2 ]/E[|an,R |] and n,R E[a2 ]/E[|an,I |], respectively. The resulting weight adap- ψG [yn ] = sgn[|yR,n | + |yI,n | − R] · (sgn[yR,n ] + sgn[yI,n ]) (19) n,I tation algorithm is called reduced constellation algorithm This algorithm attempts to drive the equalizer output to reside (RCA), as it attempts to resolve the output of the channel on a 45o rotated-square (diamond) contour as depicted in Fig. to belong to one of the four statistical symbols of a reduced 3(a). The ﬁnal equalizer output is obtained by the removal of constellation. Those four points are (R R , RI ), (RR , −RI ), this rotation. It has been reported in [7] that it performs better (−RR , −RI ) and (−RR , RI ). than CMA (if properly initialized). Kim et al. [8] observed Picchi and Prati [6] developed the ﬁrst ever stop-and-go that Weerackody-Kassam algorithm can be transformed into algorithm for blind equalization. They observed that a simple a stop-and-go algorithm. The ψ L [yn ] proposed in Kim’s decision-directed adaptation can open a closed eye, provided diamond-contour stop-and-go algorithm is as follows: the adaptation is stopped for the small proportion of the instances when the decision-directed and the RCA errors ψL [yn ] = sgn[|yR,n | + |yI,n | − |ˆR,n | − |ˆI,n |] a a have different sign. The weight adaptation rule is given as · (sgn[yR,n ] + sgn[yI,n ]) (20) Weerackody−Kassam’s where Rk is the square of the radii of the nearest constel- Kim et al. Diamond−Contour Hard−Limited Algorithm (1992) Stop−and−Go Algorithm (1996) 5 5 lation symbol for each equalizer output. To improve the RDE’s convergence, different techniques have been sug- 0 0 gested. In [12], the RDE is modiﬁed by incorporating stop- and-go ﬂags such that adaptation takes place only when the sign of ψG [yn ] = (|yn |2 − R) and ψL [yn ] = (|yn |2 − −5 −5 0 5 −5 −5 0 5 {D[yn ]}2 ) match. It was named Stop-and-Go Decision- (a) (b) Directed Multiple-modulus Algorithm (SAG-DDMMA). The Fig. 3. Contours in (a) WK-HLA (b) Kim’s DC-SAGA. resulting reliable regions are depicted in Fig. 4(c). In this paper, a straightforward “stop-and-go” version of CMA, similar to one given in [12], is proposed. The pro- Due to 45o rotation, the sliced symbol a n = aR,n + aI,n is ˆ ˆ ˆ posed Circular-Contour Stop-and-Go Algorithm (CC-SAGA) obtained differently as a n = D[yn ·e−π/4 ]·eπ/4 . The single ˆ is based on the framework described in Section II, and is ﬂag f is obtained by comparing the sign of [|y R,n | + |yI,n | − given as wn+1 = wn − µ · ψL [yn ] · f · x∗ , where f is 1 n |ˆR,n |−|ˆI,n |] and [|yR,n |+|yI,n |−R]. It should be noted that a a when the sign of ψ G [yn ] (22) and ψ L [yn ] (23) match, and 0 no closed form expression for the computation of R has been otherwise. The value of R in ψ G [yn ], is computed as reported in [8]. However, based on the proposed framework, the value of R (that maximizes the reliable regions) can be R = max[{Rk }] = max[{|a2 | + |a2 |}] R I obtained as: 2 · β2, square QAM 2 √ √ = 2 1 (24) R = 2 · max[{|aR |}, {|aI |}] = 2 · β (21) β− + β 2 , cross QAM 3 3 where β is obtained from (18). Fig. 3(b) depicts multiple where β is computed form (18). The proposed algorithm diamond contours generated by Kim’s stop-and-go algorithm. differ from [12] in the formulation of reliable regions; oth- Kim’s algorithm is thus named Diamond-Contour Stop-and- erwise, the weight adaptation process is similar. The reliable Go Algorithm (DC-SAGA). regions formed by the proposed algorithm are depicted in V. CIRCULAR-CONTOUR STOP-AND-GO Fig. 4(d). ALGORITHM Godard’s Algorithm [p=2] (1980) Ready−Gooch (1990) Constant Modulus Algorithm (1983) Radius−Directed Equalization The Godard algorithm [9] is one of the best known blind 5 5 equalization algorithms and is a stochastic gradient algorithm 1 for the cost function, J p = 2p E(|y(n)|p − R)2 , where p ∈ {1, 2, · · ·} and E denotes statistical expectation. The corre- 0 0 sponding algorithm is w n+1 = wn − µyn |yn |p−2 (|yn |2 − R)x∗ . The constant modulus algorithm (CMA) proposed in n (a) (b) [10] is a special case of Godard algorithm for p = 2. The −5 −5 0 5 −5 −5 0 5 corresponding algorithm is w n+1 = wn −µψG [yn ]x∗ , where n Cheolwoo−Hong Stop−and−Go Proposed Circular−Contour ψG [yn ] is given as Multimodulus Algorithm (1998) 5 5 Stop−and−Go Algorithm ψG [yn ] = yn (|yn |2 − R) (22) and R = E[|a(n)|4 ]/E[|a(n)|2 ]. CMA forces the equalizer 0 0 output yn to reside on a circular contour as depicted in Fig. (c) (d) 4(a). −5 −5 A decision-directed type CMA was proposed in [11], −5 0 5 −5 0 5 called radius-directed equalization (RDE), for QAM signals Fig. 4. (a) CMA, (b) RDE, (c) SAG-DDMMA, (d) Proposed CC-SAGA. based on the known modulus (circular contour) of the con- √ radii stellation symbol radii. For example, 16 QAM has three √ √ √ √ ( 2, √10 and √ √ 18); and 32 QAM has ﬁve radii ( 2, 10, VI. LINE-CONTOUR STOP-AND-GO ALGORITHM 18, 26 and 34). The algorithm uses the error between the equalizer output modulus and the nearest symbol radius The constant modulus algorithm (CMA) is blind to car- to update the equalizer weights, as depicted in Fig. 4(b). rier phase offset error. The CMA has been modiﬁed to It provides faster convergence than the CMA. However, the incorporate phase information, it resulted in multimodulus convergence of RDE is not guaranteed, as it operates totally algorithm (MMA). The multimodulus algorithm was pro- in decision-directed mode. RDE uses error function ψ L [yn ] posed independently by many authors [13], [14], [15]. This which is given as, algorithm minimizes the dispersion of real and imaginary parts, yR,n and yI,n , and forces them to lie on straight-line ψL [yn ] = yn (|yn |2 − Rk ) (23) contours. MMA error function ψ G [yn ] is given as ψG [yn ] = 2 2 2 2 yR,n (yR,n −RR )+ yI,n (yI,n −RI ). Recently, a complexity QAM constellations. Figure 6(a) depicts the square-contour efﬁcient MMA is proposed, named soft-constraint satisfac- generated by TW-GSCA for 16-QAM constellation. tion multimodulus algorithm (SCS-MMA) [16], [17]. The Thaiupathump−Kassam’s SCS-MMA error function ψ G [yn ] is expressed as Proposed Square−Contour Square−Contour Algorithm (2003) Stop−and−Go Algorithm 4 4 ψG [yn ] = yR,n (|yR,n | − RR ) + yI,n (|yI,n | − RI ) (25) 2 2 3 where RR = E[|aR | ]/E[a2 ] and RI = E[|aI | R 3 ]/E[a2 ]. I 0 0 The (straight) line-contours exhibited by SCS-MMA are −2 −2 shown in Fig. 5(a). In order to obtain a stop-and-go version of SCS-MMA, the dispersion constants R R and RI are −4 −4 −2 0 (a) 2 4 −4 −4 −2 0 (b) 2 4 a replaced with |ˆ R,n | and |ˆ I,n |, respectively, to get a new a error function ψ L [yn ] as follows Fig. 6. Contours in (a) TW-GSCA (b) Proposed SC-SAGA. a a ψL [yn ] = yR,n (|yR,n | − |ˆR,n |) + yI,n (|yI,n | − |ˆI,n |) (26) In order to obtain a stop-and-go version of TW-GSCA, The proposed Line-Contour Stop-and-Go Algorithm (LC- the dispersion constant R p in TW-GSCA are replaced with SAGA) can be expressed as w n+1 = wn −µ [ψL [yn ]R · fR + (|ˆR,n + aI,n | + |ˆR,n − aI,n |)p to get ψL [yn ] as follows a ˆ a ˆ ψL [yn ]I ·fI ] x∗ , where fR and fI are binary-ﬂags which n ψL [yn ] = {(|yR,n + yI,n | + |yR,n − yI,n |)p indicate the match of signs of the real and the imaginary parts −(|ˆR,n + aI,n | + |ˆR,n − aI,n |)p } a ˆ a ˆ of ψG [yn ] and ψL [yn ], respectively. ψG [yn ] and ψL [yn ] are · (|yR,n + yI,n | + |yR,n − yI,n |)p−1 computed from (25) and (26), respectively. For the proposed algorithm for QAM signals, R R and RI are computed as the · {sgn[yR,n + yI,n ] + sgn[yR,n − yI,n ] outermost line contour as follows: + (sgn[yR,n + yI,n ] − sgn[yR,n − yI,n ])} (29) RR = RI = max[{|aR |}] = max[{|aI |}] = β (27) The proposed Square-Contour Stop-and-Go Algorithm (SC- SAGA) can be expressed as w n+1 = wn − µ · f · ψL [yn ]x∗ , n where β is computed form (18). The reliable regions obtained where f is the binary-ﬂag which indicates the match of by the proposed algorithm are depicted in Figure 5(b) for 16- signs of ψG [yn ] and ψL [yn ]. For the proposed algorithm, the QAM. dispersion constant R is computed as the outermost square Proposed Line−Contour Stop−and−Go Algorithm contour as follows: Abrar et al. Soft Constraint Satisfaction Multimodulus Algorithm (2004) Reliable regions for yR R = 2 · max[{|aR |}, {|aI |}] = 2 · β Reliable regions for yI 4 4 4 (30) 2 2 2 where β is computed form (18). The reliable regions obtained 0 0 0 by the proposed algorithm are depicted in Figure 6(b) for 16- −2 −2 −2 QAM. −4 −4 −2 0 2 4 −4 −4 VIII. SIMULATION RESULTS −4 −2 0 2 4 −4 −2 0 2 4 (a) (b) (c) In this section, the performance of existing and pro- Fig. 5. Contours in (a) SCS-MMA and (b)-(c) Proposed LC-SAGA. posed stop-and-go algorithms are compared. In simulations, a complex-valued seven taps transversal equalizer was used and it was initialized so that the center tap was set to one VII. SQUARE-CONTOUR STOP-AND-GO and other taps were set to zero. The channel used in the ALGORITHM simulation was taken from [6]. The signal to noise ratio Recently, Thaiupathump and Kassam presented an inter- (SNR) was taken as 30dB at the input of the equalizer. esting family of generalized square-contour algorithms (TW- The residual ISI [19] and MSE are measured for 16-QAM GSCA) [18]. Unlike WK-HLA (as described in Section IV), signaling and compared as performance parameters. Each these algorithms are free from unnecessary 45 o rotation. The trace is the ensemble average of 200 independent runs with weight adaptation process is w n+1 = wn − µψG [yn ]x∗ , n random initialization of noise and data source. where ψG [yn ] is computed as, Fig. 7 depicts traces of residual ISI convergence. Examin- ing this result, we observe that PC-SAGA and DC-SAGA are ψG [yn ] = ((|yR,n + yI,n | + |yR,n − yI,n |)p − Rp ) relatively the fastest and the slowest converging algorithms, · (|yR,n + yI,n | + |yR,n − yI,n |)p−1 respectively. The SC-SAGA is next to the PC-SAGA in · {sgn[yR,n + yI,n ] + sgn[yR,n − yI,n ] performance. Fig. 8 depicts traces of MSE convergence, where we observe that PC-SAGA is consistently performing + (sgn[yR,n + yI,n ] − sgn[yR,n − yI,n ])} (28) the best and SC-SAGA is next to it. While the CC-SAGA (Refer to [18] for the computation of R p .) The beauty of is performing worst by giving nonconverging MSE ﬂoor this algorithm is that it forces the equalizer outputs to reside because of its incapability to remove (almost 45 o ) phase- on a square-contour; which is a desirable contour for square offset error (introduced by the channel). −5 [6] G. Picchi and G. Prati. “Blind Equalization and Car- −10 rier Recovery using a ‘Stop-and-Go’ Decision-Directed Algorithm”. IEEE Trans. Commun., COM-35:877–887, Residual ISI (dB) −15 1987. Diamond contour SAG [7] V. Weerackody and S.A. Kassam. A Simple Hard- −20 Point contour SAG Limited Adaptive Algorithm for Blind Equalization. Line contour SAG IEEE Trans. Circuit and Systems-II, 39(7):482–487, −25 Circular contour SAG July 1992. −30 Square contour SAG [8] Youngkyun Kim, Sungjo Kim, and Mintaig Kim. “The Derivation of a New Blind Equalization Algorithm”. −35 ETRI Journal, 18(2):53–60, July 1996. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iterations [9] D.N. 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