Blind start-up of a dual mode CAP-QAM receiver

A receiver has a dual mode of operation--a carrierless amplitude modulation/phase modulation (CAP) mode and a quadrature amplitude modulation (QAM) mode-yet only requires a single equalizer structure for both the CAP mode of operation and the QAM mode of operation during blind start-up. The receiver uses the same blind equalization updating algorithm independent of the type of received signal for converging the equalizer structure. The blind equalization updating algorithm incorporates a constant R, whose value is a function of the type of received signal, e.g., a QAM signal or a CAP signal. The type of received signal is determined as a function of the in-phase component of the mean-squared error, E[e.sup.2.sub.n.

CROSS-REFERENCE TO RELATED APPLICATIONS
 Related subject matter is disclosed in the co-pending, commonly assigned,
 U.S. Patent application of L. M. Garth, entitled "Automatic Constellation
 Phase Recovery in Blind Start-Up Of A Dual Mode CAP-QAM Receiver,".
 FIELD OF THE INVENTION
 The present invention relates to communications equipment, and, more
 particularly, to the use of blind equalization in a receiver.
 BACKGROUND OF THE INVENTION
 Carrierless amplitude modulation/phase modulation (CAP) is a
 bandwidth-efficient two-dimensional passband line code. (Additional
 information on a CAP communications system can be found in J. J. Werner,
 "Tutorial on Carrierless AM/PM-Part I-Fundamentals and Digital CAP
 Transmitter," Contribution to ANSI X3T9.5 TP/PMD Working Group,
 Minneapolis, Jun. 23, 1992.) CAP is closely related to the more familiar
 quadrature amplitude modulation (QAM) transmission scheme. In voiceband
 modems, QAM has been used for over 25 years, while CAP has been used for
 over 15 years. However, CAP is simpler to implement digitally.
 Illustrative prior art transceiver structures for the QAM and CAP
 transmission schemes are shown in FIGS. 1 and 2, respectively. Both FIGS.
 1 and 2 illustrate two-dimensional encoding where a complex symbol,
 A.sub.n, is applied to the transmitter portion (where A.sub.n =a.sub.n
 +jb.sub.n), and a recovered complex symbol, A.sub.n, is provided by the
 receiver portion, where A.sub.n =a.sub.n +jb.sub.n. With respect to other
 notation used in these FIGS., g(t) (e.g., see FIG. 1) is an impulse
 response of a baseband shaping filter, e.sub.in (t) and e.sub.qu (t) are
 equalizers for the in-phase and quadrature components, respectively, and
 p(t) and p(t) are impulse responses of a shaping filter which form a
 Hilbert pair (e.g., see FIG. 2).
 As can be observed from FIG. 1, the conventional QAM transceiver structure
 requires a modulator 1 and a demodulator 2 at the transmitter and
 receiver, respectively. In contrast, the CAP transceiver of FIG. 2 does
 not require a modulator and a demodulator. Generally speaking, the CAP
 system of FIG. 2 is not compatible with the QAM system of FIG. 1, but
 provides the same theoretical performance as QAM and is simpler to
 implement digitally than QAM. Indeed, the CAP structure of FIG. 2 can be
 modified into a simpler QAM-only transceiver, which is shown in FIG. 3.
 Currently, some broadband access applications, such as VDSL (Very high rate
 Digital Subscriber Line), may require either a CAP receiver or a QAM
 receiver. Some in the art have proposed simply putting both the CAP
 receiver and the QAM receiver into one receiver--in effect having a dual
 structure receiver with a CAP section (having its own equalizer) and a
 separate QAM section (with its own equalizer). To further complicate
 matters, this dual structure receiver may require the use of blind
 equalization techniques in both the QAM section and the CAP section. In
 this case, there is no training signal for the dual structure receiver to
 use to identify the type of modulation. As such, the dual structure
 receiver must first independently converge both the equalizer in the QAM
 section and the equalizer in the CAP section, and then make a decision as
 to the type of modulation--all of which may cause significant timing
 overhead.
 SUMMARY OF THE INVENTION
 We have developed a receiver that has a dual mode of operation--a CAP mode
 and a QAM mode-yet only requires a single equalizer structure for both the
 CAP mode of operation and the QAM mode of operation during blind start-up.
 In an embodiment of the invention, a receiver comprises an adaptive filter.
 The same blind equalization updating algorithm is used independent of the
 type of received signal. The blind equalization updating algorithm
 incorporates a constant R, whose value is a function of the type of
 received signal, e.g., a QAM signal or a CAP signal. The type of received
 signal is determined as a function of the in-phase component of the
 mean-squared error, E[e.sup.2.sub.n ]. As such, this receiver can be
 started blindly without knowing whether the received signal is a QAM
 signal or a CAP signal.

DETAILED DESCRIPTION
 An illustrative high-level block diagram of a portion of a communications
 system embodying the principles of the invention is shown in FIG. 4. For
 illustrative purposes only, it is assumed that receiver 10 receives either
 a CAP signal or a QAM signal. It is assumed that the CAP or QAM signal has
 been distorted while propagating through communications channel 9 and
 experiences intersymbol interference (ISI). The purpose of receiver 10 is
 to remove the ISI and minimize the effect of any additive noise .zeta.(t)
 to provide signal r'(t). The inventive concept will illustratively be
 described in the context of a receiver that has a dual mode of
 operation--a CAP mode and a QAM mode-that utilizes only a single equalizer
 for both the CAP mode of operation and the QAM mode of operation.
 However, before describing the inventive concept, some background
 information on adaptive filters is presented. Also, as used herein, an
 adaptive filter is, e.g., a fractionally spaced linear equalizer, which is
 hereafter simply referred to as an FSLE or, simply, an equalizer. Further,
 the term "single equalizer structure" encompasses an adaptive filter for
 equalizing a received signal. As known in the art, this equalizer
 structure, or adaptive filter, itself may comprise other filters for
 different components, or combinations, of the received signal. For
 example, a "single equalizer structure" is a cross-coupled equalizer,
 which comprises four filters, or a phase-splitting equalizer, which
 comprises two filters, etc.
 Adaptive Filters, CMA and MMA
 An illustrative phase-splitting FSLE equalizer 100 is shown in FIG. 5. It
 is assumed that FSLE equalizer 100 operates on an input signal that can be
 characterized as having N dimensions. In this example, N=2, i.e., the
 input signal comprises two component dimensions: an in-phase component and
 a quadrature component. (It should also be noted that the term "channel"
 is also used herein to refer to each dimension, e.g., the in-phase
 dimension is also referred to as the in-phase channel.) FSLE equalizer 100
 comprises two parallel digital adaptive filters implemented as finite
 impulse response (FIR) filters 110 and 120. Equalizer 100 is called a
 "phase-splitting FSLE" because the two FIR filters 110 and 120 converge to
 in-phase and quadrature filters. Some illustrative details of the
 equalizer structure are shown in FIG. 6. The two FIR filters 110 and 120
 share the same tapped delay line 115, which stores sequences of successive
 Analog-to-Digital Converter (A/D) 125 samples r.sub.k. The sampling rate
 1/T' of A/D 125 is typically three to four times higher than the symbol
 rate 1/T and is chosen in such a way that it satisfies the sampling
 theorem for real signals. It is assumed that T/T'=i, where i is an
 integer.
 The outputs of the two adaptive FIR filters 110 and 120 as shown in FIG. 6
 are computed at the symbol rate 1/T. The equalizer taps and input samples
 can be represented by a corresponding N-dimensional vector. As such, the
 following relationships are now defined:
EQU r.sub.n.sup.T =[r.sub.k,,r.sub.k-1,, . . . , r.sub.k-N, ]=vector of A/D
 samples in delay line; (1)
EQU c.sub.n.sup.T =[c.sub.0,,c.sub.1,,c.sub.2,, . . . , c.sub.N, ]=vector of
 in-phase tap coefficients; and (2)
 d.sub.n.sup.T =[d.sub.0,,d.sub.1,,d.sub.2,, . . . , d.sub.N, ]=vector of
 quadrature phase tap coefficients; (3)
 where the superscript T denotes vector transpose, the subscript n refers to
 the symbol period nT, and k=(i)(n).
 Let y.sub.n and y.sub.n be the computed outputs of the in-phase and
 quadrature filters, respectively, and:
EQU y.sub.n =c.sub.n.sup.T r.sub.n ; and (4)
EQU y.sub.n =d.sub.n.sup.T r.sub.n. (5)
 An X/Y display of the outputs y.sub.n and y.sub.n or, equivalently, of the
 complex output Y.sub.n =y.sub.n +jy.sub.n, is called a signal
 constellation. After convergence, ideally the signal constellation
 consists of a display of the complex symbols A.sub.n =a.sub.n +jb.sub.n
 corrupted by some small noise and ISI.
 Referring back to FIG. 5, FSLE equalizer 100 can be characterized as having
 two modes of operation, a normal mode (steady state) and a start-up mode
 (non-steady state). In the normal mode of operation, the decision devices,
 i.e., slicers 130 and 135, compare the equalizer complex output samples,
 Y.sub.n, (where Y.sub.n =y.sub.n +jy.sub.n) with all the possible
 transmitted complex symbols, A.sub.n (where A.sub.n =a.sub.n +jb.sub.n),
 and select the symbol A.sub.n which is the closest to Y.sub.n. The
 receiver then computes an error, E.sub.n, where:
EQU E.sub.n =e.sub.n +je.sub.n =Y.sub.n -A.sub.n, (6)
 which is used to update the tap coefficients of equalizer 100. The most
 common tap updating algorithm is the LMS algorithm, which is a stochastic
 gradient algorithm that minimizes the mean square error (MSE), which is
 defined as:
EQU MSE.DELTA.E[.vertline.E.sub.n.vertline..sup.2 ]=E[.vertline.Y.sub.n
 -A.sub.n.vertline..sup.2 ]=E[e.sub.n.sup.2 ]+E[e.sub.n.sup.2 ]. (7)
 In equation (7), E[.multidot.] denotes expectation and e.sub.n and e.sub.n
 are the following in-phase and quadrature errors:
EQU e.sub.n =y.sub.n -a.sub.n, and (8)
EQU e.sub.n =y.sub.n -b.sub.n. (9)
 The tap coefficients of the two adaptive filters are updated using the
 above-mentioned least-mean-square (LMS) algorithm, i.e.,
EQU c.sub.n+1 =c.sub.n -.alpha.e.sub.n r.sub.n, and (10)
EQU d.sub.n+1 =d.sub.n -.alpha.e.sub.n r.sub.n, (11)
 where .alpha. is the step size used in the tap adjustment algorithm.
 In contrast to the steady state mode of operation, the start-up mode is
 used to converge the tap coefficient values to an initial set of values.
 In some systems a training sequence is used during start-up (i.e., a
 predefined sequence of A.sub.n symbols), from which the receiver can
 compute meaningful errors E.sub.n by using the equalizer output signal
 Y.sub.n and the known sequence of transmitted symbols A.sub.n. In this
 case, tap adaptation is said to be done with respect to an "ideal
 reference."
 However, when no training sequence is available, equalizer 100 has to be
 converged blindly. This usually comprises two main steps. First, a blind
 equalization algorithm is used to open the "eye diagram." Then, once the
 eye is open enough, the receiver switches to, e.g., the above-described
 LMS tap adaptation algorithm. The philosophy of blind equalization is to
 use a tap adaptation algorithm that minimizes a cost function that is
 better suited to provide initial convergence of equalizer 100 than the MSE
 represented by equation (7).
 As known in the art, there are three general techniques for blind
 equalization: one is referred to herein as the "reduced constellation
 algorithm" (RCA) (e.g., see Y. Sato, "A Method of Self-Recovering
 Equalization for Multilevel Amplitude-Modulation Systems," IEEE Trans.
 Commun., pp. 679-682, June 1975; and U.S. Pat. No. 4,227,152, issued Oct.
 7, 1980 to Godard); another technique is the so-called "constant modulus
 algorithm" (CMA) (e.g., see D. N. Godard, "Self-Recovering Equalization
 and Carrier Tracking in Two-Dimensional Data Communications Systems," IEEE
 Trans. Commun., vol. 28, no. 11, pp. 1867-1875, Nov. 1980; and N. K.
 Jablon, "Joint Blind Equalization, Carrier Recovery, and Timing Recovery
 for High-Order QAM Signal Constellations", IEEE Trans. Signal Processing,
 vol. 40, no. 6, pp. 1383-1398, 1992); and the final technique is referred
 to as the "multimodulus algorithm" (MMA) (e.g., see Yang, J. and Werner,
 J. J., The Multimodulus Blind Equalization Algorithms, Proceedings of
 DSP97, Santorini, Greece, 1997).
 Dual Mode CAP-QAM Receiver
 We have developed a receiver that has a dual mode of operation--a CAP mode
 and a QAM mode-yet only requires a single equalizer structure for both the
 CAP mode of operation and the QAM mode of operation during blind start-up.
 In accordance with the inventive concept, a portion 200 of a dual-mode
 receiver for CAP and QAM, such as receiver 10 of FIG. 4, (also referred to
 herein as a CAP-QAM receiver) is shown in FIG. 7. Other than the inventive
 concept, the elements shown in FIG. 7 are well-known and will not be
 described in detail. CAP-QAM receiver portion 200 comprises rotator 215,
 filter updating elements 220 and 225, slicers 230 and 235, element 240,
 and a single phase-splitting FSLE as represented by FIR filters 205 and
 210. CAP-QAM receiver portion 200 illustrates the tap updating of FIR
 filters 205 and 210. As can be observed from filter updating elements 220
 and 225 of FIG. 7, CAP-QAM receiver portion 200 uses the same filter
 updating algorithm independent of the type of received signal. For the
 purposes of this example, it is assumed that the MMA blind equalization
 algorithm is used. However, the inventive concept is not so limited and
 other forms of blind equalization can be used, such as, but not limited
 to, RCA and CMA (described below). As described further below, the value
 of the constant R (shown in filter updating elements 220 and 225, and
 described further below) is adjusted as a function of the type of received
 signal. As such, this receiver can be started blindly without knowing
 whether the received signal is a QAM signal or a CAP signal.
 Reference now should also be made to FIG. 8 which shows an illustrative
 flow diagram of a method embodying the principles of the invention for use
 in a CAP-QAM receiver, such as the structure of FIG. 7. In step 505, the
 receiver initially starts in the CAP mode of operation. A variable,
 S(.theta.), is set equal to zero and the value for the constant R is
 initially set to R.sub.C (described below). In step 510, a decision is
 made as to whether a CAP signal or a QAM signal is being received by
 comparing the corresponding in-phase component of the mean-squared error,
 E[e.sub.n.sup.2 ],to a predetermined MSE value, M (which is determined
 experimentally and depends upon the signal constellation). (The in-phase
 error, e.sub.n, (developed by element 240 of FIG. 7) is computed and a
 running average is determined over a number of time periods.) Step 510 of
 the blind start-up procedure can be schedule-driven, event-driven, or
 both. For example, a schedule-driven approach occurs after some fixed
 number, Z, of iterations (which can be determined by a counter, for
 example). If the value of E[e.sub.n.sup.2 ] is larger than the value of M
 then it is assumed that a QAM signal is being received and execution
 proceeds to steps 515 and 520. In step 515, a variable, S(.theta.), is set
 equal to one, and, in step 520, the value of R is set equal to R.sub.Q
 (described below). On the other hand, if the value of E[e.sub.n.sup.2 ] is
 less than the value of M, then it is assumed that a CAP signal is being
 received and execution proceeds to steps 545 and 550. In step 545, the
 variable, S(.theta.), is set equal to zero. In step 550, the value of R is
 set equal to R.sub.C (described below). Once the value of R has been
 selected, the filter coefficients (in this example, FIR filters 205 and
 210 of FIG. 7) are updated in step 525. (It should be obvious to those
 skilled in the art that the flow chart of FIG. 8 can be further simplified
 when step 510 determines a CAP signal is being received, e.g., by the
 elimination of step 545, etc.).
 As described above, a variable, S(.theta.) is either set to zero or one.
 This variable controls the operation of rotator 225 of FIG. 7. When
 S(.theta.) is equal to zero, there is effectively no rotation of the
 filter, or equalizer, output signal Y.sub.n, i.e., Y.sub.n ' is equal to
 Y.sub.n. In this case, the in-phase error, e.sub.n, is equal to y.sub.n
 -a.sub.n. However, when S(.theta.) is equal to one, the equalizer output
 signal, Y.sub.n, is demodulated by:
EQU Y.sub.n '=Y.sub.n e.sup.j.omega..sup..sub.c .sup.nT,or (12)
EQU Y.sub.n '=[y.sub.n cos(.omega..sub.c nT)]+j[y.sub.n cos(.omega..sub.c
 nT)-y.sub.n sin(.omega..sub.c nT)], (13)
 where Y.sub.n '=y.sub.n '+jy.sub.n ', and Y.sub.n =y.sub.n +jy.sub.n. In
 this case, the in-phase error, e.sub.n, is equal to y.sub.n '-a.sub.n.
 As noted above, filter updating elements 220 and 225 of FIG. 7 use the same
 filter updating algorithm (here, illustratively MMA-based) independent of
 the type of received signal. The tap updating algorithms are:
EQU c.sub.n+1 =c.sub.n -.alpha.(y.sub.n.sup.2 -R.sup.2)y.sub.n r.sub.n,and
 (14)
EQU d.sub.n+1 =d.sub.n -.alpha.(y.sub.n.sup.2 -R.sup.2)y.sub.n r.sub.n. (15)
 It should be noted that in the case of receiving a QAM signal additional
 phase compensation may be required. In this case, Y.sub.n '=Y.sub.n
 e.sup.-j[.omega..sup..sub.c .sup.nTS(.theta.)+.phi..sup..sub.n .sup.],
 where .phi..sub.n is obtained through conventional phase recovery
 circuitry used in QAM. (Also, see the above-referenced co-pending commonly
 assigned, U.S. Patent application of L. M. Garth, entitled "Automatic
 Constellation Phase Recovery in Blind Start-Up Of A Dual Mode CAP-QAM
 Receiver.")
 Other illustrative embodiments of the inventive concept are shown in FIGS.
 9 and 10 for use in receiver 10 of FIG. 4. FIG. 9 illustrates an
 embodiment representative of a digital signal processor 700 that is
 programmed to implement an FSLE in accordance with the principles of the
 invention. Digital signal processor 700 comprises a central processing
 unit (processor) 705 and memory 710. A portion of memory 710 is used to
 store program instructions that, when executed by processor 705, implement
 CAP-QAM blind equalization (described above). This portion of memory is
 shown as 711. Another portion of memory, 712, is used to store tap
 coefficient values that are updated by processor 705 in accordance with
 the inventive concept It is assumed that a received signal 704 is applied
 to processor 705, which equalizes this signal in accordance with the
 inventive concept to provide a output signal 706. For the purposes of
 example only, it is assumed that output signal 706 represents a sequence
 of output samples of an equalizer. (As known in the art, a digital signal
 processor may, additionally, further process received signal 704 before
 deriving output signal 706.) An illustrative software program is not
 described herein since, after learning of the CAP-QAM blind equalization
 as described herein, such a program is within the capability of one
 skilled in the art. Also, it should be noted that any equalizer
 structures, such as that described earlier, can be implemented by digital
 signal processor 700 in accordance with the inventive concept.
 FIG. 10 illustrates another alternative embodiment of the inventive
 concept. Circuitry 600 comprises a central processing unit (processor)
 605, and an equalizer 610. The latter is illustratively assumed to be a
 phase-splitting FSLE as described above. It is assumed that equalizer 610
 includes at least one tap-coefficient register for storing values for
 corresponding tap coefficient vectors (e.g., as shown in FIG. 6).
 Processor 605 includes memory, not shown, similar to memory 710 of FIG. 9
 for implementing CAP-QAM blind equalization. Equalizer output signal 611,
 which represents a sequence of equalizer output samples, is applied to
 processor 605. The latter analyzes equalizer output signal 611, in
 accordance with the inventive concept, to adapt values of the tap
 coefficients in such a way as to converge to a correct solution.
 A blind start-up procedure in accordance with the principles of the
 invention for use in receiver 10 of FIG. 4 is shown in FIG. 11. In step
 580, receiver 10 uses CAP-QAM blind equalization with its corresponding
 tap updating algorithms to begin blind convergence of an equalizer, e.g.,
 steps 505 to 525 of FIG. 8 and equalizer 610 of FIG. 10. In step 585, a
 decision is made whether to switch from CAP-QAM blind equalization to the
 LMS adaptation algorithm or to continue using CAP-QAM blind equalization
 to converge the equalizer. Typically, this is referred to in the art as
 determining if the eye is open enough (as noted above). Step 585 of the
 blind start-up procedure can be schedule-driven, event-driven, or both.
 With a schedule-driven approach, the switch between two different tap
 updating algorithms occurs after some fixed number, K, of iterations
 (which can be determined by a counter, for example). This approach
 presumes a certain amount of eye-opening after K iterations. With an
 event-driven approach, the switch occurs when a certain quality of eye
 opening is achieved. This can be done, for example, by continuously
 monitoring the MSE and making the switch when the MSE is below some
 threshold S. If the eye has been opened enough, receiver 10 switches to
 the LMS Adaptation algorithm in step 590. (It should be noted that the
 focus of the inventive concept is the use of a single equalizer during
 blind start-up. As such, once the transition to the LMS algorithm occurs,
 further updates occur as known in the art. For example, in the QAM mode of
 operation, an additional rotator may have to be used when updating the
 equalizer using LMS.)
 As described further below, and in accordance with the principles of the
 invention, the value of the constant R (shown in filter updating elements
 220 and 225) is adjusted as a function of the type of received signal. In
 particular, the value of R is either
EQU R=R.sub.Q,or (16)
EQU R=R.sub.C. (17)
 The value of R is also affected by what type of blind equalization method
 is being used. Below, values for R are given for the CMA, MMA, and RCA,
 blind equalization algorithms. (Additional information on these blind
 equalization algorithms can be found in the above-cited articles.)
 CMA
 The CMA algorithm for a CAP signal minimizes the following cost function
 (CF):
EQU CF=E[(.vertline.Y.sub.n.vertline..sup.L -R.sup.L).sup.2 ], (18)
 where L is a positive integer, Y.sub.n are the equalized samples, and R is
 the radius of a circle. The case L=2 is the most commonly used in
 practice. The cost function in equation (18) is a true two-dimensional
 cost function which minimizes the dispersion of the equalizer complex
 output signal Y.sub.n with respect to a circle with radius R.
 In accordance with the principles of the invention, it can be shown that
 for the CMA blind algorithm, the CAP-QAM dual mode receiver has the same
 filter updating algorithm as that of a CAP-only equalizer. As such, when
 the CMA algorithm is used,
EQU R.sub.Q =R.sub.C =R. (19)
 Consequently, assuming a perfectly equalized channel the following value
 for R results:
 ##EQU1##
 where the expression on the right holds for the usual case where the
 statistics of the symbols a.sub.n and b.sub.n are the same. As note above,
 typically, L=2.
 MMA
 As noted earlier, the MMA algorithm was used in the illustrative example of
 FIG. 7. The MMA algorithm for a CAP signal minimizes the following cost
 function:
EQU CF=E[(y.sub.n.sup.L -R.sup.L (Y.sub.n)).sup.2 +(y.sub.n.sup.L -R.sup.L
 (Y.sub.n)).sup.2 ], (21)
 where L is a positive integer and R(Y.sub.n) and R(Y.sub.n)take discrete
 positive values, which depend on the equalizer outputs Y.sub.n. The MMA
 algorithm minimizes the dispersion of the equalizer output samples y.sub.n
 and y.sub.n around piecewise linear in-phase and quadrature contours.
 For square constellations, R(Y.sub.n)=R(Y.sub.n)=R=constant, so that the
 cost function of equation (21) becomes:
EQU CF=CF.sub.i +CF.sub.q =E[(y.sub.n.sup.L -R.sup.2).sup.2 +(y.sub.n.sup.L
 -R.sup.L).sup.2 ]. (22)
 Unlike the cost function for CMA represented by equation (18), equation
 (22) is not a true two-dimensional cost function. Rather, it is the sum of
 two independent one-dimensional cost functions CF.sub.i and CF.sub.q. For
 L=2, the cost functions of MMA can be represented as:
EQU CF=E[(y.sub.n.sup.2 -R.sup.2).sup.2 +(y.sub.n.sup.2 R.sup.2).sup.2 ]. (23)
 As such, the values for R.sub.C and R.sub.Q are different.
 For a CAP signal, assuming a perfectly equalized channel, the following
 value for R=R.sub.C results:
 ##EQU2##
 If equation (23) is applied to a QAM signal, in accordance with the
 principles of the invention, it can be shown that the following value for
 R=R.sub.Q results:
 ##EQU3##
 From equation (25), it can be observed that for receiving a QAM signal, the
 constant R is not only a function of the transmitted symbols, but also of
 the angle .theta.=.omega..sub.c T, where .omega..sub.c is the radian
 carrier frequency and T is the symbol period. The constant R can be
 numerically computed from equation (25). However, the constant R can also
 be expressed as a function of the number m of symbol levels. As such,
 equation (25) it can be shown that equation (25) can be alternatively
 expressed as:
EQU R.sub.Q.sup.2 =R.sup.2 =1/54m.sup.2 (3+4.beta.)+4.beta.-7], (26)
 where .beta.=cos.sup.2 +L .omega..sub.c +L nTsin.sup.2 +L .omega..sub.c +L
 nT. From equation (26), it can be observed that R is a function of m and
 .beta. with .omega..sub.c =0.fwdarw..theta.=0.fwdarw..beta.=0, this leads
 equation (26) to become:
EQU R.sub.Q.sup.2 =R.sup.2 =1/5(12m.sup.2 -7), (27)
 which can be shown to be equivalent to the same value as for MMA with a CAP
 signal.
 The constant R can be computed from either equation (25) or equation (26).
 Since R is dependent on .theta., R can be different even for the same
 constellation. For instance for 16-QAM with .theta.=3/5, .function.c=15.55
 MHz and 1/T=25.92 MHz, we obtain .beta.=0.1. Illustrative values of the
 constant R for CAP and QAM signals are shown in Table One, below for
 different signal point constellations.
 TABLE ONE
 Parameter/Point 16-Point 64-Point 256-Point 1024-Point
 m 2 4 8 16
 R.sub.C 2.86 6.08 12.34 24.76
 R.sub.Q 3.1 6.5 13.1 26.36
 RCA
 The CAP-QAM receiver for RCA is similar to the one for MMA. The cost
 function of RCA is:
 ##EQU4##
 For a CAP signal, assuming a perfectly equalized channel, the following
 value for R=R.sub.C results:
 ##EQU5##
 If equation (28) is applied to a QAM signal, in accordance with the
 principles of the invention, it can be shown that the following value for
 R=R.sub.Q results:
 ##EQU6##
 The numerator of equation (30) can be simplified to E[a.sub.n.sup.2 ].
 However, the denominator of equation (30) is difficult to simplify because
 the expression involves absolute values. So equation (30) can be rewritten
 as:
 ##EQU7##
 In this case, a numerical solution is required to compute R for each
 application. Illustratively, for a 16-QAM constellation, R.sub.Q =2.6.
 The foregoing merely illustrates the principles of the invention and it
 will thus be appreciated that those skilled in the art will be able to
 devise numerous alternative arrangements which, although not explicitly
 described herein, embody the principles of the invention and are within
 its spirit and scope. For example, although the inventive concept was
 illustrated herein as being implemented with discrete functional building
 blocks, e.g., equalizer 610, etc., the functions of any one or more of
 those building blocks can be carried out using one or more appropriately
 programmed processors or processing circuitry, e.g., a digital signal
 processor; discrete circuit elements; integrated circuits; etc. In
 addition, although the inventive concept was described in the context of a
 single phase-splitting equalizer, other forms of equalizers can also be
 used.