Generalized Partial Response for Equalized ... - Semantic Scholar

IEEE TRANSACTIONS ON COMMUNICATIONS,

VOL. C O M - ~ ~NO. ,

11,

NOVEMBER

1975

1251

Generalized Partial Response for Equalized Channels with Rational Spectra DAVID G. MESSERSCHMITT,

Abstract-Harashima and Miyakawa [I] and Tomlinson [2] have described a generalized partial response technique which achieves the performance of the decision-feedback equalizerwithout the error propagation problem. We show here that when the equalized and baud-rate sampled channel assumes the special rational z transform M

g;zi

i-0

F(2) = , l , L k l some M , the remaining f m are integers, and a = - 1. There feedback of Fig. 1( b ) (where past binary decisions are is then no peak power penalty a t the transmitter.

+

+

+

(" + +

-.

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IEEE TRANSACTIONS ON COMMUNICATIONS, NOVEMBER

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digits (Bk-1, * * ,Bk-N) and N past values Of Lk, (Lk-1,- * * , Lk-N). It thendetermines Lk andthe precoded data sample x k by the standard modulo p operation using its knowledge of the ( di}. Implementation becomes especially easy in the binary case when the coefficients ( d i ) are chosen judiciously. Let there be integers L, li such that

mind that implementation difficulties in resolving the multilevel received levels compound as L increases. 111. GPR FOR AUTOREGRESSIVE EQUALIZEDCHANNELS Let F ( Z ) be of the autoregressive form

F(2)

=

1

,

~

& = l

li d. = -

N

diZi

l

i=o

L'

O 0. By

1 < 1,

I 1 + A

the method of (9) , we have

(a) B, = I

I-----,

0, = - I r----1

However, we know that in this case rk = f1, and the bound of (29) becomes progressively weaker as A -+ 1. Determining the actual range of L k is ‘quite straightforward, a t least inprinciple. Define the state of the system at time k as Sk

=

(Lk-i,Bk-i, 1 5 i

IN ).

Then, based on (6) and (13), the new state defined function of S k and Bk,

Sk+l

=

f(Bk,Sk).

(30) Sk+l

(31)

-

= Lk =

0,

IC

I

is a well

We can divide the states into mutually exclusive groups corresponding to each possible (Bk-1,* ,BL-N),as illustrated for the binarycase with N = 2 by the four boxes in Fig. 4 ( a ) . When the state is updated, depending on which of the two values of Bk occurs, a state in one of the two new boxes will result. The movement of states between boxes as a functionof Bk is shown in Fig.4(a). Themotion of the system is also determined by the initial condition, which we specify as

Bk

r

< 0.

-1

I

(32)

Based on ( 3 2 ) , the state SN is determined solely by the data sequence (B N - - l , -,Bo)and not by any past history of the system; that is, there is only one state SN for each of the boxes in Fig. 4 ( a ) . These initial states are shown in Fig. 4(b), and marked with a LLzero’’ to indicate that they are initialconditions. Now, as illustrated in Fig. 4 (c) , for each initial “zero” state there are two new states S N +generated ~ in boxes determined by Fig. 4 ( a ) . One or both of these new states may, in fact, be an already existing state, as illustrated by the transition from the “zero” state in the lower right box to the“zero” state in thelower left box corresponding t o Bt = 1. All the truly new states are marked bya “one” to indicate the first iteration. At the next iteration, illustrated in’Fig. 4 ( d ) , transitions from all the states marked “one” are determined. I n this case, all the transitions end in already existing states. This must happen eventually

-

(dl of Fig. 4. Determining number of received levels.(a)Movement states when Bk = 1 or B k = -1. (b) Initial states. (c) Generation of new states. (d) Termination of states.

(although usually notatthe second iteration!), since the total number of states is finite, and we cannot continue to generate new states indefinitely. The result is a complete and minimal set of states for the system, and the largest L k can be determined accordingly. The extension of this Markov chainapproach to the systemwithtransmission zeros in (21) is possible, but

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only at .the expense of a 'far greater number of states. Since L k is chosen such that

10 log

2Rod N".

~

(37)

where u2 is given by (26). When N = 0 (the zero forcing equalizer case), (37) is 30.3 d B for y = 60 dB. The - - mini_ _ mum value of (37) which is attainable by GPR ---is the @'for /3 = 4,it follows that value fordecisionrKdback the _equalizer, and is 19.9 d B j z s j when y = 60 dB [ll]. Thus, there is a potential 10 dB -2L 1 5 mk 5 2L. performance advantage t o be gained by GPR. Thus the number of possible sequences (mkPL, ,mkPM) When the A of (36) was chosen optimally, it was found is finite,andthestate of (30) canbeaugmentedtothattheresulting L,,,.* was excessive. Therefore, it is necessary t o sacrifice some noise advantage to limit the " = (Lk--i,Bk--i,l 5 5 N 7 m k - i , 1 5 5 M , ' (33) number of received levels. The numericalresults are Thenumber of statesremainsfiniteand(31)remains shown inFig. 5, where therange of received levels was valid, SO thatthe remainder of the analysisproceedsdetermined byprogrammingthealgorithm of Section without major modification. V. The regions of fixed L,,., delimited are dashed by lines. It should also bestraightforward,although we have It is also interestingtocompare GPR with Class I not attempted to do so, to determine exactly the peak P R [5] which has been shown to have a noise advantage5 and average transmitted power for our autoregressive on coaxial cable [SI, [SI. The attractive feature of Class GPR case by exploiting the finite state structure just I PR is that it has no peak or average transmitted power-. described. penalty. As s h o w n x g . 5 , Class I PR a has 5 d B adi vantage over zero forcing equalization(wefoundClass VI. APPLICATION TO THE fl CHANNEL I1 to have a virtually identical advantage). I n this section we will determine the reduction in noise The results of Fig. 5 are summarized below. enhancement which can be obtained on the fl channel 1) Asubstantial ( 5 dB)advantage is obtainedby characteristic of coaxial cables and some wire pairs. The GPRwithoutany increase in thenumber of received levels (L,,., = 0). However,thissameadvantagecan frequency response is given by [ll] be obtained with Class I PR, with no transmitted power. I H ( w ) l2 = 2rK2Roexp ( - 2 K ~ l ' ~ ) (34) penalty, and three received kevels for binary transmission. N-2TF .. . .~ ----- .--- - -. - - .. .. .. 4 or L,,, 2 1, theadvantage of GPR increases where Rois the energy in an isolated pulse, K is a constant proportional to the line length, and transmitted impulses with the number of poles ( N ) , which is related to the ___are assumed. If we define y to be the loss of the linea t complexity of thetransmitter. v3, I L . 1 , . ._;- !the half-baud rate in decibels, then 3) About 9 of the potential 10 dB advantage.can be achieved with L,,, = 1 or 2 . However, if GPR is penalized for its maximumpossible increase in peak transmitted power, i t has anadvantage overClass I P R only for multilevel (but not binary) Finally, the reader is cautioned that those results apply A . Optimum Pole Placement only t o coaxial cable with y = 60 dB. In particular, on Theminimization of (26) wasaccomplished for an channels with a more __ .-modest roll-off with f r e q u e n c y , j y , l , autoregressive F ( 2 ) using an iterative optimization . -.._ (not advantage). \ program. The program was initially run for y = 60 and Class I PR often incurs a noise psnalty

c.

+

---

7

1

~

I

_ I _ L

I

_

I , -

~

2

with one to four complex poles constrained t o lie outside the unit disk; the result in eachcase was a set of real and equal poles. While this may be a local rather than global pinimum,the performance figures to follow indicate that it is a good solution in the sense that the maximum attainable improvement is approached rapidly. The resulting equalized channel is given by F(2)

=

(1 - AZ)-N

(36)

where A-' is the pole location,

7

-

A matched filter output has noise variance No/2Ro(when the signal is normalized to unit height), so that the noise penalty relative to the matchedfilter bound for an isolated pulse is given by

VII. CONCLUSIONS Wehave describedamodification of GPR whichis valid for an overall equalized channel with rational spectrum and which eliminates the necessity of storing analog samples in the transmitter. This technique can come close t o achievingtheimprovementin noise penalty of the decision-feedback equalizer without the error propagation problem of that receiver. The inevitable price that is paid for this benefit is 1) apotentialdoubling of the peak transmitted voltage in -the binary case and 2 ) (usually) an increase in the number of received levels. The technique can be applied to channels with a moderate amount This noise advantage for optimum receiver equalizationis easily evaluated by letting F(2) = 1 2 in (26). Harashima and Miyakawa [16] have previously shown that on coaxial cable a large number of transmitting levels is optimum with GPR.

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IEEE TRANSACTIONS ON

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0 I-

W

I? 0

z 0.001 ,

0.1

0.01

0.2

0.3 0.4 0.5 0.6

1.0

POLE, A

Fig. 5.

Noise penalty versus pole position, N equal real poles, y K.

of timevariationbymakingthe receiverequalization adaptive. Application t o a coaxial cable channel reveals that in this case GPR can yield asubstantialadvantageover zero-forcing equalization exclusively at the receiver, but that if GPR is penalized for the upper bound on its increase inpeaktransnlittedpower,it is moreattractive than Class I PR onlyformultilevel (butnotbinary) data. Decision-feedback equalization remains an attractivealternative, since it incursnoincreasein transmitted power or number of received decision levels, and for random data its error propagation can be controlled by limiting the number of feedback taps [13]. ACKNOWLEDGMENT The authoris indebted t o Dr. R. Price formany valuable comments on the manuscript. REFERENCES [I] H. Harashima and H. Miyakawa, “Matched-transmission technique for channels with intersymbol interference,” I E E E Trans. Commun. (Special Issue on Communications in Japan), vol. COM-20, pp. 774-780, Aug. 1972. [2] M. Tomlinson,“New automatic equalizer employing modulo arithmetic,” Electron. Lett., vol. 7, Mar. 1971. [3] A. Lender, “Theduobinary technique for high-speed data transmission,” I E E E Trans. Commun. Electron., vol. 82, pp. 214-218, May 1963. [4] ---, “Correlative level coding for binary-data transmission,” I E E E Spectrum, vol. 3, pp. 104-115, Feb. 1966. [5] E. R. Kretzmer, “Generalization of a technique for binary data communication,” I E E E Trans. Commun. Technol. (Concise Papers), vol. COM-14, pp. 67-68, Feb. 1966. [6] A. M. Gerrish and It. D. Howson, “Multilevel partial response signaling,” presented a t the 1967 Int. Conf. Commun., Minneapolis, Minn., June 1967. [7] D. W. Tuftsand C. V. Ramamoorthy, “Modulo-m linear sequential circuits, partial response signaling formats,and signal-flow graphs,”Div. Eng.and Appl. ’Physics, Harvard Univ., Cambridge,Mass.,Tech. Rep. 519 (AD-650958), Feb. 1967. IT [8] T. Ericson, “Optimum PAM-filters and someapplications,” Telecommunication Theory, Royal Inst. Technol., Stockholm, Sweden, Tech. Rep. 50, pp. 27-29, May 1972. [9] S. A. .Fredricsson, “Joint optimization of transmitter and re-

1

-=

=

60 dB.

ceiving filters indigitalPAM-systems with a Viterbi detector,” TelecommunicationTheory,Royal Inst. Technol.,Stockholm, Sweden, Tech. Rep. 37, 89, pp. 20-21, Oct. 1974.,, [lo] L. A. McColl, ‘‘Signaling methodandapparatus, u. s. Patent 2 056 284, Oct. 1936. [11] D. G. Messerschmitt, “A geometric theory of intersymbol i n t e r f e r e n c e p a r t 11: Performance of the maximum likelihood detector,” Bell Syst. Tech. J., vol. 52, pp. 1521-1539, Nov. 1973. [12] D. L. Duttweiler, J. E. Mazo, and D. G. Messerschmitt, “An upperbound on theerrorprobability in decision-feedback equalization,,, I E E E Trans. InfornL. TTheory, vel. IT-z0, pp, 490-497, July 1974. [13] D. G. Messerschmitt, “Design of a finiteimpulse response for the Viterbialgorithm and decision-feedback equalizer,” presented attheInt. Conf. Commun., Minnemolis.Minn..June 1974. [14] R. Price,“Nonlinearly fyedback-equalized PAM vs. capacity for noisy filter channels, presented at the 1972 Int. Conf. Commun., Philadelphia, Pa., June 1972. [15] D. G. Messerschmitt, “A geometric theory of intersymbol i n t e r f e r e n c e p a r t I: Zero-forcing and decision feedback equalization,” Bell Syst. Tech. J., vol. 52, pp. 1483-1519, Nov. 1973. [16] H. Harashima and H. Miyakawa, “Capacity of channels with matched transmission technique for peak transmitting power limitation,” in Nat. Conv.Ree. Inst. Electron. Commun. Eng. Japan, no. 1268, A ~ g , ~ l 9 6(English 9 translation available from theauthor)” i .E . \ C S > ~ ~ ~ . ~ C ! ~ L : . ~ ~ % [17] D. D. Falconer and F. R. Magee, Jr., “Adaptive channel memory truncation for likelihood sequenceestimation,” Bell Syst. Tech. J . , vol. 52, pp. 1541-1.562, Nov. 1973. *

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