Absfruct-The advantages with regard to noise enhancement of certain suboptimum nonlinear techniques for "equalization" of linear distortion on channels used for digital data transmission have recently received considerable attention. These nonlinear schemes fall into two classes: decision feedback and nonlinear precoding. Yet each of these methods has an associated peculiarity which has impeded attempts to give f i i bounds on performance. Thus, for decision feedback error propagation has caused the analytical pains, while for nonlinear precoding of an L-level alphabet the unknown increase in transmitter power has been culpable. The former problem has recently received successful attention by one of the present authors and some colleagues. Here we address the concomitant problem for nonlinear precoding and its extension to quadrature amplitude modulation (QAM), and show that the transmitted normalized power P satisfies (L2 -1)/3 < P < (L2 -1)/3 + 1. INTRODUCTION IYAKAWA AND HARASHIMA [ 13 and Tomlinson [2] have suggested a nonlinear data precoding scheme for eliminating intersymbol interference caused by past data symbols in pulse-amplitude modulation (PAM) digital communication systems.l Partial response techniques [3] , [4] constitute a special case. Price [5] optimized such a system operating over a noisy frequency attenuating channel, while implementation simplifications have been considered by Messerschmitt [6]. ' The purpose of this paper is to examine this precoding scheme and its extension to quadrature amplitude modulation (QAM). We pay particular attention to the transmitted signal power when independent symbols are encoded in this non-, linear manner. This question has heretofore been neglected Paper approved by the Associate Editor for Data Communication Systems of the IEEE Communications Society for publication without oral presentation. Manuscript received July 11, 1975; revised October 2, 1975. J. E. Mazo is with Bell Laboratories, Murray Hill, NJ 07974. J. Salz is with Bell Laboratories, Holmdel, NJ 07733. We have sometimes heard these schemes being called the "Tomlinson filter" and sometimes the "modulo-transmitter (MT)" technique. and our results provide tight bounds on the average transmitted signal power. We begin with a brief review of the precoding scheme which is best explained with the aid of a block diagram shown in  otherwise u is t o be reduced to this range by algebraically adding an appropriate multiple of 2L and then the result being called f ( u ) . Also in (I), pi are real constants and x k is the data symbol modeled as an L-valued random variable (L fixed throughout) which takes successive values independently and with equal probability from the set The utility of this filter in data transmission can be explained by considering the filter output sequence ( Y k ) l m to represent data symbols which are transmitted by PAM over a linear time-invariant channel. Let the overall impulse response, including transmitting and receiving filters, be h(t). The system output sample at time k is then given by where hj = h(j). MAZO AND SALZ: TRANSMITTED POWER IN PARTIAL RESPONSE The .first term is the desired output quantity while the additional terms constitute undesired intersymbol interference. The. second term in (3) depends on future data symbols and is caused by the nonvanishing samples of the overall impulse response for i < 0 (precursors) while the third term is a function of previously detected data symbols and conceptually may be eliminated by decision feedback techniques (see [7]). This type of intersymbol interference is termed causal, because it is due solely to the postcursor samples of the impulse response. The prefilter which we are examining here can at most affect this causal intersymbol interference and for purposes of this study we shall assume nique without incurring a noise penalty. It is known that the same can be achieved by decision feedback techniques at the receiver. But since this feedback must be done in the presence of noise, error propagation may result (for a discussion of decision feedback equalization and error propagation see Duttweiler et al. [8] ). We have seen that a linear filtering scheme at the transmitter could also eliminate postcursors while at the same time avoid error propagation, but the transmitted signal power may be unbounded. The chief advantage then of the nonlinear filter discussed here is that the transmitted symbols give rise to bounded power because they are restricted to the same range as the original data symbols. Our main concern here is to derive bounds on the transmitted signal power after precoding, and thus establish the penalty in transmitted signal power that must be paid for accomplishing the equalization at the transmitter. Since some data communication systems employ QAM, we extend this nonlinear precoding scheme to this modulation technique by generalizing to a two-input-two-(4) output coupled digital filter. where nk is a suitable integer. Substituting (4) into (3) with 11. THE CENTRAL RESULTS ho = 1 yields An interesting mathematical problem is to determine the recursion (1). In general one would not expect a probability density to exist; the distribution could very well be singular. It is easy to demonstrate that ( Y k ) l m is uniform when ( x k } l m ( 5 ) is uniform. Moreover, it can be shown that if the x k ' s are independently and identically uniformly distributed, the Y k ' s are also independent and uniform. Halfin [ 9 ] has recently Now if hi = pi, i = 1, ... N and hi = 0 i > N we see that examined the first-order probability distribution when pi = 0, i > 1. Our main concern however is to determine the average (6) power in the transmitted signal2 Since by hypothesis x k E [ -L , L ] , reducing sk mod 2L yields x k . This, then, is the key idea behind the precoding scheme. We also note here that reducing the range ( -L , L ] of the func-  where 6 k is a symmetric random variable, independent of x, the latter being of the L-valued variety described above. Our approach is to find the distribution for that maximizes Eyk2 in (IO). At this stage even the symmetry of will not be exploited; it will be shown later that including this requirement will not improve the bounds obtained. Clearly, if F(6) is the distribution function of 6k ; then Since is bounded for any k, and f 2 is bounded, we can easily approximate the probability distribution of 8 in (1 1) by a finite sum of delta functions while taking t; = 1, a = 1 in (1 3) yields The result (19) At this point we note that the symmetric distribution for 8 yields the same upper bound for the power and hence symmetrization of the density of 0 will not improve the bound. Equation (19) says that at most a 3 dB penalty is incurred for a binary system, 1.38 dB for a 3-level system, 0.8 dB for a 4-level one, etc. Since the largest uncertainty in the bound is for the binary case, let us consider this further. Consider the special binary case with similarly, Thus (25) which implies that the average power will be (2 + 1)/2 times greater than without the nonlinear filter (or 1.76 dB greater). In summary, for the binary case there are situations where use of the nonlinear filter can incur a penalty of 1.76 dB, but it cannot incur more than 3 dB. In QAM data .transmission there are two independent streams of data symbols, and The idea now is to design a two-input port and two-output port filter which encodes the pair of data sequences ( C k ) A block diagram is shown in Fig. 2 . As seen the output symbols satisfy where f(.) is defined as before. It can be verified that when (27a) and (27b) are substituted into (26a) and (26b), respectively, and the results reduced mod 2L, Sk(') = a k and s k ( 2 3 = bk. As in the baseband case the transmitted power is determined by Eak2 and Ebk2 since the signals are sent on orthogonal carriers. Because there is no cross coupling between the channels, we need make no additional arguments about the bounds on the transmitted power and they are the same here as in the baseband situation. REFERENCES H. Miyakawa and H. Harashima, "Capacity of channels with matched transmission technique, for peak transmitting power limitation," in Nat. Conv. Rec. Inst. Elec. Commun. Eng. Japan, Aug. 1969, no. 1268, p. 1269. M. Tomlinson, "New automatic equalizer employing modulo arithmetic,"Electron. Lett., vol. 7, pp. 138-139, Mar. 1971 J.,Salz, "Optimum mean-square decision feedback equalization," BellSyst. Tech. J., vol. 52, no. 8, Oct. 1973. D. L. Duttweiler, J. E. Mazo, and D. G. Messerschmitt, "On the error probability of decision-feedback equalization in the presence of error propagation," IEEE Trans. Inform. Theory, vol. S. Halfin, "Explicit construction of invariant measures for a class of continuous state Markov processes," Ann. Probability, to be published. D. Slepian, "On maxentropic discrete stationary processes," BellSyst. Tech. J., vol. 51, no. 3, Mar. 1972. COM-23, pp. 1251-1258 , NOV. 1975 . IT-20, pp. 490-497, July 1974 * James E. Mazo ("6.5)  Concise Papers A Simulation Model of the TIDAS Computer Network Abstract-To study the performance of a full-duplex messageswitched computer network, called TIDAS-T, during various conditions a simulation model has been built in order to simulate the procedure. The model describes the communication and the line procedure in the TIDAS network very closely. The only simplification made is that the processing time in the node computers is disregarded as being insignificant in 'comparison with the transmission and queueing times. The model, programmed in SIMSCRIPT 1.5, is very flexible because all important parameters are given by input data. The outputs hom a simulation differ from run to run due to the gbject defiqed by input data parameters. The statistical results are mostly presented as complete statistical distributions. It has'mrned out that the model has been a splendid tool in order to Study andview the consequences of different design approaches. Some of the problems in building such a complex model are elucidated and some simulation runs are given as illustrations. As the model is very flexible it may be used to study computer networks other than TIDAS.
The associated increase in transmitted signal power (see REF ) is the counterpart of quantization error in this paper.
On the Transmitted Power in Generalized Partial Response
IEEE Transactions on Communications
Mathematics