A great deal of research has been done on a precoding technique, namely, generalized partial response signaling (GPRS). For details on GPRS, one can refer to publications by M. Tomlinson, "New Automatic equalizer employing modulo arithmetic," Electron. Lett., Vol. 7, nos. 5/6, Mar. 1971, pages 138-139; H. Harashima and H. Miyakawa, "Matched-transmission technique for channels with intersymbol interference," IEEE Trans. Commun., Vol. COM-20, August 1972, pages 774-780 and J. Mazo and J. Salz, "On the Transmitted Power in Generalized Partial Response," IEEE Trans. Commun., Vol. Com-24, March 1976, pages 348-352, all of which are hereby incorporated by reference.
The GPRS technique enables one to prevent signals from being adversely affected by intersymbol interference caused by a signal transmission through a channel having a finite memory and fixed characteristics. Knowing the impulse response of such a channel, one can design, in accordance with GPRS, a nonlinear filter in the transmitter for precoding the signals to be transmitted. This precoding compensates for, inter alia, intersymbol interference caused by the nonideal characteristics of the channel. GPRS, however, is not particularly useful for intersymbol interference compensation in actual communication system applications. This stems from the fact that actual communication channels have a virtually infinite memory and time-variant characteristics, as opposed to the finite memory and fixed characteristics required by GPRS.
Much attention has been focused in recent years on signal-space codes which provide so-called "coding gain." Prominent among these are the so-called "trellis" codes described in such papers as G. Ungerboeck, "Channel Coding with Multilevel/Phase Signals," IEEE Trans. Information Theory, IT-28, 1982, pages 55-67; A. R. Calderbank and N. J. A. Sloane, "A New Family of Codes for Dial-Up Voice Lines," Proc. IEEE Global Telecomm. Conf., November 1984, pages 20.2.1-20.2.4; A. R. Calderbank and N. J. A. Sloane, "Four-Dimensional Modulation With an Eight-State Trellis Code," AT&T Technical Journal, Vol. 64, No. 5, May- June 1985, pages 1005-1018; A. R. Calderbank and N. J. A. Sloane, "An Eight-Dimensional Trellis Code," Proc. IEEE, Vol. 74, No. 5, May 1986, pages 757-759; and L-F Wei, "Rotationally Invariant Convolutional Channel Coding with Expanded Signal Space--Part I: 180 Degrees and Part II: Nonlinear Codes," IEEE J. Select. Areas Commun., Vol. SAC-2, September 1984, pages 659-686all of which are hereby incorporated by reference. Commercial use of these codes has, for the most part, been concentrated in voiceband data sets and other carrier data communication systems. The term "coding gain" refers to the increased performance of a system resulting from the use of a particular code. It is defined as the amount by which the signal-to-noise ratio (SNR) may deteriorate for a system utilizing that particular code before the bit error rate for this system equals that of the same system without using the code.
The trellis codes that have been developed to date provide full coding gain in the presence of "white" noise, i.e., noise that contains components of virtually every frequency in the spectrum. However, the noise that appears in a received signal to be decoded is dependent upon the characteristics of the channel through which the signal was transmitted. Many communication channels, for example, a single two-wire pair or "local loop" of a telephone cable network that connects customer premises to a central office, impart a received signal with non-white or "colored" noise. This being so, the resulting coding gain in systems using such channels is less than the full coding gain. Indeed, in many system applications the difference between the full coding gain and that actually realized is significant. The failure to realize the full coding gain poses a problem with, for example, the proposed implementation of the Integrated Services Digital Network (ISDN). Relying on the full coding gain of, for example, the trellis code, ISDN purports to provide high signal rates on the local loops while a nominal SNR is maintained. Not having this full coding gain realizable makes the implementation of ISDN extremely difficult. Accordingly, it is desirable to have such a coding gain fully or substantially realized.