# Trellis modulation

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In telecommunication, trellis modulation (also known as trellis coded modulation, or simply TCM) is a modulation scheme which allows highly efficient transmission of information over band-limited channels such as telephone lines. Trellis modulation was invented by Gottfried Ungerboeck working for IBM in the 1970s, and first described in a conference paper in 1976; but it went largely unnoticed until he published a new detailed exposition in 1982 which achieved sudden widespread recognition.

In the late 1980s, modems operating over plain old telephone service (POTS) typically achieved 9.6 kbit/s by employing 4 bits per symbol QAM modulation at 2,400 baud (symbols/second). This bit rate ceiling existed despite the best efforts of many researchers, and some engineers predicted that without a major upgrade of the public phone infrastructure, the maximum achievable rate for a POTS modem might be 14 kbit/s for two-way communication (3,429 baud × 4 bits/symbol, using QAM).[citation needed] However, 14 kbit/s is only 40% of the theoretical maximum bit rate predicted by Shannon's Theorem for POTS lines (approximately 35 kbit/s).

## A new modulation method

Trellis diagram

The name trellis was coined because a state diagram of the technique, when drawn on paper, closely resembles the trellis lattice used in rose gardens. The scheme is basically a convolutional code of rates (r,r+1). Ungerboeck's unique contribution is to apply the parity check on a per symbol basis instead of the older technique of applying it to the bit stream then modulating the bits. The key idea he termed Mapping by Set Partitions. This idea was to group the symbols in a tree like fashion then separate them into two limbs of equal size. At each limb of the tree, the symbols were further apart. Although hard to visualize in multi-dimensions, a simple one dimension example illustrates the basic procedure. Suppose the symbols are located at [1, 2, 3, 4, ...]. Then take all odd symbols and place them in one group, and the even symbols in the second group. This is not quite accurate because Ungerboeck was looking at the two dimensional problem, but the principle is the same, take every other one for each group and repeat the procedure for each tree limb. He next described a method of assigning the encoded bit stream onto the symbols in a very systematic procedure. Once this procedure was fully described, his next step was to program the algorithms into a computer and let the computer search for the best codes. The results were astonishing. Even the most simple code (4 state) produced error rates nearly one one-thousandth of an equivalent uncoded system. For two years Ungerboeck kept these results private and only conveyed them to close colleagues. Finally, in 1982, Ungerboeck published a paper describing the principles of trellis modulation.

A flurry of research activity ensued, and by 1990 the International Telecommunication Union had published modem standards for the first trellis-modulated modem at 14.4 kilobits/s (2,400 baud and 6 bits per symbol). Over the next several years further advances in encoding, plus a corresponding symbol rate increase from 2,400 to 3,429 baud, allowed modems to achieve rates up to 34.3 kilobits/s (limited by maximum power regulations to 33.8 kilobits/s). Today, the most common trellis-modulated V.34 modems use a 4-dimensional set partition which is achieved by treating two 2-dimensional symbols as a single lattice. This set uses 8, 16, or 32 state convolutional codes to squeeze the equivalent of 6 to 10 bits into each symbol sent by the modem (for example, 2,400 baud × 8 bits/symbol = 19,200 bit/s).

Once manufacturers introduced modems with trellis modulation, transmission rates increased to the point where interactive transfer of multimedia over the telephone became feasible (a 200 kilobyte image and a 5 megabyte song could be downloaded in less than 1 minute and 30 minutes, respectively). Sharing a floppy disk via a BBS could be done in just a few minutes, instead of an hour. Thus Ungerboeck's invention played a key role in the Information Age.[original research?]

## In popular culture

In the December 8, 1991 edition of the Dilbert comic strip, Scott Adams refers to the mere mentioning of trellis code modulation as a means for stopping a casual conversation cold.[1]

## Relevant papers

• G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inform. Theory, vol. IT-28, pp. 55–67, 1982.
• G. Ungerboeck, "Trellis-coded modulation with redundant signal sets part I: introduction," IEEE Communications Magazine, vol. 25-2, pp. 5–11, 1987.