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.

In the late 1980s, modems operating over plain old telephone service ("POTS") typically achieved 9.6 to 14.4 kbit/s. 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 was 19.2 kbit/s. However, 19.2 kbit/s is only 60% of the theoretical maximum bit rate predicted by Shannon's Theorem for POTS lines (approximately 30 kbit/s).

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 ot the tree, the symbols were further apart. Although in multi-dimensions, it is hard to visualize, 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 1000 times lower than 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 1984, the International Telecommunication Union published modem standards utilizing trellis modulation to achieve rates of 19.2 kbit/s and higher. Today, the most common TCM modems (V.34) use a 4-Dimensional set partition (achieved by treating two 2-Dimensional symbols in time as a single 4-Dimension Lattice) with 16, 32, or 64 state convolutional codes

Once manufacturers introduced modems with trellis modulation, transmission rates increased dramatically to the point where interactive transfer of multimedia over POTS became feasible. Thus, it can be argued that Ungerboeck's invention played a key role in the telecommunications revolution and the explosive growth of the World Wide Web.