The name
trellis derives from the fact that a state diagram of the technique closely resembles a
trellis lattice. The scheme is basically a
convolutional code of rates (
r,
r+1). Ungerboeck's unique contribution is to apply the parity check for each
symbol, instead of the older technique of applying it to the bit stream then modulating the bits. He called the key idea
mapping by set partitions. This idea groups symbols in a tree-like structure, then separates them into two limbs of equal size. At each "limb" of the tree, the symbols are further apart. Though hard to visualize in multiple dimensions, a simple one-dimension example illustrates the basic procedure. Suppose the symbols are located at [1, 2, 3, 4, ...]. Place all odd symbols in one group, and all 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 symbol in 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 1984 the
International Telecommunication Union had published a standard, V.32, for the first trellis-modulated modem at 9.6 kilobit/s (2,400 baud and 4 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—achieved by treating two two-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 the modem sends (for example, 2,400 baud × 8 bits/symbol = 19,200 bit/s). TCM is one member of a broader family of coded modulation techniques. In
multilevel coding, introduced by Hideki Imai and Shuji Hirakawa in 1977, several component error-correcting codes protect different levels of a multilevel modulation signal. ==Relevant papers==