# Trellis modulation

Refers to the trellis coded modulation, as Ungerboeck code, trellis coding, trellis modulation, abbreviated as TCM is a combination used in the digital signal processing channel coding for forward error correction of transmission errors, and a modulation technique to provide digital information on electric lines such as telephone lines transferred to.

The trellis code modulation was developed in 1982 by Gottfried Ungerboeck and found in the following years in telephone modems that operate according to the ITU- T standard V.32, V.32bis, V.34 and V.fast, wide application. It is also used in newer transmission systems and is used for example in Gigabit Ethernet ( 1000Base -T) in combination with a 5 -PAM modulation technique. But even in the symmetric DSL access to the standards and G.SHDSL SHDSL is the trellis code in combination with a 16 -PAM and 32 -PAM application.

The trellis code modulation is a very efficient channel coding and modulation technique, which is just at the theoretical limit of channel capacity and, depending on the specific implementation, only just out of the low- density parity -check code ( LDPC ) and by only a few years later developed turbo codes ( Turbo convolutional code and Turbo Product code ) is exceeded.

## Method

The trellis coded modulation is one of the coded modulation techniques, and is divided into two major functional blocks:

The essential difference of the trellis coded modulation to another separate channel coding and the method of digital modulation is that the channel coding and the modulation in the TCM are set operatively linked. A code word may be only just as long at the TCM to be assigned as a whole a transmission symbol in the modulation can.

One consequence of this, which only gives the additional code gain of the trellis code modulation, is that to evaluate possible error can not be considered as designed on its own channel coding method of the minimum Hamming distance between two code words. But instead of the Euclidean distance, which describes the geometrical distance between two points in a complex plane. This plane is defined by the amplitude and phase of the carrier wave and allocates transmission symbols to specific points in this plane.

### Encoder

At the encoder, the allocation of the individual bit combinations from which a code word is formed is made to the respective symbol of the modulation. Instead of a conventional modulation techniques, such as with other mapping, such as via the Gray code Ungerböck chose a structure that is referred to in mathematics as a binary tree. Here are the top node, the individual branching points are referred to in a binary tree, all 2k 1 symbols as before. The least significant bit of the code word is taken as the decision to the binary tree a step to climb down: depending on whether the respective bit of the code word is logic 0 or logic 1. This gives rise to the underlying level two nodes, each comprising half of the total possible symbols.

The division of each symbol is selected so that the Euclidean distance between neighboring symbols is maximized. With an 8 -PSK modulation with 8 symbols to the unit circle icons with straight index to the right and left symbols with odd index in the binary tree will be contacted. In most English-language literature, this method is called set partitioning: In each level division ( halving ) are the available transmit symbols.

Then is moved to the next bit of the code word according to the same scheme, so long until all the codeword bits corresponding transitions are assigned in the binary tree. In a convolutional code with a 3 -bit code word, ie two data bits of the input, a modulation with 8 symbols ( 23 ), such as 8 -PSK must be used. This results in the binary tree 3 transitions between levels. Only in the lowest level there is the concrete assignment of a particular symbol which is selected in this example of the 3 bits of a code word.

The special feature is that it increases at each step down one level in the binary tree, the Euclidean distance between remaining symbols at this level. The larger is the Euclidean distance between the individual symbols, the greater must be a fault on the transmission channel to come to a wrong decision at the decoder.

Is now chosen as least significant code bit the added by the convolution encoder redundancy, in the example with a three -bit code word the 3rd bit, this bit has to be the biggest mistake probability incorrectly decoded during transmission because it has the shortest distance to adjacent symbols. At the same time it contributes the least information as it is derived from only the other two data bits. The high-order bits in the code word are in the TCM, depending on the selected convolution, often not specifically coded but correspond directly to the user data bits. When these bits is through the division symbol (set partitioning ) before already, a much larger Euclidean distance between the transmit symbols, and thus a significantly lower probability of error in decoding.

### Decoder

During the decoding of the TCM signal of the known techniques such as convolutional code, the Viterbi algorithm may be used. The decoding process can be represented in a so-called trellis diagram, as is shown alongside a convolutional encoder having four states. A trellis diagram is an illustration of a state transition diagram which is " rolled " over the time axis. The transitions from one state to the next get different probability values assigned, which in a row across most clearly is emerging across multiple states, a single path in the trellis, which has the smallest sum probability of error over all other paths. The symbols associated with this path are then considered by the decoder as the symbols most likely transmitted.

A special feature to note in the decoding of TCM that by the uncoded, significant data bits result in the trellis diagram parallel branches. ( In the adjacent figure this circumstance occurring in TCM is not shown. ) These ambiguities can convolutional higher order are multi-state avoided.

Generally, the length of the convolutional code significant impact on the code gain, provided that the longer of the convolutional code, the more inner states it comprises, the greater the gain code associated therewith. Since the code gain in the TCM also depends on the modulation used, the determined gain code for the 8-PSK modulation, wherein a bit error rate (BER) of 10-6 and a function is indicated by the specific convolution in the following table. For other modulations, similar values are obtained, and detailed tables this can be found in literature specified below.

Generally speaking, that offer convolution with only four internal states at TCM no advantage since a convolutional code with four states already has a code gain of 3.6 dB for yourself. From a convolutional code of eight states up however, the TCM is always superior to combination the sole convolutional code in the code gain.

## Extensions of real implementations

In actual implementations of the trellis coded modulation, as in the ITU-T V.34 standard, another method for improving the transmission properties are used. These extensions include, among others, the following:

## Naming

Trellis is the English name for a trellis ( en: Garden trellis ) - for example, crossed at right angles wooden slats on walls as support for climbing plants such as ivy and Virginia creeper. The graphic representation of the trellis graph as a two-dimensional lattice structure corresponds to the arrangement of the slats of the framework. The crossing points of the slats corresponding to the node states, the slats than the edge state transitions during trellis coding.