Trellis-Coded Modulation

A very successful method of reducing the power requirements without an increase in the requirements on bandwidth was published by Gottfried Ungerboeck in 1987 [Ung87]. Trellis coded modulation (TCM) allows the achievement of significant coding gains over conventional uncoded multilevel modulation without compromising bandwidth efficiency. TCM schemes employ redundant nonbinary modulation in combination with a finite-state encoder which controls the selection of modulation signals to generate coded signal sequences.

We start our lesson with an illustrative example of this method of combining coding and modulation. Trellis encoder example with an 8-state finite-state machine driving a 3-bit to 8-PSK signal mapper is shown in Fig. 1. We use the same notation as in [Ung87], i.e. a denotes a complex-valued discrete channel signal transmitted at modulation time nT. Uncoded 4-PSK modulation is regarded as a reference system.

Fig. 1 Realization of 8-PSK code by means of systematic convolutional encoder with feedback.

The encoder shown in Fig. 9 consists of two parts, the first of which is a finite-state machine (FSM) with a total of eight states. State of the FSM is defined by the contents of the delay cells. The second part is called a signal mapper. Its function is a memoryless mapping of the three bits (yn2, yn1, yn0) into one of the eight symbols of an 8-PSK signal set. The FSM accepts two bits (xn2, xn1) at each symbol time n and moves from a state Sn to one of four successor states Sn+1.Assuming that the encoder in Fig. 1 operates continuously, there are four choices at each time n, which allows us to transmit two information bits (xn2, xn1) per symbol, the same as the reference uncoded 4-PSK system.