Various efforts to improve the signal-to-noise ratio in channels with additive noise and intersymbol interference, which is characteristic of severely bandlimited channels (e.g., the telephone channel), led to the original development of set partitioning techniques that culminated in Ungerboeck’s development (1982) of trelliscoded modulation (TCM).

Ungerboeck showed how large Hamming distance between differing data sequences does not necessarily imply large Euclidean distance between modulated data sequences, unless the assignment of coded signals to modulated signals is cleverly made.

He proposed a new method of set partitioning,which not only provided a method for the assignment of coded signals to channel signals, but also provided a simple formula for the lowerbound of the Euclidean distance between modulated data sequences.

The trellis code replaces the QPSK signal constellation with an 8-ary PSK signal constellation so as to introduce eight points in the complex plane that represents phase modulation. Instead of using the eight points to increase the data rate by modulating three bits at a time into a channel symbol, the trellis code modulates only two bits of data at a time into a three-bit channel symbol.

The three code bits define one of eight possibilities that are labeled in the phase diagram. Every pair of incoming data bits is mapped into three code bits by the trellis encoder in order to create a waveform.

The transmitted trellis-coded waveform has the same data rate as a QPSK waveform and uses the same bandwidth, but the power requirement is reduced by 2.5 times.

TCM can also be used to increase the transmitted data rate at a fixed transmitted power. Much more complicated signal constellations are required for this purpose.

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