Shaping and Coding for Additive Gaussian Noise Channels
An Enumerative Approach to Modulation
A classical result obtained in our group is the `Pascal-triangle algorithm' (Schalkwijk,
1972). It initiated our enumerative source coding research. The idea to
enumerate all sequences that satisfy certain constraints is also useful in
channel coding. We can e.g. enumerate all the sequences with odd integer
components that have energy which is not more than say Emax. It turns out that
these enumerative techniques give efficient ways of computing the index of a
certain sequence within this set of energy-limited sequences and vice versa.
These enumerative methods can be used to construct signal constellations
consisting of all points on a rectangular lattice within a sphere of given
radius. Using these hyperspherical constellations we are able to decrease the
required energy for modulation purposes. The gain with respect to hypercubic
constellations can be as much as 1.53 dB. Moreover it is possible to combine
these shaping methods with coding techniques. Coding techniques enlarge the
distance between signal points by forbidding certain signal points.
PUBLICATION:
At the IEEE First Symposium on Communications and Vehicular Technology in the
Benelux, that took place in Delft, the Netherlands on October 27-28, 1993, F.M.J.
Willems and J.J. Wuijts presented the paper "A Pragmatic Approach to Shaped
Coded Modulation".
Abstract :
We discuss shaping codes in an elementary way, which is from the standpoint of
enumerative coding. Then we show how to combine our shaping methods with a error
correcting codes. It turns out that we can easily gain a decibel by using
shaping techniques, with or without error correction coding. Finally we describe
the combination of our shaping method with the `pragmatic' approach to coded
modulation, which can be realized with VLSI circuits that are widely available.
A postscript version of the full paper (341 kB) is
available.