Center-of-gravity information feedback

We consider
M
signals in a
D
-dimensional signal space. These
M
signals are used to communicate over an additive Gaussian white noise channel. It is shown that if a noiseless feedback channel is available one could use this feedback channel to inform the transmitter of the location of the center of gravity of the signal structure and thus obtain a very efficient signaling scheme. Each signal point is assigned a mass proportional to its posterior probability at the particular instant. The center of gravity is used by the transmitter as a new origin for the signal space. It is shown that some previously considered coding schemes for channels with feedback are particular cases of center-of-gravity feedback. The probability of error decreases as a double exponential function of the coding delay as opposed to an exponential decrease for one-way systems. The effect of noise in the feedback path is briefly considered.