Bound on the average transmission time for sequential estimation systems containing an additive Gaussian noise channel
Toshihito Kadota, Abraham J. Wyner, J. Ziv · IEEE Transactions on Information Theory · 1971
We consider the class of sequential estimation systems containing additive Gaussian noise channels. Letmbe a random variable representing the source andx(t),y(t)be stochastic processes representing the signal and the channel output, respectively. Denote by\hat{m}an estimate ofmbased on observingy(t)from 0 to\tau, where\tauis a random variable determined by a certain stopping rule of the decoder depending on a realization ofy. Letd(m,\hat{m})be the distortion of\hat{m}relative tomandR(\cdot)be the rate distortion function ofmwith respect to the distortion measured. Denote byP_oandN_othe available average signal power and the noise power level, respectively. We show thatE_{\tau} \geq (2 N_o / P_o)R. (Ed(m,\hat{m}))and henceEd(m,\hat{m}) \geq R ^ {-1} (P_o E_ \tau /2N_o ).. That is, given the average distortionEd(m,\hat{m}), the average transmission time required can be no smaller than(2N_o/P_o)R(Ed(m,\hat{m})). Conversely, given the average transmission timeE_{\tau}, the average distortion can be no smaller thanR ^ {-1} (P_o E _ {\tau} /2N_o).