Bound on the average transmission time for sequential estimation systems containing an additive Gaussian noise channel

We consider the class of sequential estimation systems containing additive Gaussian noise channels. Let
m
be a random variable representing the source and
x(t),y(t)
be stochastic processes representing the signal and the channel output, respectively. Denote by
\hat{m}
an estimate of
m
based on observing
y(t)
from 0 to
\tau
, where
\tau
is a random variable determined by a certain stopping rule of the decoder depending on a realization of
y
. Let
d(m,\hat{m})
be the distortion of
\hat{m}
relative to
m
and
R(\cdot)
be the rate distortion function of
m
with respect to the distortion measure
d
. Denote by
P_o
and
N_o
the available average signal power and the noise power level, respectively. We show that
E_{\tau} \geq (2 N_o / P_o)R. (Ed(m,\hat{m}))
and hence
Ed(m,\hat{m}) \geq R ^ {-1} (P_o E_ \tau /2N_o )
.. That is, given the average distortion
Ed(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 time
E_{\tau}
, the average distortion can be no smaller than
R ^ {-1} (P_o E _ {\tau} /2N_o)
.