Error Rate for Peak-Limited Coherent Binary Channels

An exponential-type bound on error rate P
e
for peaklimited binary coherent channels operated at low signal-to-noise ratio (SNR) is presented. The bound depends exponentially only on the first and second moments of the channel output and serves to justify, in part, the use of SNR calculations for error-rate performance. It is assumed that the receiver output
V
is given by a simple sum of
n( = [TW])
identically distributed, independent random variables w
i
, each of which is decomposable into the sum of two independent random variables z
i
and η
i
, i.e.,
w_{i} = z_{i}+\eta_{i}
. The z
i
are peak limited by
B_{z},|z_{i}| \leq B_{z}
, whereas η
i
are normal (
0, \sigma^{2}
). The z
i
represent the output of a peak-limited channel and the η
i
represent any post channel receiver thermal noise (which may be zero,
\sigma^{2} = 0
). For example, the z
i
may represent the output of a bandpass-limited satellite repeater, with an interference input in addition to the desired signal, and η
i
the front-end noise in a receiving ground station. No assumption as to the channel-limiting characteristic or interference model, other than stated above, is made. Defining α as the ratio
[Ew]^{2}/var w
(i.e., twice the receiver input average SNR) and β as the ratio
B_{z}^{2}/var w
(i.e., twice the receiver input peak SNR), then for
\alpha < 1
,
P_{e} < e^{-n\mu}
\mu=\frac{\alpha}{2}(1-\alpha)-\frac {\gamma^{3}}{6} e^{\gamma}
\gamma= \sqrt{\alpha\beta}
, i. e., twice the geometric mean of average and peak SNR. If all odd moments of
z
have the same sign then a larger μ is obtained:
\mu=\frac{\alpha}{2}(1-\alpha)-\frac{\gamma^{4}}{24}e^{\gamma^{2}}/28
Upper bounds on the size of α and β are provided to guarantee
\mu > 0
. An example of a captured limiter that exhibits SNR suppression effects on the error rate is presented.