Asymptotics and Error Rate Bounds for M-ary DPSK

The symbol error probability
P_{E}(M)
for
M
-ary DPSK is shown to be bounded in terms of a recent asymptotic approximation
P_{asym}(M)
by the inequalities
P_{asym}(M) < P_{E}(M) < 1.03P_{asym}(M)
;
M \geq 4, E_{b}/N_{0} \geq 1
where
E_{b}/N_{0}
is the bit energy-to-noise spectral density ratio. Aside from the wide range of validity and the closeness of the lower and upper bounds, this result is striking in light of the often held view that such asymptotic approximations are primarily of value only in the limit
E_{b}/N_{0} \rightarrow \infty
; thus, one of the goals of this note is to demonstrate that asymptotic methods can lead to extremely good error rate approximations in lieu of the more traditional and more widely used bounding techniques. The results are also noted to be applicable in other similar situations which commonly occur.