B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

10.1 1 Noncoherent Detection 581
error probability ( or cost) that occurs under the worst possible a priori probability distribution.
Let us choose a = a1 , as shown in Fig. 10.35b. In this case the worst possible case occurs
when P(O) = 1 and P(l) = 0, that is, when the signal s(t) is always absent. The type of
error in this case is false alarm. These errors have a cost C 1 . On the other hand, if we choose
a = a2, the worst possible case occurs when P(O) = 0 and P(l) = 1, that is, when the signal
is always present, causing only the false-dismissal type of errors. These errors have a cost C 2 •
It is evident that for the setting a = a, the costs of false alarm and false dismissal are equal,
namely, C, . Hence, for all possible source statistics the cost is C a . Because C a < C1 and C 2 ,
this cost is the minimum of the maximum possible cost (because the worst cases are considered)
that accrues for all values of a. Hence, a = a represents the minimax setting.
It follows from this discussion that the minimax receiver is rather conservative. It is
designed under the pessimistic assumption that the worst possible source statistics exist. The
maximum likelihood receiver, on the other hand, is designed on the assumption that all messages
are equally likely. It can, however, be shown that for a symmetrical signal set, the
maximum likelihood receiver is in fact the minimax receiver. This can be proved by observing
that for a symmetrical set, the probability of error of a maximum likelihood receiver (equal a
priori probabilities) is independent of the source statistics [Eq. (10.137)]. Hence, for a symmetrical
set, the error probability P e M = a of a maximum likelihood receiver is also equal
to its [P e M lmax• We now show that no other receiver exists whose [P e M lmax is less than the
a of a maximum likelihood receiver for a symmetrical signal set. This is seen from the fact
that for equiprobable messages, the maximum likelihood receiver is optimum by definition.
All other receivers must have P e M > a for equiprobable messages. Hence, [P e M lmax for these
receivers can never be less than a. This proves that the maximum likelihood receiver is indeed
the minimax receiver for a symmetrical signal set.
10.1 1 NONCOHERENT DETECTION
If the phase e in the received RF pulse v'2,p'(t) cos (cv c t + 0) is unknown, we can no longer
use coherent detection techniques. Instead, we must rely on noncoherent techniques, such as
envelope detection. It can be shown 9 • 1 0 that when the phase 0 of the received pulse is random
and uniformly distributed over (0, 2n), the optimum detector is a filter matched to the RF
pulse v'2,p 1 (t) cos W e t followed by an envelope detector, a sampler (to sample at t = T b ), and
a comparator to make the decision (Fig. 10.36).
Amplitude Shift Keying
The noncoherent detector for ASK is shown in Fig. 10.36. The filter H (f) is a filter matched to
the RF pulse, ignoring the phase. This means the filter output amplitude A P
will not necessarily
be maximum at the sampling instant. But the envelope will be close to maximum at the sampling
Fi g ure 10.36
Noncoherent
detection of
digital modulated
signals
for ASK.
"'1 fi fl fl fl fi fi fir,
P a (t)
l
O ,A11Kf ,J A Al• 2r,.
0 V V V V V V V v:v,v_v_v_ YY .
J Y,Y) V
- - ,
r
-H (-
__ __,.--, /
)
Threshold
p(t) . -,I . detector
p 0
(t)
deveice ,__ _____
,--IEṉveḻop-,e
t = nT 0
,_____ Decision