A uniformly most powerful decision rule may not exist when an
unknown parameter appears in a nonlinear way in the signalmodel. Most pertinent to array processing is the unknown time
origin case: the signal has been subjected to an unknown delay(
,
) and we must determine the signal's presence. The
likelihood ratio cannot be manipulated so that the sufficientstatistic can be computed without having a value for
. Thus, the search for a
uniformly most powerful test ends in failure and other methodsmust be sought. As expected, we resort to the generalized
likelihood ratio test.
More specifically, consider the binary test where a signal is
either present (
) or not (
). The signal waveform is known, but
its time origin is not. For all possible values of
, the delayed signal is
assumed to lie
entirely in the observations
(
). This signal model is ubiquitous in
active sonar and radar, where the reflected signal's exacttime-of-arrival is not known and we want to determine whether a
return is present or not
and the value of
the delay.
For a much more realistic (and
harder) version of the active radar/sonar problem, see
this problem .
Additive
white Gaussian noise is assumed present. The conditionaldensity of the observations made under
is
Despite uncertainties in the signal's delay
, the signal is assumed
to lie entirely within the observation interval. Hence thesignal's duration
, the
duration
of the observation
interval, and the maximum expected delay are assumed to berelated by
. The figure shows a signal
falling properly within the allowed window and a grey onefalling just outside. The exponent contains the only portion of this conditionaldensity that depends on the unknown quantity
. Maximizing the
conditional density with respect to
is equivalent to
maximizing
. As the signal is assumed to be contained entirely in
the observations for all possible values of
, the second term does not
depend on
and equals half
of the signal energy
. Rather
than analytically maximizing the first term now, we simply writethe logarithm of the generalized likelihood ratio test as
where the non-zero portion of the summation is expressed
explicitly. Using the matched filter interpretation of thesufficient statistic, this decision rule is expressed by
This formulation suggests that the matched filter having a
unit-sample response equal to the zero-origin signal beevaluated for each possible value of
and that we use the
maximim value of the resulting output in the decision rule. Inthe known-delay case, the matched-filter output is sampled at
the "end" of the signal; here, the filter, which has a duration
less than the observation
interval
, is allowed to continue
processing over the allowed values of signal delay with themaximum output value chosen. The result of this procedure is
illustrated
here .
There two signals, each having the same energy, are passedthrough the appropriate matched filter. Note that the index at
which the maximim output occurs is the maximim likelihoodestimate of
. Thus,
the detection and the estimation problems are solved
simultaneously . Furthermore,
the amplitude
of the signal need not be known as it enters in
expression for the sufficient statistic in a linear fashion andan UMP test exists in that case. We can easily find the
threshold
by establishing
a criterion on the false-alarm probability; the resulting simplecomputation of
can be
traced to the lack of a signal-related quantity or an unknownparameter appearing in
.
