Page 446 - Fundamentals of Radar Signal Processing

(6.12)

Because of its simpler form, the log-likelihood ratio will be used. Substituting
Eq. (6.12) into Eq. (6.8) and rearranging gives the decision rule:

(6.13)

where all the right-hand-side constants have been combined into a single
constant T. Note that the right-hand side of the equation consists only of

constants, though not all are yet known. Equation (6.13) thus specifies that the
available data samples y be integrated (summed) and the integrated data
n
compared to a threshold. This integration is an example of how the LRT
specifies the data processing to be performed on the measurements. Note also
that Eq. (6.13) does not require specifically evaluating the PDFs, let alone
determining what exactly is the region in N-space comprising or whether the
1
observation y is in it.
In many cases of interest, the specific form of the log-likelihood ratio can
be further rearranged to isolate on the left-hand side of the equation only those
terms explicitly including the data samples y , moving all other constants to the
n
right-hand side. Equation (6.13) is such a rearrangement of Eqs. (6.8) and
(6.12). The term Σy is called a sufficient statistic for this problem, and is
n
denoted by ϒ(y). The sufficient statistic, if it exists, is a function of the data y
that has the property that the likelihood ratio (or log-likelihood ratio) can be
written as a function of ϒ(y), i.e., the data appear in the likelihood ratio only
through ϒ(y) (Van Trees, 1968). This means that in making a decision that is

optimal under the Neyman-Pearson criterion, knowing the sufficient statistic
ϒ(y) is as good as knowing the actual data y. In particular, the decision
criterion in Eq. (6.8) can be expressed as

(6.14)