Page 133 - Radar Technology Encyclopedia

detection, sequential detection of a signal with unknown parameters 123

Sequential detection is “a method of automatic detection in Detection of a signal with known parameters applies when
two or more steps, normally the first using a high probability the signal parameters are completely known. If the parame-
of false alarm and the last a low probability of false alarm. In ters of signal u(t): amplitude A(t), frequency w , and phase
0
radars with controllable scanning, the first detection can be f(t), are completely known, one can write
used to order the scan to return to, stop at, or stay longer at the u(t) = A(t) cos [w t + f(t)]
0
suspected target position.” If the white-noise model is applicable to the interference,
In sequential detection the sample size is not fixed (as the output of the detector in time t there will consist of a sig-
opposed to fixed-sample detectors). Typically, it uses likeli- nal y(t), which is an additive mixture of the useful signal u(t)
hood ratio detection in applications where an agile-beam and white noise n(t), that is
antenna is used and the dwell times correspond to the y(t) = u(t) + n(t).
sequence of random sample sizes used by the detector. The likelihood ratio for a completely known signal is
Such a detection process can be practically implemented E s 2
with a phased-array radar, obtaining a power saving of 3 to 4 L = exp – ------ + ------ ZT ()
N 0 N 0
dB as compared with uniform scanning. The term sequential
detection sometimes is used synonymously with the sequen- where
tial observer criterion. SAL T
Ref.: IEEE (1990), p. 27; Barton (1988), p. 483; Skolnik (1980), p. 381; Bar- E = ò 2 d
u t () t
kat (1991), p. 162; DiFranco (1980), p. 547. s
0
The sequential observer (detection) criterion gives the is the signal energy,
minimum time for detection when the parameters of probabil-
T
ity density are sequentially checked under the conditions of
ut () yt () t
ZT () = ò d
fixed values of probability of detection P and probability of
d
false alarm P . The discrete values, Y , ..., Y of the signal- 0
1
n,
fa
plus-noise combination, Y(t) = u(t) + n(t), are supplied at the is the correlation integral, and
detector input. The likelihood ratio is L(y) = [W (y)]/W (y)] N is the noise power spectral density.
0
0
1
(see likelihood ratio detection criterion). Sequential detec- To derive a solution one must compare a L with the
tion consists of the comparison of likelihood ratio L(Y ) with threshold L :
0
n
two fixed thresholds A and B (A > B), applied at every step, k. If L ³ L there is a signal.
0
If L(Y ) ³, B, then the decision of signal presence is made. If If L < L there is no signal.
0
n
L(Y ) £ A, then the decision of signal absence is made. If This comparison is difficult to implement in practice.
n
B £L(Y ) £A, the decision is not made and the next (k + 1) Therefore, insofar as E = const, and the likelihood ratio L
s
n
step is performed. The threshold values can be determined as depends on the correlation integral Z(T), one must compare
Z(T) with the threshold Z :
A » P /P fa 0
d
Z
If Z(T) ³ there is a signal.
0
If Z(T) < Z there is no signal.
( 1 – P ) 0
d
B » --------------------- An optimum detector, which computes the correlation
( 1 – P )
fa integral and compares it with the threshold, can be of two
kinds. In the first case it consists of a multiplier and integrator
Sometimes this criterion is called the Wald criterion after the
(which make up the correlator) and a threshold unit and is
person who first to introduced it. AIL
called a correlation detector. In the second case the detector
Ref.: Dulevich (1978), p. 62; Barkat (1991), p. 162. consists of a matched filter and threshold unit. Such a detector
Detection of signals with full polarization reception occurs is called a filter detector. Signals with known parameters
when two orthogonal polarized components of the reflected hardly exist in practice and are used mainly in theoretical
electromagnetic waves are received. For conventional radar analysis. AIL
operating with fixed polarization, the intensity of the received Ref.: DiFranco (1968), p. 291; Dymova (1975), pp. 51–57; Sosulin (1992),
signal depends much on how the antenna polarization coin- pp. 48–55.
cides with polarization optimum for observed target. This Detection of a signal with unknown parameters is the
shortcoming can be eliminated in the radars with full polar- usual case where a signal is present in a background of inter-
ization reception giving a greater increase in signal detection. ference. A signal with unknown parameters is one in which
For example, for monopulse radar with full polarization either the phase f(t) or amplitude A(t) and phase are random.
reception, the probability of detection can be two to four A signal with unknown random initial phase can be written in
times that of analogous radar employing horizontal polariza- the form
tion only. The value of this increase depends on the polariza- u(t,y) = A(t) cos [w t + f(t) + y]
0
tion properties of the target. AIL where y is a random quantity, uniformly distributed in the
Ref.: Tuchkov (1985), pp. 213–215. interval 0, ... , 2p.