IEEE Signal Processing - July 2018 - 121

(3)

where H k +1 is the Jacobian of h ^x k +1 h evaluated at the state
x k +1 . The recursion in (3) starts with the initial Fisher information matrix J 0 that is equal to the inverse of the covariance
matrix of x 0 , assuming that the initial distribution of x 0
is Gaussian.
The recursive equation in (3) is the sum of two terms:
the first, [F -1] T J k-1 F - 1, is the a priori information given by
the previous target state, while PD H Tk + 1 R k-+1 1 H k +1 is the
information gained by the radar measurements. The term
PD can be intuitively justified, considering that there is a
measurement information contribution only if the target has
been detected.
As shown, the CRLB depends on the target trajectory,
sensor accuracy, transmitted waveform, probability of detection, and bistatic geometry. This dependence is given by the
measurement information term, which is itself dependent
on the probability of detection PD and on the measurement
covariance matrix R k . In particular, the target dependent
quantity PD det ^R -k 1 h can be considered to be a measure of
the measurement information, that is, the higher PD det ^R -k 1 h
is, the higher the information gained by the radar measurement. Therefore, this quantity can be used to select the channel with the best performance for each point of the analyzed
area. Knowing the transmitter positions and the estimated
state vector of the target at each time k, the receiver is able to
calculate the quantity PD det ^R -k 1 h for each point of the target
trajectory, and therefore it is able to evaluate the channel with
the best performance.
The target trajectory being deterministic, it is easy to verify
that the CRLBs of the receiver that dynamically selects the
best channel are given by
J k + 1 = 6F -1@ J k F -1 + J CH
k+1,
T

(4)

where the value J CH
k + 1 represents (5), shown in the box at the
bottom of the page. At each step k + 1 the measurement
information term J CH
k + 1 is given by selecting the channel
with the higher measurement information. Upper index 1

J CH
k+ 1 = )

PCRLB of x (m)

T
J k + 1 = 6F -1@ J k F -1 + PD H Tk + 1 R k-+1 1 H k + 1,

100

10

UMTS
FM
Selected

1

0.1

0

200

400

600

1,000

FiguRe 11. The root of the PCRLB of the target state, with the UMTS
channel, FM channel, and dynamic selection channel (x coordinate).

100

10

UMTS
FM
Selected

1

0.1

0

200

400

600

800

1,000

k

FiguRe 12. The root of the PCRLB of the target state, with the UMTS
channel, FM channel, and dynamic selection channel (y coordinate).

refers to the UMTS channel, while upper index 2 applies to
the FM channel.
The CRLBs of the receiver that selects the best transmitter are shown in Figures 11 and 12. From the results, it is evident that there is a substantial gain with respect to each bistatic
channel and that the obtained performance is equal to or better
than the performance of the channel with the lowest CRLB. As
is apparent in the figures, the initial performance is the same
as that of the FM channel. This is due to the fact that, in the
far range, the FM channel has the best performance, thanks
to its higher SNR. When the target approaches the harbor, the
UMTS channel has better resolution and, exploiting this channel, the proposed receiver is able to improve the performance
in estimating the target trajectory [61].
Active and passive technologies can be used jointly, as
proposed in [65], where a new idea of symbiotic radar (SR)

(1)
(1)
(1)
(1)
(2)
(1 )
(1)
(2)
P D 6H k + 1@ 6R k + 1@ H k + 1, if P D det ^6R k + 1@ h 2 P D det ^6R k + 1@ h
(2) T
(2) - 1 (2)
(1) - 1
(2) - 1 .
(2)
(1)
(2)
P D 6H k + 1@ 6R k + 1@ H k + 1, if P D det ^6R k + 1@ h 1 P D det ^6R k + 1@ h
T

800

k

PCRLB of y (m)

vector at time k. Its inverse is the bistatic CRLB that bounds
the error variance of the target state estimate at time k, that is,
J k-1 # E {(- x k) (- x k) T }, where xt k | k is an unbiased estimator
of the state vector based on all of the available measurements
up to time k. Using the results shown in [53] and [68], it is possible to demonstrate that J k can be computed recursively using
the following equation:

-1

IEEE Signal Processing Magazine

-1

|

July 2018

-1

|

(5)

121



Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018

Contents
IEEE Signal Processing - July 2018 - Cover1
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IEEE Signal Processing - July 2018 - Cover3
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