6.2 Coordinates and Movement Parameters Measuring …
143
When prior density p ps (α) has Gaussian distribution law, estimations obtained
via mentioned methods are matched.
Criteria of maximum likelihood are widely used at radar measurements, since
its application does not require of priori data during both at solution of detection
tasks and radar signals resolution, and at estimation of its informative parameters.
In estimation theory, it is proofed that in this case estimations, obtained via method
of maximum likelihood, ensure an unbiasedness of estimations. Along with estimations obtaining of informative parameters of radar signal, it is important to know the
effectiveness of these estimations, i.e., what is an error of measurements, what is a
value of mean-square root error of estimation, how it is related with characteristics
of the radar system itself and used radar signal. In statistical theory of estimation,
an estimation error dispersion in accordance with Rao–Cramer bound is commonly
used as quantitative characteristic of estimation effectiveness. As it was mentioned
before, in parameters of reflected from targets electromagnetic oscillations, the coordinates and data on target movement are “encoded.” Depending on solving tasks by
radar system, the different assumptions on coordinates changing behavior and targets
movement parameters can be examined.
In some cases, for example, during air space observation, it is reputed that during
dwell time the movement coordinates and parameters are not changed in time, they
are constant. In another case, at continuous target tracking, the targets coordinates
are continuously changed, but even in this case during signals receiving at some
period (e.g., pulse duration or pulse burst duration), it is also reputed that movement
coordinates and parameters are constant. In all abovementioned cases, the coordinates measurement is reduced to estimation problem of constant signals parameters,
constant values of coordinates and target movement parameters.
When assumption concerning coordinates constancy is prohibited to apply, an
estimation of changing in time signals parameters is examined, i.e., filtering of
parameters is performed. The two-stage estimation method of targets coordinates
and parameters is used in radar location. At the first stage, the HF radar signals
processing is carried out as well as estimation of “instantaneous” read-out values of
movement coordinate and parameters for a time when assumption on its constancy
is fulfilled. At the second stage, the processing of “radar data” is provided, i.e., an
estimation of variables at time of read-out values of coordinates, the filtering of
coordinates changing processes is fulfilled. This second stage is carried out in digital
read-out values and, as a rule, corresponds to video-frequency or low-frequency
spectral region. Usually, the filtering in this case is related to the linear type of
filtering.
To determine a structure of optimum meter (discriminator/pulse filter), we
examine the correlation integral Ψ (α s ), which for maximum determination is
necessary to be differentiated in shift parameter α s and equaled to zero:
dΨ (α s )
dα s
= 0, or
d
dα s
⎡
⎣
∞
−∞
y(t)S(t, α s )dt
⎤
⎦ = 0
(6.36)
143
When prior density p ps (α) has Gaussian distribution law, estimations obtained
via mentioned methods are matched.
Criteria of maximum likelihood are widely used at radar measurements, since
its application does not require of priori data during both at solution of detection
tasks and radar signals resolution, and at estimation of its informative parameters.
In estimation theory, it is proofed that in this case estimations, obtained via method
of maximum likelihood, ensure an unbiasedness of estimations. Along with estimations obtaining of informative parameters of radar signal, it is important to know the
effectiveness of these estimations, i.e., what is an error of measurements, what is a
value of mean-square root error of estimation, how it is related with characteristics
of the radar system itself and used radar signal. In statistical theory of estimation,
an estimation error dispersion in accordance with Rao–Cramer bound is commonly
used as quantitative characteristic of estimation effectiveness. As it was mentioned
before, in parameters of reflected from targets electromagnetic oscillations, the coordinates and data on target movement are “encoded.” Depending on solving tasks by
radar system, the different assumptions on coordinates changing behavior and targets
movement parameters can be examined.
In some cases, for example, during air space observation, it is reputed that during
dwell time the movement coordinates and parameters are not changed in time, they
are constant. In another case, at continuous target tracking, the targets coordinates
are continuously changed, but even in this case during signals receiving at some
period (e.g., pulse duration or pulse burst duration), it is also reputed that movement
coordinates and parameters are constant. In all abovementioned cases, the coordinates measurement is reduced to estimation problem of constant signals parameters,
constant values of coordinates and target movement parameters.
When assumption concerning coordinates constancy is prohibited to apply, an
estimation of changing in time signals parameters is examined, i.e., filtering of
parameters is performed. The two-stage estimation method of targets coordinates
and parameters is used in radar location. At the first stage, the HF radar signals
processing is carried out as well as estimation of “instantaneous” read-out values of
movement coordinate and parameters for a time when assumption on its constancy
is fulfilled. At the second stage, the processing of “radar data” is provided, i.e., an
estimation of variables at time of read-out values of coordinates, the filtering of
coordinates changing processes is fulfilled. This second stage is carried out in digital
read-out values and, as a rule, corresponds to video-frequency or low-frequency
spectral region. Usually, the filtering in this case is related to the linear type of
filtering.
To determine a structure of optimum meter (discriminator/pulse filter), we
examine the correlation integral Ψ (α s ), which for maximum determination is
necessary to be differentiated in shift parameter α s and equaled to zero:
dΨ (α s )
dα s
= 0, or
d
dα s
⎡
⎣
∞
−∞
y(t)S(t, α s )dt
⎤
⎦ = 0
(6.36)