Local trajectory parameters estimation and detection of moving targets in Rayleigh noise
The problem of detection of moving targets and estimation of local trajectory parameters based on the analysis of sensor data in the form of two-dimensional image is considered. In accordance with the target and sensor models, probability distribution of noise at the output of the detector is Raylei...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2014
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| Цитувати: | Local trajectory parameters estimation and detection of moving targets in Rayleigh noise / I.G. Prokopenko, V.Iu. Vovk, I.P. Omelchuk, Yu.D. Chirka, K.I. Prokopenko // Технология и конструирование в электронной аппаратуре. — 2014. — № 1. — С. 23-35. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-705372017-04-10T13:09:31Z Local trajectory parameters estimation and detection of moving targets in Rayleigh noise Prokopenko, I.G. Vovk, V.Iu. Omelchuk, I.P. Chirka, Yu.D. Prokopenko, K.I. Системы передачи и обработки сигналов The problem of detection of moving targets and estimation of local trajectory parameters based on the analysis of sensor data in the form of two-dimensional image is considered. In accordance with the target and sensor models, probability distribution of noise at the output of the detector is Rayleigh distribution, while probability distribution of signal is Rice distribution. Two trajectory parameters estimation techniques are considered: ordinary least squares and Hough transform. A detection stage based on the integration of an input signal along estimated trajectory is proposed. Statistical modeling was performed and detection characteristics were obtained. Рассмотрена проблема и предложен алгоритм обнаружения и локальной оценки параметров траектории движущихся целей на основе анализа данных в форме двумерного изображения. Исходя из принятых моделей цели и детектора, фоновая помеха имеет распределение Рэлея, а сигнал — распределение Райса. Рассмотрены два метода оценки параметров траектории: метод наименьших квадратов и метод преобразования Хафа. Предложена процедура обнаружения цели, основанная на интегрировании отраженной мощности вдоль вероятной траектории. Проведено статистическое моделирование, по результатам которого построены характеристики обнаружения для предложенного алгоритма. Розглянуто проблему та запропоновано алгоритм виявлення та локальної оцінки параметрів траєкторії рухомих цілей на основі аналізу даних у формі двовимірного зображення. Виходячи з прийнятих моделей цілі та детектора, фонова завада має розподіл Релея, а сигнал — розподіл Райса. Розглянуто два методи оцінки параметрів траєкторії: метод найменших квадратів і метод перетворення Хафа. Запропоновано процедуру виявлення цілі, що ґрунтується на інтегруванні відбитої потужності вздовж імовірної траєкторії. Проведено статистичне моделювання, за результатам якого побудовано характеристики виявлення для запропонованого алгоритму. 2014 Article Local trajectory parameters estimation and detection of moving targets in Rayleigh noise / I.G. Prokopenko, V.Iu. Vovk, I.P. Omelchuk, Yu.D. Chirka, K.I. Prokopenko // Технология и конструирование в электронной аппаратуре. — 2014. — № 1. — С. 23-35. — Бібліогр.: 29 назв. — англ. 2225-5818 DOI: 10.15222/tkea2014.1.23 http://dspace.nbuv.gov.ua/handle/123456789/70537 629.7.058.54(043.2)/629.7.086:004.932.2 en Технология и конструирование в электронной аппаратуре Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Системы передачи и обработки сигналов Системы передачи и обработки сигналов |
| spellingShingle |
Системы передачи и обработки сигналов Системы передачи и обработки сигналов Prokopenko, I.G. Vovk, V.Iu. Omelchuk, I.P. Chirka, Yu.D. Prokopenko, K.I. Local trajectory parameters estimation and detection of moving targets in Rayleigh noise Технология и конструирование в электронной аппаратуре |
| description |
The problem of detection of moving targets and estimation of local trajectory parameters based on the analysis of sensor data in the form of two-dimensional image is considered. In accordance with the target and sensor models, probability distribution of noise at the output of the detector is Rayleigh distribution, while probability distribution of signal is Rice distribution. Two trajectory parameters estimation techniques are considered: ordinary least squares and Hough transform. A detection stage based on the integration of an input signal along estimated trajectory is proposed. Statistical modeling was performed and detection characteristics were obtained. |
| format |
Article |
| author |
Prokopenko, I.G. Vovk, V.Iu. Omelchuk, I.P. Chirka, Yu.D. Prokopenko, K.I. |
| author_facet |
Prokopenko, I.G. Vovk, V.Iu. Omelchuk, I.P. Chirka, Yu.D. Prokopenko, K.I. |
| author_sort |
Prokopenko, I.G. |
| title |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise |
| title_short |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise |
| title_full |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise |
| title_fullStr |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise |
| title_full_unstemmed |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise |
| title_sort |
local trajectory parameters estimation and detection of moving targets in rayleigh noise |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2014 |
| topic_facet |
Системы передачи и обработки сигналов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/70537 |
| citation_txt |
Local trajectory parameters estimation and detection of moving targets in Rayleigh noise / I.G. Prokopenko, V.Iu. Vovk, I.P. Omelchuk, Yu.D. Chirka, K.I. Prokopenko // Технология и конструирование в электронной аппаратуре. — 2014. — № 1. — С. 23-35. — Бібліогр.: 29 назв. — англ. |
| series |
Технология и конструирование в электронной аппаратуре |
| work_keys_str_mv |
AT prokopenkoig localtrajectoryparametersestimationanddetectionofmovingtargetsinrayleighnoise AT vovkviu localtrajectoryparametersestimationanddetectionofmovingtargetsinrayleighnoise AT omelchukip localtrajectoryparametersestimationanddetectionofmovingtargetsinrayleighnoise AT chirkayud localtrajectoryparametersestimationanddetectionofmovingtargetsinrayleighnoise AT prokopenkoki localtrajectoryparametersestimationanddetectionofmovingtargetsinrayleighnoise |
| first_indexed |
2025-07-05T19:44:33Z |
| last_indexed |
2025-07-05T19:44:33Z |
| _version_ |
1836837422969651200 |
| fulltext |
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2014, ¹ 1
23
SIGNALS TRANSFER AND PROCESSING SYSTEMS
UDC 629.7.058.54(043.2)/629.7.086:004.932.2
I. G. PROKOPENKO, Dr.Sc., V. Iu. VOVK, I. P. OMELCHUK, Ph.D.,
Yu. D. CHIRKA, K. I. PROKOPENKO, Ph.D.
Ukraine, Kiev, National Aviation University
E-mail: prokop-igor@yandex.ru, vitalii.vovk@nau.edu.ua
LOCAL TRAJECTORY PARAMETERS ESTIMATION AND
DETECTION OF MOVING TARGETS IN RAYLEIGH NOISE
Moving targets detection and estimation of
their trajectory parameters are the tasks of sec-
ondary stage of radar signal processing. Classical
approaches to tracking and detection of moving
low-snr (signal-to-noise ratio, SNR) targets are
based on multi-survey analysis of radar data with
thresholding procedure applied to it. In the case
of rapid target detection tasks, it is necessary to
reduce the number of surveys before decision about
target presence can be made. Such rapid targets
may produce on a radar image a few samples
instead of one. In this case energy backscattered
from the target is distributed along the entire
trajectory, forming a track. This “blurring” effect
causes the decreasing of SNR. For low-snr targets
the threshold must be low (or even should be
absent at all) to allow sufficient probability of
target detection. A low threshold also gives a high
rate of false detections which cause the tracker to
form false tracks. In this case information about
target trajectory may be very useful to provide an
opportunity to integrate all backscattered from the
target power along the entire trajectory.
The commonly used group of algorithms work-
ing with raw unthresholded data to estimate target
state is known in literature as the track-before-
detect (TBD) [1]. These methods are used when
classical thresholding approaches are not suitable
for tracking and detecting targets due to low SNR,
typically stealthy military aircraft and cruise mis-
siles, for which thresholding has an undesirable
effect of disregarding potentially useful data [2].
Sensor image is often a highly nonlinear func-
tion of the target state (which describes kinematic
and power parameters of the target). There are
The problem of detection of moving targets and estimation of local trajectory parameters based on the
analysis of sensor data in the form of two-dimensional image is considered. In accordance with the
target and sensor models, probability distribution of noise at the output of the detector is Rayleigh
distribution, while probability distribution of signal is Rice distribution. Two trajectory parameters
estimation techniques are considered: ordinary least squares and Hough transform. A detection stage
based on the integration of an input signal along estimated trajectory is proposed. Statistical modeling
was performed and detection characteristics were obtained.
Keywords: target detection, moving target, estimation, trajectory, track-before-detect, Rayleigh noise,
Hough transform, least squares.
a number of methods to deal with this nonlinear-
ity. First of all, such estimation techniques as
the Viterbi algorithm [3] and the hidden Markov
model (Baum-Welsh) filter or smoother [4] can be
applied to the discretized state space, when nonlin-
earity is irrelevant due to such discretization. Such
technique is used in several approaches to TBD,
e.g. [5—7]. The main disadvantage of discretiza-
tion of the state space leads to high computation
and memory resource requirements.
To avoid the problems mentioned above, an
alternative approach called “particle filtering”
can be used to solve the nonlinear estimation
task [2, 8]. This method (also called Sequential
Monte Carlo) uses Monte Carlo techniques to
solve the analytically intractable estimation in-
tegrals. It uses randomly placed samples instead
of fixed samples as is the case for the discretized
state space. Particle filtering has been used by
a number of authors for TBD, e.g. [9—11]. In
many cases it is possible to achieve close or even
similar estimation performance for lower cost by
using less sampling points than would be required
for a discrete grid.
An alternative approach to estimation of the
target state is called the histogram probabilistic
multi-hypothesis tracking (PMHT) algorithm
[12]. In this method, a parametric representation
of the target state pdf is used rather than a nu-
merical one. This makes it possible to reduce the
computational load of the algorithm. The main
idea of maximum likelihood probabilistic data
association (PDA) is to reduce the threshold to a
low level and then apply a grid-based state model
for estimation [13, 14] rather than using the whole
DOI: 10.15222/TKEA2014.1.23
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24
SIGNALS TRANSFER AND PROCESSING SYSTEMS
sensor image. The association of the high number
of measurements is handled using PDA.
In addition to estimating the target state, the
TBD algorithm needs to detect the presence or
absence of targets. One method to do this is to
extend the state space by including target pres-
ence state. In this case, null-presence state will
correspond to the case when there is no target
[9, 15]. A target is detected when any state other
than the null state has the highest probability.
Another closely related concept is to use a sepa-
rate Markov chain to describe the target presence
state as originally introduced for PDA in [16].
This approach has been used for the particle filter
[10] and a generalized version was applied to the
PMHT [17]. Comparison of detection performance
for TBD algorithms mentioned above has been
done by Davey, et al. [18].
Many of the articles mentioned above deal with
radar data in the form of a two-dimensional im-
age. The x and y axes correspond to range-bearing
subspace of a global range-bearing-elevation-
doppler measuring hypercube, or to linear space
coordinates, which are obtained from these mea-
surements. It is usually assumed that equal size
of resolution bins (cells) may be feasible in case
when space being studied is a small part of the
observed sector and is far enough from the sensor.
A sequence of radar images is usually processed
before the decision about target presence or ab-
sence can be made in TBD approaches. A target is
usually a point scatterer which produces a blurred
spot on the radar image (Fig. 1), moving from
frame to frame according to the kinematic param-
eters of the target.
In this paper
we describe the
approach, which
has two main dis-
tinctions from the
methods mentioned
above. Firstly, to
decrease computa-
tional efforts, we
try to avoid us-
ing any grid-based
algorithms, either
uniform or non-uni-
form (as in particle
filtering). For the
same reason we do not estimate the target state
pdf. Secondly, our method can be applied to ana-
lyze a radar image, which may contain not just a
single sample, but a track of the moving target.
This fact gives additional information which can
be used to reduce a number of analyzed radar im-
ages before decision about target presence can be
made. Such images can be formed from a sequence
of radar plots in case of slow moving targets, or
even a single radar plot in case of rapid moving
targets (“rapid” compared with the radar image
forming time). Here several assumptions are made:
1) the target is a point scatterer; 2) the target has
enough speed to produce track while the radar
image (frame) is being formed; 3) the trajectory
of the target is close to linear; 4) all resolution
cells (bins) are of equal size. This can be relevant
in case of small radar target (e.g. cruise missile)
tracked by phased-array or multiple beams systems
with wideband signal (e.g. MIMO radars with
pseudo-noise signals or systems with frequency
modulated pulse compression). The similar radar
images can be obtained by observing ionized trails
in atmosphere produced by meteors in radar meteor
detection systems [19]. However, in this case we
deal with a linearly distributed target. Another
interpretation of the xy-plane could be a projec-
tion of space volume onto the sensor in optical or
infrared systems (e.g. forward looking infrared,
FLIR [20]). Such systems are used in some cases
when a high resolution is needed (e.g. for detection
of meteors or fast cruise missiles) [21]. Hereinafter
we will assume that the target is a fast moving
point radar object.
The proposed algorithm contains two main
stages [22]. At the first stage geometric parameters
of a potential trajectory should be estimated. There
are two commonly used methods to perform such
estimation: ordinary least squares technique [23]
and Hough transform [24]. At the second stage a
decision about target presence is made by simple
detection technique based on the integration along
the track of a power backscattered from the target.
This algorithm can also be used for more accurate
forming of proposed distributions in particle filter
TBD approaches, or as initiation procedure for
complex trajectory detection by analysis of its
small linear part.
Target, signal and sensor models
Target model
As it was mentioned earlier we assume the target
to be a rapid point-scattering object. We take into
account two models of target moving: linear and
non-linear (parabolic). The linear target moving
model can be described in Cartesian x-y-plane in
the form r = x(t)cosq + y(t)sinq or, which is equal
to r = x(t)sinj + y(t)cosj (see Fig. 2), where r
is the distance from the origin to a straight line,
to which trajectory belongs; q is the anticlockwise
angle between the x-axis and the perpendicular
from the origin to the straight line; j = –(q – p/2);
y
15
10
5
5 10 15 x
Fig. 1. Example of a radar image
in some TBD applications [11]
Fig. 2. Geometric representation of a straight line
y
x
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2014, ¹ 1
25
SIGNALS TRANSFER AND PROCESSING SYSTEMS
x and y are the space coordinates; t is time (either
continuous or discrete).
The parabolic target moving model is described
by the equation y(t) = a2x2 + a1x + a0. We should
note that such representation does not take into
account the possible rotations of the parabolic
trajectory.
Signal model
The signal at the input of the sensor is a se-
quence of radio pulses with constant carrier fre-
quency w, random phase j and constant amplitude
vS. The noise is an additive Gaussian random
process. Thus, along the trajectory, the signal-
noise model is
( ) ,cosS v t wS ω ϕ= + + (1a)
and outside the trajectory, the model is
S = w. (1b)
Here w~N(0,sN), sN is a standard deviation of
noise; N(m,s) denotes Gaussian distribution with
mean m and standard deviation s.
Sensor model
The analysed image is a grayscale image
formed from the output of the amplitude detector.
According to (1), values of each pixel (sample) of
the image are distributed with Rice distribution,
if such sample belongs to the trajectory of the
target [25, p. 19]:
–( )
,expz
w w v
I
w v
2,
, , ,
x y
N
x y
N
x y S
N
x y S
2 2
2 2
0 2σ σ σ
=
+e co m
(2a)
and with Rayleigh distribution, if it does not:
–( )
,expz
w w
2,
, ,
x y
N
x y
N
x y
2 2
2
σ σ
= e o
(2b)
where I0(•) is the modified Bessel function of zero order;
x and y are coordinates of the sample.
Some examples of images which correspond to
the model described by (2) are presented in Fig. 3
(hereinafter the measure units for x and y axes
will be the same nominal units, the size of image
pixel corresponds to the size of the resolution cell).
Selection of important samples
The trajectory parameters estimation procedure
consists of two subtasks. At first we need to select
a set of samples (hereinafter “important samples”)
from the source image, which will be used later
for estimation of the trajectory parameters. Then,
the parameters of the trajectory must be estimated
in some way. We shall consider Hough transform
and ordinary least squares techniques to do this.
Both techniques in a 2D-image case require set of
samples’ coordinates as the input. We consider a
several methods to select such samples, based on
order statistics.
The first method involves the independent ap-
plying of two Boolean masks to the original image.
Both masks are of the same size as the original
image. The first mask is formed in the following
way. For every row of it we set “true” in the bin,
which holds the maximum value in the correspond-
ing row of the source image (R-mask) and “false”
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
Fig. 3. Images with sim-
ulated linear tracks of
moving target for signal-
to-noise ratio equals
4 dB (a), 7 dB (b) and
10 dB (c)
a) b) c)
Fig. 4. Procedure for selecting important samples (RC-mask) (a) and example of its application:
b — source image; c — resulting image
y
12
8
4
4 8 12 x
y
12
8
4
4 8 12 x
a) b) c)
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2014, ¹ 1
26
SIGNALS TRANSFER AND PROCESSING SYSTEMS
elsewhere. The second mask is formed in the same
way, but for columns (C-mask). These masks are
then independently applied to the source image.
Thus, we obtain two sets of important samples for
further analysis. The second method is based on
applying to the source image the result of a con-
junction of R- and C-masks RC = R∧C (Fig. 4).
The third method assumes additional use of two
binary masks formed in the same way as R-mask
and C-mask but for diagonal (D-mask) and cross-
diagonal (CD-mask) elements. Thus, the resulting
mask RCD = (R∧C) ∨ (D∧CD).
The normalized power properties for different
sample selection techniques are shown in Fig. 5
(PSEL is the power that corresponds to selected im-
portant samples; PSIG is the full power backscattered
from target power; PSP is the power backscattered
from the target power that corresponds to selected
important samples). These plots were obtained as a
result of computer modeling with number of simu-
lations N=2,000 and image size 16×16 pixels (all
simulated tracks were built in the way they crossed
the center of the image). In Fig. 5, a the power that
corresponds to selected important samples normal-
ized to full signal power (full power backscattered
from target) is shown. It can be clearly seen that
in low-snr region important samples mainly corre-
spond to noise. Fig. 5, b shows that for all proposed
techniques the amount of used signal power (the
backscattered power that corresponds to the true
trajectory) is increased with the increase of SNR,
but RC technique rejects the half of useful data. At
the same time, as it can be seen from Fig 5, c, RC
technique shows the lowest level of noise samples
power among other sample selection techniques. This
is important for trajectory parameters estimation
procedures, because, due to low order of trajectory
nonlinearity, its parameters still can be estimated by
small number of important samples (2 or 3). Due to
this fact, it is better to lose some useful information
at this step, but reduce more noise samples. The av-
erage number of selected noise samples for different
sample selection technique in case when there is no
actual track presented in the radar plot of size M×M
is shown in Table 1.
We should note that such approach cannot
be applied in case of two or more closely-spaced
targets, because of missing and mixing data from
such targets.
To additionally reduce the number of noise sam-
ples being selected, an additional clearing procedure
Fig. 5. Normalized power properties for different sample
selection techniques
4
3
2
1
0
–2 0 2 4 6 8 10 12
Signal-to-noise ratio, dB
0,8
0,6
0,4
0,2
0
–2 0 2 4 6 8 10 12
Signal-to-noise ratio, dB
0,8
0,6
0,4
0,2
0
–2 0 2 4 6 8 10 12
Signal-to-noise ratio, dB
Sample selection
technique R(C) RC RCD
Average number
of selected samples M
( – )M
M
2 1
2 –
, – , ,
– , ;k l
k M i j i j
M j i1
1 if
otherwise
>
, ,
,
i j i j
i j
j
M
i
M
11 +
= +
+==
)//
– , – ,
– – , .
l i j M j i
M i j
1 1
2 1
if
otherwise
>
,i j = + +
+
)
Table 1
Average number of selected samples for different sample selection techniques
a)
b)
c)
P
S
E
L
/
P
S
IG
P
S
P
/
P
S
IG
P
S
P
/
P
S
E
L
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2014, ¹ 1
27
SIGNALS TRANSFER AND PROCESSING SYSTEMS
can be introduced in case of linear trajectories. This
procedure contains following steps:
1. Select important samples from the source
image.
2. For every important sample estimate the
trajectory parameters without considering this
sample, and evaluate the deviation of other sam-
ples around estimated trajectory.
3. Remove the sample which, if not taken into
account, minimizes deviation.
4. Go to step 1 until stop-condition is not met.
This stop-condition could be the number of
important samples left (at least two samples must
be left to estimate linear trajectory parameters) or
the deviation of the samples is less than predefined
threshold. The example of usage of such procedure
is shown in Fig. 6.
Of course, in some cases, especially for low-snr
signals, this procedure can fail. But, as shown
below, in general, it gives a gain in quality of
trajectory parameters estimation and, as a result,
in detection performance.
Estimation of the geometric parameters of the
trajectory
Once important samples have been selected,
trajectory parameters can be estimated. There is
a well-described in literature technique, called
Hough transform (HT) [26, 27], commonly used
in image processing for estimating geometric pa-
rameters of parametric curves [24]. Nowadays it
is widely used in different image processing tasks,
from radar and sonar signal processing [28] to
object recognition tasks and machine vision [29].
The main idea of Hough transform technique is
to map points from the local coordinate space to
curve’s parameters space and find a maximum in
this new space. This maximum will correspond
to the most likely parameters of the curve. To do
this, the so called accumulator array is built by
discretization the space of parameters with steps
Di, where i = 1...N, and N is the number of pa-
rameters of the curve. Then, for every important
sample we look through possible parameters of
the curve which satisfies curve equation, and add
one (or some other value corresponding to the
weight of a sample) to the corresponding bin of the
accumu lator. This can be illustrated in Fig. 7 for
the case of two straight lines (Fig. 7, a), which
can be described in Cartesian xy-plane in the form
r = x⋅cosq + y⋅sinq [24] (see Fig. 2). In Fig. 7,
r, x and y are measured in the same nominal units,
q is measured in degrees.
An alternative well-known technique for esti-
mating parameters of parametric curves is ordinary
least squares (OLS) [23]:
( ),arg min S KSTθ =
θ
S
(3)
In case of straight lines in some two-dimension-
al space without any preferred directions, (3) can
be rewritten as:
[ , ]
( – ) .arg min sin cos
n
x y1
,
i i
i
n
2
1
θ ρ ϕ
ϕ ϕ ρ
= =
= +
ρ ϕ =
'
S S S
/
(4)
Denoting
( – ) ,sin cose
n
x y1
i i
i
n
2
1
ϕ ϕ ρ= +
=
/
(5)
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
a) d)
g)
e)b) c)
f) h) i)
Fig. 6. Clearing procedure for linear trajectory (SNR = 5 dB):
a — source image; b, c, d, e — important samples at steps 1, 4, 8 and 12; f, g, h, i — true trajectory (solid), estimated
trajectory before clearing step (dash), estimated trajectory after clearing step (dash-dot) at the same steps; the crossed
sample is the sample which is removed at the current clearing step
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
[s1,…,sL];
vector of errors;
the error or the “distance” between the
model and the jth data point, sj = f(xj, yj, q);
covariance matrix;
vector of model (curve) parameters.
where S =
T is
sj is
K is
q is
can be treated as the “distance” from the
origin to the line;
is the angle between the Ox axis and the
straight (Fig. 2);
are coordinates of the ith sample;
is the number of important samples.
where r
–j
xi, yi
n
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
and using (4) OLS for fitting data by a straight
line model can be written as follows:
– ( – ) ,
( – )
( ) ,
sin cos
sin cos
cos sin
e
n
x y
e
n
x y
x y
2 0
2
0
i i
i
n
i i
i
n
i i
1
1
#
#
2
2
2
2
ρ
ϕ ϕ ρ
ϕ
ϕ ϕ ρ
ϕ ϕ
= + =
= +
+ =
=
=
Z
[
\
]
]]
]
]]
/
/
(6)
whence we obtain
[ ] [ ],
– ,
sin cosE x E y
A B p A B p A4 4 02 2 2 2 2 2
$ $ρ ϕ ϕ= +
+ + + =^ ^h h)
(7)
where E[•] is the mean value of the expression in brackets;
p = cos2r;
A = E[xy] – E[x]⋅E[y];
B = (E[y2] – E2[y]) – (E[x2] – E2[x]).
The sign of cosj can be obtained as follows. If
– ,sin p1 0$ϕ = then
,
,
, ,
cos sin
sin
sin
sin
cos
p p A B
p A B
p A B
p A B
p
0 0
0 0
0 0
0 0
if
otherwise
/ / 0
0 / / 0
0 / / 0
0 / /
1 1
1 1
1 1
$ $ $
$
$
$
ϕ ϕ
ϕ
ϕ
ϕ
ϕ
=
=
^
^
^
^
h
h
h
h
Z
[
\
]
]
]
]
]]
(8)
while r can be either positive or negative.
To determine which of the roots of the qua-
dratic equation in (7) minimizes (5), the second-
order differential must be analyzed. Roots which
minimize (5) must satisfy the following conditions:
– 0 0,A B
B C
AC B Aand2 2 2∆ = =
(9)
where
2; – ( – );cos sinA e B e
n
x y2
i i
i
n
2
2 2
12
2
2 2
2
ρ ρ ϕ
ϕ ϕ= = = =
=
/
– –
( ) .
cos sin
sin cos
C e
n
x y
x y
2
i i
i
n
i i
2
2
2 2 2 2
12
2
ϕ
ϕ ϕ
ρ ϕ ϕ
= = +
+ +
=
^ ^h h6
@
/
If D=0 then minimum of (5) can be found by
a small variation of estimated parameters and
analyzing the behavior of (5).
For parabolic trajectories OLS can be simply
written as
,arg minA e
, ,a a a2 1 0
= " ,S
where ( – ) ,e
n
y a x a x a1
i i i
i
n
2
2
1 0
2
1
= + +
=
/
or, as a system of linear equations:
( – ) ,
( – ) ,
( – ) .
a
e
n
a x a x a y x
a
e
n
a x a x a y x
a
e
n
a x a x a y
2 0
2 0
2 0
i i i i
i
n
i i i i
i
n
i i i
i
n
2
2
2
1 0
2
1
1
2
2
1 0
1
0
2
2
1 0
1
2
2
2
2
2
2
= + + =
= + + =
= + + =
=
=
=
Z
[
\
]
]
]]
]
]
]]
/
/
/
(10)
Thus, the parameters of parabolic trajectory
form the vector [ , , ]A a a a2 1 0=S which satisfies
(10). It should be noted, that such technique
does not take into account possible rotation of
parabolic trajectory.
Detection step
To make a decision about the presence of a tar-
get in the analyzed image, an integration of values
over obtained trajectory is performed. The integra-
tion is carried out over a certain band around the
estimated trajectory (Fig. 8). Thus, decision rule
for target detection is S > VD, where
,S
m
w1
i
i
m
2
1
=
=
/
(11)
In this paper we consider simple parametric
detection procedure, which does not provide a
constant probability of false alarms with change
of noise power. To provide constant false alarm
y
10
0
–10
–20
–20 –10 0 10 x
q
270
180
90
0 10 20 r
Fig. 7. Source image, containing two straight lines (a)
and parametric space with two maximums (b)
a)
b)
the decision threshold, defined for previously
specified probability of false alarm;
the value of the ith sample within the integration
strip;
the number of samples within integration strip.
VD is
wi is
m is
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
ratio additional procedure for estimation of the
noise power should be introduced, and detection
threshold for statistics (11) should be changed
depending on such estimation.
Since the result of a detection procedure de-
pends on the estimated parameters, some arrange-
ments can be made to improve the accuracy of
the estimation, for example, preprocessing of the
source image by smoothing. The effect of prepro-
cessing in case of Gaussian kernel smoothing and
mean-smoothing will be shown in the next section.
Statistical modeling and simulation results
To obtain detection characteristics 5,000 simula-
tions for each value of SNR were made. Dimensions
of test images are 16×16 pixels. Integration strip
width is set to 2. Probability of false alarms is set
to 0,01. According to the signal and sensor models,
signal-to-noise ratio in each bin of the image is
defined as 10 ( /(2 )),log vS N10
2 2σ where vS is the
amplitude of the modeled signal; N
2σ is the noise
variance. All linear tracks were modeled in the
way they crossed the center of the image.
In Fig. 9 detection characteristics and estima-
tion errors are shown for different sample selection
techniques (discussed above): by independently
applying R- and C-masks and finding a maximum
of (11) among two sets of samples (RCM); by
applying RC-mask (RC); by applying RCD-mask
(RCD). Ideal detection characteristic in case of
known trajectory parameters is also shown for
comparison. OLS technique was used for trajec-
tory parameters estimation.
The unit of y-axis for RMSE of r̂ is one resolu-
tion cell. The unit of y-axis for RMSE of ĵ is one
radian. A “hat” symbol means that these values are
Fig. 8. Source image (a);
real (dashed line) and
es timated (solid line) tra-
jectories (b); integration
strip (c)
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
a) b)
y
12
8
4
0 4 8 12 x
c)
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
Fig. 9. Detection characteristics (a) and parameters
estimation errors (b, c) for different sample selection
techniques
a)
b)
ñ)
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
estimates of the true parameters. For comparison,
in Fig. 10 the same characteristics are shown for
the case when thresholding is applied as a sample
selection technique with probabilities of exceeding
the threshold for noise equal to 0.003, 0.01 and
0.3 (T_0.003, T_0.01 and T_0.3 respectively).
OLS technique was used for trajectory parameters
estimation.
Fig. 11 shows the detection characteristics and
estimation errors for different techniques of esti-
mating trajectory parameters: OLS and HT. For
HT the next initialization parameters were used:
step by “distance” Dr=1 resolution cell (rc), step
by angle Dj=1°.
Fig. 12 shows the influence of the following
initialization parameters for the HT: Dr=1 rc,
Dj=1° (HT_1); Dr=1 rc, Dj=2° (HT_2); Dr=2 rc,
Dj=1° (HT_3); Dr=0,5 rc, Dj=0,5 deg (HT_4).
RCM-technique is used for sample selection.
In Fig. 13 the effects of preprocessing smooth-
ing step in case of Gaussian kernel smoothing and
mean smoothing are shown. The radius of smooth-
ing is 1 cell (i.e. the size of smoothing region for
each pixel is 3×3), parameter of Gaussian kernel
was set to sSM=1,5. RCM is used for sample se-
lection, OLS is used for parameters estimation.
It can be seen that Gaussian kernel smoothing
gives a gain on the correct detection probability
as well as on the accuracy of trajectory parameters
estimation.
In Fig. 14 the effect of applying the clearing
step for reducing the number of false important
samples is shown. OLS technique was used for tra-
jectory parameters estimation. The next complex
stop-condition was used: either number of samples
is less than 3 or their variance around estimated
trajectory is less than 1.
Fig. 15 shows detection characteristics in case
of parabolic trajectory model. It can be seen, that
in this case the detection algorithm in general
shows worse performance in comparison with
the linear trajectory case. In some cases, as for
the one in Fig. 16, a, the use of a linear model
for integration (PT-LE_1) instead of a parabolic
model (PT-PE_1) may give better detection per-
formance. But in most cases, as in Fig. 16, b, the
parabolic model should be used (PT PE_2). In
general, linear trajectory model is more robust to
noise samples than second or higher orders models.
In reality, the trajectory of moving target can
often be approximated by a sequence of linear paths.
This fact allows us to use proposed algorithm to
detect such paths, which can be then joined into a
full target trajectory. As an example, an applica-
tion of the proposed algorithm for tracking moving
targets is shown in Fig. 17. In this figure a sequence
of simulated radar plots as well as results of target
detection for each plot are shown.
The size of a radar plot is 256×256 pixels. Each
plot is divided into subplots of 16×16 pixels which
are used for further analysis.
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
a)
b)
ñ)
Fig. 10. Detection characteristics (a) and parameters
estimation errors (b, c) for thresholding procedure as an
important samples selection technique (RCM is shown
for comparison)
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
a)
b)
ñ)
Fig. 11. Detection characteristics (a) and parameters
estimation errors (b, c) for OLS and HT techniques of
estimating the trajectory parameters
a)
b)
ñ)
Fig. 12. The influence of initialization parameters of
the HT on detection characteristics (a) and parameters
estimation errors (b, c)
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
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32
SIGNALS TRANSFER AND PROCESSING SYSTEMS
a)
b)
ñ)
Fig. 13. Detection characteristics (a) and estimation
errors (b, c) for different smoothing techniques
Fig. 14. Detection characteristics (a) and parameters
estimation errors (b, c) with introducing the clearing
procedure for different sample selection techniques
a)
b)
ñ)
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
6
5
4
3
2
1
0
R
M
S
E
o
f
r̂
–2 0 2 4 6 8
Signal-to-noise ratio, dB
1,0
0,8
0,6
0,4
0,2
0
R
M
S
E
o
f
ĵ
–2 0 2 4 6 8
Signal-to-noise ratio, dB
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
Target moves through observed region in a
parabolic trajectory. At every scan all radar sub-
plots are analyzed. Local trajectory paths, for
which value of (11) exceeds threshold (which
can be fixed or adaptive), are joined to previously
obtained results to build the full trajectory. The
task of further cancellation of false local paths
should then be solved. This task is not considered
in this paper and can be solved, for example, by
any of classical algorithms of secondary radar
signal processing [25].
Conclusion
It was shown that usage of Gaussian kernel
smoothing as the preprocessing step can give some
gain in target detection performance and accuracy
of the estimation of trajectory parameters.
The additional clearing procedure for reducing
false important samples can gives about 1 dB gain
in target detection performance (with probability
1,0
0,8
0,6
0,4
0,2
0C
or
re
ct
d
et
ec
ti
on
p
ro
ba
bi
li
ty
–2 0 2 4 6 8
Signal-to-noise ratio, dB
a)
y
12
8
4
0 4 8 12 x
y
12
8
4
0 4 8 12 x
b)
Fig. 16. Images with parabolic trajectories for
A=[0.1, –0.6, 1.9] (a) and for A=[0.2, –2.8, 10.8] (b)
y
204,8
153,6
102,4
51,2
0
y
204,8
153,6
102,4
51,2
0
y
204,8
153,6
102,4
51,2
0
y
204,8
153,6
102,4
51,2
0 102,4 204,8 x 102,4 204,8 x 102,4 204,8 x 102,4 204,8 x
y
204,8
153,6
102,4
51,2
0 102,4 204,8 x
y
204,8
153,6
102,4
51,2
0 102,4 204,8 x
y
204,8
153,6
102,4
51,2
0 102,4 204,8 x
y
204,8
153,6
102,4
51,2
0 102,4 204,8 x
Fig. 17. True trajectories (top) and estimated local trajectories (bottom) of moving target at time step 1 (a),
6 (b), 12 (c) and 17 (d) at SNR = 9 dB
a) b) c) d)
of correct detection 0,9) and increases accuracy
of estimation of the trajectory parameters for
SNR>0 dB.
In general, the proposed algorithm loses to an
optimal algorithm for detection of targets with
known trajectory about 3 dB in a threshold signal
with the probability of correct detection 0,9 and
probability of false alarms 0,01.
The proposed algorithm can be used in radars
for detection and local trajectory parameters
estimation of moving radar objects. Such local
trajectory paths can then be used to build full
trajectory of a target. Another field of use of this
algorithm can be the tasks of radar meteor detec-
tion and analysis of visual or infrared images (e.g.
FLIR images). It can also be used for forming
more precisely proposed distributions in some PF
TBD approaches, or as initiation procedure for
complex trajectory detection by analysis of its
small linear part.
Fig. 15. Detection characteristics
for parabolic trajec tories
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SIGNALS TRANSFER AND PROCESSING SYSTEMS
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Äата ïостóïлениÿ рóкоïиси
в редакциþ 30.09 2013 ã.
І. Г. ПРОКОПЕНКО, В. Ю. ВОВК, І. П. ОМЕЛЬЧУК, Ю. Ä. ЧИРКА, К. І. ПРОКОПЕНКО
Уêðàїíà, м. Кèїâ, Нàціîíàëьíèé àâіàціéíèé óíіâåðñèòåò
E-mail: prokop-igor@yandex.ru, vitalii.vovk@nau.edu.ua
ЛОКАЛЬНА ОЦІНКА ПАРАМЕÒРІВ ÒРАЄКÒОРІЇ ÒА ВИЯВЛЕННЯ
РУХОМИХ ЦІЛЕЙ НА ФОНІ РЕЛЕЇВСЬКОЇ ЗАВАДИ
Розãлÿнóто ïроблемó та заïроïоновано алãоритм виÿвленнÿ та локальної оцінки ïараметрів траєкторії
рóхомих цілей на основі аналізó даних ó формі двовимірноãо зображеннÿ. Виходÿчи з ïрийнÿтих мо-
делей цілі та детектора, фонова завада має розïоділ Релеÿ, а сиãнал — розïоділ Райса. Розãлÿнóто
два методи оцінки ïараметрів траєкторії: метод найменших квадратів і метод ïеретвореннÿ Хафа.
Òåõíîëîãèÿ è êîíñòðóèðîâàíèå â ýëåêòðîííîé àïïàðàòóðå, 2014, ¹ 1
35
SIGNALS TRANSFER AND PROCESSING SYSTEMS
И. Г. ПРОКОПЕНКО, В. Ю. ВОВК, И. П. ОМЕЛЬЧУК, Ю. Ä. ЧИРКА, К. И. ПРОКОПЕНКО
Уêðàèíà, ã. Кèåâ, Нàцèîíàëьíыé àâèàцèîííыé óíèâåðñèòåò
E-mail: prokop-igor@yandex.ru, vitalii.vovk@nau.edu.ua
ЛОКАЛЬНАЯ ОЦЕНКА ПАРАМЕÒРОВ ÒРАЕКÒОРИИ И ОБНАРУЖЕНИЕ
ДВУЖУЩИХСЯ ЦЕЛЕЙ НА ФОНЕ РЭЛЕЕВСКОЙ ПОМЕХИ
Рассмотрена ïроблема и ïредложен алãоритм обнарóжениÿ и локальной оценки ïараметров траектории
движóщихсÿ целей на основе анализа данных в форме двóмерноãо изображениÿ. Исходÿ из ïринÿтых
моделей цели и детектора, фоноваÿ ïомеха имеет расïределение Рэлеÿ, а сиãнал — расïределение
Райса. Рассмотрены два метода оценки ïараметров траектории: метод наименьших квадратов и
метод ïреобразованиÿ Хафа. Предложена ïроцедóра обнарóжениÿ цели, основаннаÿ на интеãрировании
отраженной мощности вдоль вероÿтной траектории. Проведено статистическое моделирование, ïо
резóльтатам котороãо ïостроены характеристики обнарóжениÿ длÿ ïредложенноãо алãоритма.
Клþчевые слова: обнарóжение, слежение, движóщийсÿ объект, оценка ïараметров, траекториÿ, track-
before-detect, расïределение Рэлеÿ, ïреобразование Хафа, метод наименьших квадратов.
Заïроïоновано ïроцедóрó виÿвленнÿ цілі, що ґрóнтóєтьсÿ на інтеãрóванні відбитої ïотóжності вздовж
імовірної траєкторії. Проведено статистичне моделþваннÿ, за резóльтатам ÿкоãо ïобóдовано характе-
ристики виÿвленнÿ длÿ заïроïонованоãо алãоритмó.
Клþчові слова: виÿвленнÿ, стеженнÿ, рóхомий об’єкт, оцінка ïараметрів, траєкторіÿ, track-before-
detect, розïоділ Релеÿ, ïеретвореннÿ Хафа, метод найменших квадратів.
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Джиган В. И. Адаптивная фильтрация сигналов: теория и алгоритмы.—
Ìîñêâà: Òåõíîñôåðà, 2013.
В мîíîãðàфèè ðàññмàòðèâàюòñÿ îñíîâíыå ðàзíîâèдíîñòè
àдàïòèâíыõ фèëьòðîâ è èõ ïðèмåíåíèå â ðàдèîòåõíèчå-
ñêèõ ñèñòåмàõ è ñèñòåмàõ ñâÿзè. Дàíî ïðåдñòàâëåíèå î мà-
òåмàòèчåñêèõ îбъåêòàõ è мåòîдàõ, èñïîëьзóåмыõ â òåîðèè
àдàïòèâíîé фèëьòðàцèè ñèãíàëîâ. Рàññмàòðèâàюòñÿ ïðèå-
мы ïîëóчåíèÿ âычèñëèòåëьíыõ ïðîцåдóð, ñàмè ïðîцåдóðы
è ñâîéñòâà òàêèõ àëãîðèòмîâ àдàïòèâíîé фèëьòðàцèè, êàê
àëãîðèòмы Ньюòîíà è íàèñêîðåéшåãî ñïóñêà, àëãîðèòмы
ïî êðèòåðèю íàèмåíьшåãî êâàдðàòà, ðåêóðñèâíыå àëãîðèò-
мы ïî êðèòåðèю íàèмåíьшèõ êâàдðàòîâ è èõ быñòðыå (âы-
чèñëèòåëьíî ýффåêòèâíыå) âåðñèè; ðåêóðñèâíыå àëãîðèò-
мы ïî êðèòåðèю íàèмåíьшèõ êâàдðàòîâ дëÿ мíîãîêàíàëь-
íыõ фèëьòðîâ è èõ âåðñèè дëÿ îбðàбîòêè íåñòàцèîíàðíыõ
ñèãíàëîâ, à òàêжå мíîãîêàíàëьíыå àëãîðèòмы àффèííыõ
ïðîåêцèé. Дàíî îïèñàíèå ñòàíдàðòíыõ è íåñòàíдàðòíыõ ïðèëîжåíèé дëÿ мîдåëèðî-
âàíèÿ àдàïòèâíыõ фèëьòðîâ íà ñîâðåмåííыõ ÿзыêàõ ïðîãðàммèðîâàíèÿ MATLAB,
LabVIEW è SystemVue, à òàêжå ðåàëèзàцèé àдàïòèâíыõ фèëьòðîâ íà ñîâðåмåí-
íыõ цèфðîâыõ ñèãíàëьíыõ ïðîцåññîðàõ îòåчåñòâåííîãî è зàðóбåжíîãî ïðîèзâîд-
ñòâà. Оñîбåííîñòью мàòåðèàëà ÿâëÿåòñÿ èзëîжåíèå òåîðåòèчåñêèõ мàòåðèàëîâ дëÿ
íàèбîëåå îбщåãî ñëóчàÿ — àдàïòèâíыõ фèëьòðîâ ñ êîмïëåêñíымè âåñîâымè êîýф-
фèцèåíòàмè, íàëèчèå ðàздåëîâ ïî мíîãîêàíàëьíым àдàïòèâíым фèëьòðàм è àëãî-
ðèòмàм àдàïòèâíîé фèëьòðàцèè íåñòàцèîíàðíыõ ñèãíàëîâ. Кíèãà ÿâëÿåòñÿ ïåðâым
ñèñòåмàòèчåñêèм èзëîжåíèåм òåîðèè àдàïòèâíîé фèëьòðàцèè íà ðóññêîм ÿзыêå.
Пðåдíàзíàчåíà дëÿ íàóчíыõ ðàбîòíèêîâ, èíжåíåðîâ, àñïèðàíòîâ è ñòóдåíòîâ ðàдè-
îòåõíèчåñêèõ è ñâÿзíыõ ñïåцèàëьíîñòåé, èзóчàющèõ è èñïîëьзóющèõ íà ïðàêòèêå
цèфðîâóю îбðàбîòêó ñèãíàëîâ è, â чàñòíîñòè, àдàïòèâíóю фèëьòðàцèю ñèãíàëîâ.
ÍÎÂÛÅ ÊÍÈÃÈ
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