Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный исследовательский университет «МИЭТ»

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Diss / 10

.pdf

Скачиваний:

120

Добавлен:

27.03.2016

Размер:

18.05 Mб

Скачать

☆

<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 1213 / 6313 14 15 16 17 18 19 20 21 22 23 24 25 > Следующая >>>

Digital Interperiod Signal Processing Algorithms

Reference to (3.109) through (3.112) allows us to obtain the algorithm to control and adjust the threshold in the following form:

Kg (ρ, PF ) = EZ[ρ] − Φ		[0.5PFσZ (ρ )],		(3.113)
ˆ	ˆ	−1	ˆ

where

EZ[ ρˆ ] is the mean of decision statistic when the input process samples correlate with each other and this mean is equal to the mean of the decision statistic of the initial nonparametric generalized signal detection algorithm

σ2Z (ρˆ ) is the variance of decision statistic when the input process samples correlate with each other

The main problem under synthesis of the adaptive-nonparametric generalized signal detection algorithms is to define the variance of decision statistic at the adaptive-nonparametric DGD output, which depends on input process stationary and the way forming the sequence of values ςi.

The adaptive-nonparametric generalized signal detection algorithms with the adaptive tuning threshold based on the sign and rank criteria possess a satisfactory stability to changes in the noise correlation function at the adaptive-nonparametric DGD input. Simulation results show us that with increase in the coefficient of noise interperiod correlation from 0 to 0.5 the probability of false alarm PF at the adaptive-nonparametric DGD output increases by 2 ÷ 5 times, but in the case of the initial sign nonparametric DGD the probability of false alarm PF increases by 100 ÷ 300 times. The adaptive-nonparametric generalized rank signal detection algorithms have analogous characteristics on the stability of the probability of false alarm PF.

In conclusion, we would like to note that the digital adaptive asymptotic optimal generalized signal detection algorithms and the generalized signal detection algorithms using the similarity and invariance principles can be employed in addition to the adaptive-nonparametric generalized signal detection algorithms for signals with the correlated and uncorrelated noise with arbitrary pdf. Each signal detection algorithm has its own peculiarities defining a practicability to use these signal detection algorithms under specific conditions. It is impossible to design the signal detection algorithms that will have the same efficacy when the input signals are not controlled.

3.4 DIGITAL MEASURERS OF TARGET RETURN SIGNAL PARAMETERS

Estimation of target return signal parameters comprising information about the coordinates and target characteristics is the main operation of radar signal preprocessing. Estimation of parameters starts after making the decision a “yes” signal; that is, the target has been detected in the direction of radar antenna scanning at the definite distance. At this time, the target detection is associated with rough calculation of target coordinates, for example, the azimuth accurate with the radar antenna directional diagram width and the target range accurate with dimension of the resolution element in radar range and so on. The main task of digital measurer is to obtain more specific information about primary data of estimated target return signal parameters to the predetermined values of target return signal parameters.

Henceforth, we assume that the totality of target return signals, which are used to solve the problem of definition of the target return signal parameters, is within the limits of “moving/tracking window” and dimensions of this “moving/tracking window” correspond to strob bandwidth by radar range and the width of radar antenna directional diagram by angular coordinates. All initial conditions about statistic of the target return signals used under the operation of digital signal processing algorithms of radar signals are kept. We consider uniform estimations of the main noenergy parameters of radar signals, namely, the estimations of angular coordinates, Doppler frequency shift (the radial velocity), and time delay. Quality indices of one-dimensional measurements are the variance of errors σθ2, where θ = {β, ε, fD, td}, and the work content of the corresponding digital signal processing and detection algorithms.

94	Signal Processing in Radar Systems

3.4.1 Digital Measurer of Target Range

Definition of target range by CRSs is carried out as a result of measurements of the time delay td of the target return signal relative to the searching signal in accordance with the formula

td =	2rt g	,	(3.114)

	c

where c is the velocity of light propagation. In surveillance radar systems with uniform circular (sector) scanning by radar antenna or phased-array radar antenna implemented to define the coordinates a lot of targets, the target range is estimated by readings of the scaled pulses from the instant to send the searching radar signal ∆td1 (the transmit antenna) to the instant to receive the target return signal ∆td2 (the receive antenna, see Figure 3.16). In doing so, with sufficient accuracy in practice, we can believe that there is no target displacement within the limits of receiving the target return pulse train. Consequently, the target range measured by all N pulses of the target return pulse train can be averaged in the following form:

N
rˆtg = ∑rtg j ,	(3.115)

j =1

where rtg j is the target range measure by a single reading. The variance of estimation is given by

σr2ˆ = N1 ∑σr2j , (3.116)

j =1

where σr2j is the variance of estimation of the target range measure by a single reading.

Error of a single reading of the time delay td produced by the digital measurer of target range can be presented as a sum of two terms under condition that positions of count pulses on time axis are random variables with respect to the searching signal:

td = td1 + td2 ,

(3.117)

where

td1 is the random shift of the first count pulse with respect to the scanning signal

td2 is the random shift of the target return signal with respect to the last count pulse, see Figure 3.16

τcp

td1

td2

Searching	Target return
signal	signal

FIGURE 3.16 Definition of the range to target—a time diagram.

Digital Interperiod Signal Processing Algorithms

These errors are independent random variables uniformly distributed within the limits of the count pulse duration τcp, that is, within the limits of the interval [−0.5τcp, +0.5τcp]. By this reason in this case, the variance of error of a single reading of the time delay td is determined in the following form:

σt2d =	1	τcp2 .	(3.118)
	6
If the initial instant of count pulses is synchronized with the searching signal, in this case,			td1 = 0, the
variance of error of a single reading of the time delay td is determined in the following form:
σt2d =	1	τcp2 .	(3.119)
σt2d =	12	τcp2 .	(3.119)
	12

The discussed flowchart of digital measurer of the target range can be realized by a specific microprocessor.

3.4.2 Algorithms of Angular Coordinate Estimation under Uniform Radar Antenna Scanning

Optimal algorithms to measure the angular coordinate are synthesized by the maximal likelihood ratio criterion, as a rule. The shape of the likelihood function depends on the statistical characteristics of the signal and noise, radar antenna directional diagram shape, and radar antenna scanning technique in the course of measuring. First, we consider the target return pulse train processing that is obtained as a result of uniform rotation of the radar antenna within the limits of radar range sampling interval.

Under multiple-level quantization of the target return signals and weight functions we obtain digital counterpart of optimal measurer of angular coordinate in the scanning plane, that is, for the two-coordinate surveillance radar system—the coordinates of target azimuth βtg. The likelihood function for azimuth estimation by N normalized nonfluctuating target return pulse train in the stationary noise take the following form:

		N
(x1, x2 ,…, xN \|q0 ,βtg ) = ∏p(xi \|qi ,βi ),				(3.120)
		i=1
where for the considered case we have
p(xi \|qi ,βi ) = xi exp	−0.5(xi2 + qi2 ) I0 (qi , xi ).			(3.121)

In turn,
qi = q0 g(βi ,βtg ),				(3.122)
where q0 is the SNR by voltage in the center of target return pulse train (see Figure 3.17a);
g(βi ,βtg ) = g		βi − βtg	= g(α)	(3.123)


		ϕ0

is the function defining the radar antenna directional diagram envelope, for receiving and transmitting in the scanning plane; φ0 is the one-half main beam width of the radar antenna directional diagram at zero level; and βi is the value of azimuth angle when receiving the ith pulse of the pulse train.

96	Signal Processing in Radar Systems

βi

γ (βi, βtg)

qi = q0 g(βi, βtg)

βtg

βi

(a)

(b)

FIGURE 3.17 (a) Target return pulse train and (b) discrete weight function.

Under the fixed value of q0, we can obtain the final expression of the likelihood function equation for estimation of the angular coordinate βtg in the following form [30,90,91]:

N
ˆ	(3.124)
∑xi γ (βi ,β tg ) = 0,	(3.124)

i=1

where

ˆ	∂g(βi	ˆ
		,β tg )
γ (βi ,β tg ) =			, i = 1,…, N	(3.125)
	ˆ
	∂β tg

is the discrete weight function, that is, a sequence of the weight coefficients, to weigh the normalized amplitudes of target return pulse train (see Figure 3.17b). This weight function has a form of discriminatory characteristic, the gain slope of which is a function of the radar antenna directional diagram shape and the zero point coincides with the maximum of the radar antenna directional diagram.

Under estimation of target azimuth angle by the quick-fluctuating target return pulse train, the likelihood function equation takes the following form:

N
2	ˆ	(3.126)
∑xi	γ ′(βi ,β tg ) = 0,	(3.126)

i=1

where

		ˆ			∂g(βi	ˆ
ˆ	g(βi ,β tg )				∂g(βi	,β tg )
γ ′(βi ,β tg ) =	2 2	ˆ	2	×	ˆ		, i = 1,…, N	(3.127)
		ˆ			∂β tg
	1 + k0 g (βi ,β tg )

is the discrete weight function to weight the normalized amplitudes of the target return pulse train from the quick-fluctuating target surface.

Comparing (3.124) and (3.126), we see that in contrast to the case of nonfluctuating target surface to estimate the azimuth angle of target with the quick-fluctuating target surface, there is a need to

Digital Interperiod Signal Processing Algorithms

sum the squared amplitudes of the target return pulse train with their weights. In this case, the weight function γ ′(βi, βˆ tg) has a more complex form, but its character and behavior do not change. Thus, the optimal algorithm for the estimation of azimuth target angle under the uniform radar antenna scanning includes the following operations:

•The target return pulse train storage in the “moving/tracking window,” the bandwidth of which corresponds to the train time

•Weighting of each target return signal amplitude in accordance with the values of corresponding weight coefficients

•Formation of half-sum of the weighted target return signal amplitudes observed by the “moving/tracking window” at the right and left sides with respect to zero value of the weight function

•Comparison of half-sums and fixation of position where the result of comparison passes through zero value

The simplest example of block diagram of realization of the target azimuth angle coordinate estimation algorithms given by (3.124) and (3.126) is presented in Figure 3.18. In accordance with this block diagram, to realize the target azimuth angle coordinate estimation algorithm there is a need to carry out N − 1 multiplications and N − 1 summations of multidigit binary data while receiving each target return pulse train. The problem of potential accuracy definition of the target azimuth angle coordinate estimation by the target return pulse train can be solved by analytical procedures or simulation with definite assumptions.

Under binary quantization of the target return signal amplitudes (the weight multidigit function), the likelihood function of the estimated target angular coordinate βtg takes the following form:

N	1− di
di		,	(3.128)
(βtg ) = ∏PSNi	(xi ) PSNi

i=1

where

PSNi is the probability that the input signal (the target return signal) exceeds the threshold of binary quantization at the ith position of the target return pulse train

PSNi = 1 − PSNi is the probability to get the unit on the jth position of the target return pulse train di is given by (3.47)

Input memory

device

Azimuth scaled

pulse counter

.....

0.5 N

Compa-

xi γi

γi

rator

i=1

0.5N+1

.....

γN

γ1

Target azimuth code

1 N

ROM

FIGURE 3.18 Block diagram of the target azimuth angle coordinate estimation algorithm given by (3.124) and (3.126).

Signal Processing in Radar Systems

In the case of nonfluctuating target return signals we can write

∞

PSNi

= ∫ xi exp −

+ qi

(qi , xi )dxi ,

(3.129)

and in the case of fluctuating target return signals we have

∞

PSNi = ∫

exp

−

dxi.

(3.130)

1+ ki2

(1+ ki2 )

After appropriate mathematical transformations we can write the following likelihood function equation for the considered case

N
ˆ	(3.131)
∑diγ ′′(βi ,β tg ) = 0,	(3.131)

i=1

where

ˆ	1		dPSNi
γ ′′(βi ,β tg ) =		×		(3.132)
γ ′′(βi ,β tg ) =		×	ˆ	(3.132)
	PSNi PSNi		dβ tg

is the weight function of target return pulse train positions under estimation of the target azimuth angle. A type of this function is analogous to the function given by (3.125) in the case of the nonfluctuated target return signal and by (3.127) in the case of the fluctuated target return signal.

Thus, the optimal estimation of target angular coordinate comes to a sum formation of the weight coefficient values γ ″(βi, βˆ tg) on positions where di = 1 at the right and left sides with respect to zero value of the weight function. Coordinate estimation is fixed when the sums accumulated in this manner are the same within the limits of accuracy given before. Realization of this algorithm of the target return angular coordinate estimation is much simple in comparison with the algorithm given by (3.124) through (3.127), since the multiplication operations of multidigit numbers are absent. Potential accuracy of the measurer designed and constructed based on this algorithm obtained using the Cramer–Rao equality is determined by the formula

σ2ˆ

(3.133)

βtg

∑

SNi

i=1

PSNi

dβtg

PSNi

Under binary quantization of the target return signal without taking the radar antenna directional diagram shape into consideration we arrive to heuristic algorithms of the target return angular coordinate estimation:

Digital Interperiod Signal Processing Algorithms

1. By a position of the first and last pulses of the target return pulse train
ˆ	+ µ − k]} β ,	(3.134)
βtg = {0.5[λ − (l − 1)

where

λis the position number with respect to direction selected as the origin and the first pulse of the target return pulse train is fixed on this position by the criterion of l from m (l/m)

μis the position number and the last pulse of the target return pulse train is fixed on this position by the criterion of k one after another omission

βis the angular resolution of pulses in train

This algorithm provides for shift compensation of the instant of target return pulse train detection at the (l − 1)th position and a shift of detection instant of the last pulse of train on k positions.

2.By a position of the last pulse of the target return pulse train and the number of positions from the first pulse to the last pulse of the target return pulse train

ˆ	(3.135)
βtg = [µ − 0.5(Np − k − 1)] β ,

where Np is the number of positions appropriate to the bandwidth of the detected target return pulse train; this algorithm is realized by digital storage of binary-quantized signals.

Heuristic generalized algorithms of the target return angular coordinate estimation are the simplest in realization but lead us to 25%–30% losses in accuracy in comparison with the optimal generalized signal detection algorithms. Functional dependence of the relative variance of target return angular coordinate estimation, which is given by (3.134) and (3.135), on SNR at the center of target return pulse train at Np = 15 and PF = 10−4 (see Figure 3.19) allows us to compare by accuracy the discussed target return angular coordinate estimation under the uniform radar antenna scanning.

σ2^ βtg

β Nonfluctuated Fluctuated

4	1	Digital storage
3		device of units
2	2	“4/5 – 2”
2	3	“3/4 – 2”
	4	“3/3 – 2”
1
1
3
4
2
1

2	3	4	5	SNR (dB)
				SNR (dB)

FIGURE 3.19 Relative variance of target return angular coordinate estimation as a function of SNR in the center of the target return pulse train.

100	Signal Processing in Radar Systems

3.4.3 Algorithms of Angular Coordinate Estimation under Discrete Radar Antenna Scanning

In target tracking and controlling radar systems the methods of monopulse direction finding employed by the multichannel radar systems or the discrete radar antenna scanning employed by the single channel radar systems are used for accurate angular coordinate measurement. In such cases, tracking or nontracking digital measurer can be used. Henceforth, we consider an algorithmic synthesis of the nontracking measurer of a single angular coordinate in the single channel radar system with discrete radar antenna scanning. Under measuring by the discrete scanning method the radar antenna takes two fixed positions (see Figure 3.20). In each fixed position, a direction of maximal radiation with respect to an initial direction corresponds to the angles θ1 and θ2(θ2 > θ1). The difference between angles θ1 and θ2, that is, Δθ = θ2 − θ1 is called the angle of discrete scanning. Under angle readings relative to radar boresight a deviation of maximal radiation at scanning is ±θ0 and a deviation of the target at scanning is θtg.

Measurement of angular coordinate in the discrete radar antenna scanning plane lies in receiving n1 target return signals from direction θ1 and n2 target return signals from direction θ2. Under target displacement θtg with respect to the radar boresight, the amplitudes of target return signals received from each direction are not the same and are equal to X1i and X2i, respectively. We are able to define the azimuth target angle by the amplitude ratio of these target return signals. As before, an optimal solution of the problem of target angular coordinate estimation is defined by maximal likelihood function criterion. In this case, the likelihood function equation takes the following form:

∂ (x1, x2	\|θtg )	= 0,	(3.136)
∂θtg		= 0,	(3.136)
∂θtg		ˆ
		θtg = θ tg

where

x1 =

X11

X12

X1n1

x11x12 x1n1

(3.137)

σn1

θ1 θ2 Radar

boresight

Target

θ0 θtg

FIGURE 3.20 Fixed positions of radar antenna under discrete scanning.

Digital Interperiod Signal Processing Algorithms

101

and

x2 =

X21

X22

X2n1

x21x22 x2n2

(3.138)

σn2

are the vectors of normalized amplitudes of target return signals received from the directions θ1 and θ2, respectively.

The likelihood function of sample ( ) is differed subjecting to accepted models of the signal and noise. When the target return signal fluctuations are absent and the samples are statistically independent, the likelihood function can be presented in the following form:

n1	n2
(x1, x2 \|θtg ) = ∏pSN1 (x1i )∏pSN1 (x2 j ),		(3.139)
i=1	j =1

where

	x12i + q12 (θtg )	× I0[x1i , a1(θtg )],
pSN1 (x1i ) = x1i exp −			(3.140)
	2

	x22i + q22 (θtg )	× I0[x2i , a2 (θtg )],
pSN1 (x2i ) = x2i exp −			(3.141)
	2

q1(θtg ) = q0 g(θtg − θ0 ) = q0 g(θtg + θ0 ),			(3.142)
q2 (θtg ) = q0g(θ0 − θtg ),			(3.143)

where

g(·) is the normalized function of the radar antenna directional diagram envelope

q0 is the SNR by voltage at the radar antenna directional diagram maximum (the same for both directions)

Substituting (3.139) in (3.136) and taking into consideration (3.140) through (3.143), in the case of powerful target return signal, we get the following result:

ln I0[x1i , a1(θtg )] ≈ x1iq1(θtg ).

(3.144)

After evident mathematical transformations we obtain the final likelihood function equation in the following form:

n1	n2	ˆ
ˆ	ˆ	ˆ	ˆ	,	(3.145)
∑x1i + υ(θ tg) ∑x2 j = q1(θ tg )n1		+ υ(θ tg )q2	(θ tg )n2	,	(3.145)
i =1	j =1

102

where

	ˆ
ˆ	dg(θ0 − θ tg )
υ(θ tg ) =	ˆ	×
	ˆ
	dθ tg

Signal Processing in Radar Systems

	ˆ		−1
	ˆ
dg(θ0 + θ tg )
		.		(3.146)
ˆ		.		(3.146)
ˆ
dθ tg

Thus, while measuring the target angular coordinate by procedure of the discrete radar antenna scanning, the digital signal processing of target return signals is used for accumulation of the normalized target return signal amplitudes at each position of radar antenna with subsequent solution of (3.145) with respect to θtg. At this time, we assume that the signal and noise parameters characterized by q1, σn1 and q2, σn2 and the function describing the radar antenna directional diagram are known. Solution of (3.145) can be found, in a general case, by procedure of sequential searching, using a partition of the interval of possible target position values with respect to radar boresight equal to 2θ0 on m discrete values. The number m = 2θ0/δθtg, where δθtg is the required accuracy of the target angular coordinate estimation.

To reduce the time in finding a solution, the function υ(θtg) can be tabulated preliminarily with the given resolution δθtg. Approximate solution of (3.145) can be obtained by the following way. Assume that the target is into the neighboring area that is very close to radar boresight, so we can think that as θtg → 0

	g′(θ0	ˆ
υ(θtg ) =		− θtg)	= −1.	(3.147)
	g′(θ0	ˆ
		+ θ tg )

In this case, (3.145) takes the following form:

n1	n2
∑x1i − ∑x2 j = q1n1 + q2n2.		(3.148)
i =1	j =1

Further, believing that

n	n
n1 = n2 = n, ∑x1i = π1, and	∑x2 j = π2	(3.149)
i =1	j =1

Equation 3.148 can be transformed into the following form:

ˆ	ˆ	(3.150)
π1 − π2 = nq0[g(θ0 + θ tg )	− g(θ0 − θtg )].	(3.150)

In the case of Gaussian radar antenna directional diagram, we can use the following approximation:

	ˆ		ˆ		ˆ
g(θ0 − θtg )		= exp[−α(θ0	− θtg )	2	] ≈ 1 − χ θ tg ,
g(θ0 − θtg )		= exp[−α(θ0	− θtg )		] ≈ 1 − χ θ tg ,	(3.151)
	ˆ		ˆ	2	ˆ	(3.151)
	ˆ		ˆ	2	ˆ
g(θ0 + θ tg ) = exp[−α(θ0			+ θ tg )		] ≈ 1 + χθ tg,

<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 1213 / 6313 14 15 16 17 18 19 20 21 22 23 24 25 > Следующая >>>

Соседние файлы в папке Diss

#
27.03.201614.79 Mб68(Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag .pdf
#
27.03.20161.14 Mб451.pdf
#
27.03.201618.05 Mб12010.pdf
#
27.03.20161.14 Mб4511.pdf
#
27.03.2016438.53 Кб4412.pdf
#
27.03.2016951.5 Кб5313.pdf
#
27.03.2016120.99 Кб4916.pdf
#
27.03.2016154.55 Кб5319.pdf