ISSN 0735-2727, Radioelectronics and Communications Systems, 2015, Vol. 58, No. 10, pp. 470–478. © Allerton Press, Inc., 2015. Original Russian Text © D.I. Popov, 2015, published in Izv. Vyssh. Uchebn. Zaved., Radioelektron., 2015, Vol. 58, No. 10, pp. 47–56.
Detection and Measurement of Coherent-Pulse Signals D. I. Popov Ryazan State Radio Engineering University, Ryazan, Russia ORCID: 0000-0003-0342-8208, e-mail:
[email protected] Received in final form May 4, 2015
Abstract—Algorithms of joint detection and measurement of coherent-pulse signals with linear adjustment of parameters have been synthesized; these algorithms make it possible to unambiguously measure the radial velocity of target in the specified range while preserving an unambiguous measurement of the range. A functional block diagram of the detecting-and-measuring device was proposed. The computer simulation was used to carry out a comparative analysis of the synthesized and known algorithms of detection and measurement. DOI: 10.3103/S0735272715100052
INTRODUCTION The measurement of coordinates of moving targets in pulse radar systems (PRS) gives rise to a well-known problem of joint unambiguous measurement of the target range and its radial velocity [1]. The limit of unambiguous measurement of both coordinates is determined by the value of repetition period T of probing pulses. In designing radar the preference is initially granted to one of the coordinates. The unequivocal measurement of radial velocity of target with high resolution and accuracy involves the need of using the probing pulses with small pulse ratio [1]. In this case special measures are used for unequivocal measurement of the target range. Coherent-pulse radars using probing pulses with high duty ratio has gained wide application that is explained by the possibility of unambiguous range measurement of a large number of targets by simple means and with high resolution. The interval of unique measurement of the Doppler frequency ±1/2T and its corresponding interval of radial velocity prove to be insufficient for real velocities of the majority of radar targets. However, a number of radars (for example radar for air traffic control, meteorological radars, etc.), besides range, require information about the radial velocity of moving object. One of the solutions of the specified problem is the use of nonequidistant coherent-pulse signals and appropriate algorithms and devices for their processing [2]. For samples U j , j =1, N at odd values of N, arriving with alternating repetition periods T1 and T2 = T1 - DT [3], the following algorithms of joint detection-and-measurement were obtained: ½(N -1)/ 2 ½ ½(N -1)/ 2 ½ u = | X 1|+| X 2 | =½ å U 2*k -1U 2k½+½ å U 2*k U 2k +1½³ u 0 , ½ k =1 ½ ½ k =1 ½
(1)
* * * Dj$ = argX 1 X 2 = arctan(Im X 1 X 2 / Re X 1 X 2 ),
(2)
where u0 is the threshold level of detection; Dj$ is the estimate of Doppler shift of signal phase over interval DT; given an appropriate selection of DT, this estimate allows us to uniquely determine the radial velocity of target [3]. Alternating repetition periods T1 and T2 correspond to the simplest case of nonequidistant signals. Coherent-pulse signals having more advanced structure for solving this problem are of interest; the detection-and-measurement of such signals will be considered below.
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STATISTICAL DESCRIPTION OF SIGNALS Let us consider the processing of coherent sequence of N radio pulses; during their radiation carrier frequencies of the pulses beginning from initial value f0 linearly hop from pulse to pulse by Df, i.e., the carrier frequency of the jth pulse f j = f 0 + ( j - 1)Df , j =1, N . During the reflection of radio pulses from the moving target their carrier frequencies acquire Doppler phase shifts j j = j + ( j - 1)Dj , in this case j = 2pf DT = 4pv r f 0T / c, Dj = 2pDf DT = 4pv r DfT / c, where f D = 2v r f 0 / c and Df D = 2v r Df / c are the Doppler frequencies, v r is the radial velocity of target, c is the velocity of radio wave propagation. A similar effect is achieved with the linear change of repetition period of probing pulses in the form T j = T + ( j - 1)DT . In this case quantity j is determined in the same way, while Dj = 2pf D DT = 4pv r f 0 DT / c. The signal reflected from the moving target represents a narrow-band random process of Gaussian type that forms with intrinsic noise of the receiving device an additive mixture, the complex envelope of which as a result of analog-to-digital transformation is represented in the form of a sequence of N digital samples: U j = x j + iy j = U sj + U nj = |U sj |exp[i(q j + j 0 )] + U nj , j =1, N , j( j - 1) Dj is the total phase shift of the jth pulse, j 0 is the initial phase. 2 form an N-dimensional column vector U ={U j }T with correlation matrix
where q j = ål =1j l = jj + j
Samples U j
R sn = UU *T / (2s 2n ), the elements of which have the form * 2 R sn jk = U jU k / 2s n = qr jk exp[i(q j - q k )] + d jk ,
where q = s 2s / s 2n is the signal-to-noise ratio, s 2s and s 2n are the dispersions of signal and intrinsic noise, r jk are the coefficients of interperiodic correlation of signal, d jk is the Kronecker symbol. In the presence of signal and noise the joint probability density of vector U has the form 1 Psn (U ) = (2p ) - N detWsn expæç - U *T Wsn U ö÷, è 2 ø where Wsn is the matrix inverse to correlation matrix Rsn. On condition of the presence of only noise, the joint probability density Pn (U ) is described by a similar expression with matrix Wn = R n = I, where I is the identity matrix. SYNTHESIS OF DETECTION-AND-MEASUREMENT ALGORITHMS Calculating the conditional likelihood ratio L(j, Dj ) = Psn (U ) / Pn (U ), we shall obtain the optimal detection algorithm
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1 L(j, Dj ) = C expé - U *T ( Wn - Wsn )U ù ³ L 0 , êë 2 úû
(3)
where C = detWsn / detWn , L 0 is the threshold level of detection. Let us find an asymptotic approximation of the desired algorithm at q >>1. Then sn
R jk = R jk » qr jk exp[i(q j - q k )] . In addition, we take into account that echo signal of the majority of radar targets, such as flying objects, has an exponential correlation function depending on the normalized signal bandwidth df sT . With due regard for conditions Df << f 0 and df sT £ 0.01, we assume that coefficients of the signal interperiod correlation are as follows: r jk = exp( -pdf sT | j - k | ) = r | j -k | . In this case inverse correlation matrix Wsn = W has a band-diagonal structure with elements: W11 = W NN =
W jj =
W j -1, j = -
re
1+r2 q(1 - r 2 )
- ij j
q(1 - r 2 )
,
1 2 q(1 - r )
,
,
j = 2, N - 1,
W j , j -1 = -
re
ij j
q(1 - r 2 )
,
j = 2, N ,
where j j = j + ( j - 1)Dj . Algorithm (3) without due regard to edge effects at j =1and N assumes the form ìï 1 æ 1 + r 2 ö÷ N * L(j, Dj ) = C expí ç 1 U U 2 ÷å j j 2ç îï è q(1 - r ) ø j =1 +
é N - ij * ùü ij * ï j U j -1U j + e j U j -1U j )ú ý. ê å (e 2q(1 - r 2 ) êë j =2 úû ïþ r
(4)
Taking into account that *
U j -2U j -1 = ae
- i (j j - Dj )
,
i (j - Dj ) , U *j -2U j -1 = ae j
where a = |U j -2U *j -1| = |U *j -2U j -1|, and making substitutions e
- ij j
= e - iDjU j -2U *j -1 / a,
ij j
= e iDjU *j -2U j -1 / a,
e RADIOELECTRONICS
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in algorithm (4), after simple transformations we obtain æ r - iDj iDj * ö L( Dj ) = CCU expç (e Y + e Y )÷ , ÷ ç 2aq(1 - r 2 ) ø è
(5)
N é1 æ ù 1 + r 2 ö÷ N 2 where CU = expê ç 1 , | U | = Y ú åU j -2U *j -1U *j -1U j . j 2 ÷å ç 2 êë è q(1 - r ) ø j =1 úû j =3 An algorithm invariant with respect to unknown quantity Dj can be found as a result of appropriate integration of L( Dj ). Assuming the uniform distribution of quantity Dj on interval [–p, p], we shall find
L=
1 2p
p
ò L( Dj )dDj = CCU
-p
= CCU
1 2p
1 2p
p
æ
-p
è
ö
r
ò expçç aq(1 - r 2 ) (cos Dj Re Y + sinDj Im Y ) ÷÷ dDj ø
p
ö æ r ÷ dDj ³ L 0 . ç exp | |cos( D j arg ) Y Y ò ç aq(1 - r 2 ) ÷ ø è -p
This integral is standard and corresponds to the modified zero-order Bessel function 2 I 0 [| Y | r / ( aq(1 - r ))]. In this case I 0 ( z ) » e z for z >>1 that usually takes place during the interperiod processing of signals. Then after taking the logarithm and algebraic manipulations the detection algorithm assumes the form: ½N ½ * * u = | Y | =½åU j -2U j -1U j -1U j½³ u 0 , ½j = 3 ½
(6)
where u 0 = aq(1 - r 2 )(ln L 0 - ln CCU ) / r. The elimination of the processing corresponding to incoherent accumulation of CU value from the algorithm leads to insignificant losses in the threshold signal-to-noise ratio that do not exceed fractions of decibel. The dependence of threshold level u0 fixing the specified probability of false alarm on the intensity of initial samples determined by the level of intrinsic noise under the real conditions implies the use of adaptive threshold device [4]. Algorithm of Doppler phase Dj estimation shall be found by using the maximum likelihood method. The probability function represents probability density Psn (U / Dj ) considered as a function of parameter Dj. Maximization of the likelihood function aimed at finding estimate Dj$ is equivalent to maximization of the conditional likelihood ratio (5) or its logarithm. Thus, the maximum likelihood equation is equivalent to equation ¶ ln L( Dj ) / ¶Dj | Dj =Dj$ = 0, the solution of which leads to the estimation algorithm æ N * * Dj$ = argY = arctan(Im Y / Re Y ) = argç åU j -2U j -1U j -1U ç j =3 è
j
ö ÷. ÷ ø
(7)
The values of arctangent lie in the interval [–p/2, p/2]; its extension to [–p, p] is based on logical operations (4) in [3]. Practical implementation of the specified logical operations was proposed in a patent specification [5]. RADIOELECTRONICS
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POPOV ´&
´&
Uj
SDT
*
SDT
SA
PC
*
´
S
DS
b MC
vr
TU
u0 Fig. 1.
In the case of linear adjustment of the carrier frequency, the estimate of target radial velocity has the form: v$ r = Dj$ c / ( 4pDfT ) = bDj$ ,
(8)
where b = c / ( 4pDfT ) is the weighting coefficient. If we choose Df £ c / ( 4v r maxT ) for the maximum possible radial velocity of target v r max in accordance with the single-valuedness condition of Doppler frequency Df D £ 1 / (2T ), the unambiguous measurement of target radial velocity will be implemented in the entire range of real target velocities. In this case, single-valuedness of the range measurement will be ensured by an appropriate selection of the value of T. With the linear variation of the repetition period the estimate of target radial velocity assumes the form: v$ r = Dj$ c / ( 4pf 0 DT ) = dDj$ , where d = c / ( 4pf 0 DT ) is the weighting coefficient. The single-valuedness condition of Doppler frequency f D £ 1 / (2DT ) leads to unambiguous measurement of radial velocity on condition of choosing DT £ c / ( 4v r max f 0 ). It should be noted that with regard to coherent-pulse sequences with non-variable parameters algorithm (7) corresponds to estimation of the radial acceleration of target. FUNCTIONAL BLOCK DIAGRAM OF DETECTING-AND-MEASURING DEVICE A functional block diagram of detecting-and-measuring device of coherent-pulse signals with linear variation of the carrier frequency is presented in Fig. 1 [6]. Products * * * * * (U j -2U j -1 ) U j -1U j = U j -2U j -1U j -1U j ,
j = 3, N
are calculated on the basis of cascade connection of assemblies consisting of storage devices SDT & Next (performing storage for repetition period T), complex conjugate units (*), and complex multipliers (´). these products are fed to synchronous accumulator SA, where they are synchronously added resulting in formation of quantity Y. The execution of detection algorithm (6) is finalized in a unit of modulus calculation (MC) and threshold unit (TU). Phase calculator (PC) implements the algorithm of estimation (7) and logical operations [3, 5] extending the measurement range of Dj$ to interval [–p, p]. The weighting unit (´) determines the estimate of radial velocity v$ r in accordance with algorithm (8); this estimate via switch S is passed to the output of detecting-and-measuring device when detection signal (DS) arrives from threshold unit (TU). Signal DS is used during the automatic recording of other coordinates of target. Under the real conditions the threshold device should be adaptive as noted above [4].
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ANALYSIS OF MEASUREMENT ACCURACY The maximum likelihood estimates determined in accordance with algorithms (2) and (7) are asymptotically effective and asymptotically normal. In this connection, for the characteristic of measurement accuracy we can use the Cramer–Rao expression [7]: s D2 j$ = -[¶ 2 ln Psn (U / Dj ) / ¶Dj 2 ]-1,
(9)
where Psn (U / Dj ) is the likelihood function representing joint probability density of vector U with correlation matrix of the sum of signal and noise Rsn = Rs + Rn = R, where Rn = I. After calculations in expression (9) we obtain 2 s Dj$
é æ ¶ 2 W öù = êspç R ÷ú 2 ÷ êë çè ¶Dj ø úû
-1
=
detR sp(BR * )
,
(10)
where symbol “sp” designates the trace of matrix, W is the matrix inverse with respect to matrix R, while elements of matrix B are formed from algebraic complements A jk of elements R jk : B jk = -( j - k ) 2 A jk . Expression (10) characterizes the potential measurement accuracy specifying the lower limit of dispersion s 2Dj$ . If the signal-to-noise ratio is above threshold (q > q thr ), for practical purposes it is possible to use heuristic expressions for the root-mean-square error of measurement. Then, in the case of signals with alternating repetition periods we have s Dj$ = 2p /(18 . q( N - 1) / 2 ), while in the case of signals with linear variation of parameters we get: s Dj$ = 2p / (18 . q( N - 2)). However, the results of computer statistical simulation of algorithms and signal processing devices in question are the most reliable that gained recognition among radar designers, because they adequately reflect the characteristics of real devices. SIMULATION OF DETECTION-AND-MEASUREMENT ALGORITHMS Statistical simulation of detecting-and-measuring devices of radar signals includes the construction of a model of initial sequence (burst) of pulses and noises, transformation of this model in accordance with the processing (detection and estimation) algorithms and the statistical determination of desired performance indicators. Simulation can be expediently performed on computer in the environment of the general-purpose mathematical system MathCAD. MathCAD system has powerful built-in capabilities for implementing numerical methods of calculation and mathematical simulation in combination with the possibility of executing numerous mathematical operations with alphanumeric symbols. An additive mixture of coherent radio pulse burst reflected from the moving target and intrinsic noise of the receiving device at the input of detecting-and-measuring device can be presented in the form of a sequence of N digital samplesU j = x j + iy j of quadrature projections of complex envelope. Simulation of the these samples with the normal (Gaussian) distribution law and the specified correlation properties within the burst limits is reduced to specifying two groups of correlated numbers (h j and x j , j =1, N ) and two-dimensional rotation of each pair by angle q j . The sequence of random numbers obeying the normal distribution law is specified by using an in-built random number generator that is called by function rnorm( n , m , s) in the MathCAD system; parameters of this function include the number of called elements n, mathematical expectation m, and root-mean-square deviation s. RADIOELECTRONICS
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Then in the case of slow fluctuations of signal described by the exponential function of correlation | j -k | , where r = exp( -pdf sT ), for initial correlated numbers we have [8]: r jk = r h j = rh j -1 + 1 - r 2 rnorm(1,0, q / 2 )üï ý, 2 x j = rx j -1 + 1 - r rnorm(1,0, q / 2 ) ïþ
j =1, N ,
where h0 = rnorm(1, 0, q / 2 ), x 0 = rnorm(1, 0, q / 2 ), and q is the signal-to-noise ratio. Next, for specifying the quadrature components x j and y j it is necessary to perform two-dimensional rotations by angle q j and add the samples of uncorrelated (intrinsic) noise that is also a normal random process with zero mathematical expectation (m = 0) and the unity total dispersion of quadrature components (s 2n = 1). Then for quadrature components x j and y j we have: x j = ( h j cos q j - x j sin q j ) + rnorm(1,0, 1 / 2 ) üï ý, y j = ( h j sin q j + x j cos q j ) + rnorm(1,0, 1 / 2 )þï j =1, N ,
(11)
Dj j åk =1( -1) k for signals with alternating repetition periods T1 = T + DT / 2, 2 j( j - 1) T2 = T - DT / 2 and q j = jj + Dj for signals with linear variation of the carrier frequency or linear 2 variation of the repetition period. Next, generated samples U j = x j + iy j are subjected to processing in accordance with the known and proposed algorithms of joint detection-and-measurement. First we shall determine the value of detection threshold level u 0 fixing the specified probability of false alarm F. To this end, uncorrelated samples of noise sequence should be generated at the input of the investigated detecting device; the quadrature components of these samples at q = 0 are specified in the form of last terms of algorithms (11). The method of statistical tests (Monte Carlo method) [9] consisting of multiple repetition of detection algorithm (6) for the output decision-making statistic u (input quantity of the threshold device) is used to obtain sampling {u s }, s =1, S , where S is the number of test repetitions. Estimate of the false alarm probability F can be generated in accordance with its statistical definition: where q j = jj -
F = p$ = S 0 / S , where S0 is the number of exceedances of the detection threshold level u 0 by realizations u s . This is usually a set of its values for the purpose of subsequent plotting of histogram. In this case, according to the practice of statistical simulation, the minimum number of test repetitions amounts to S min =10 / p. With the specified probability of false alarm F = p = 10 -6 the minimum number of test repetitions amounts to S min =10 7 that can be unacceptable even with due regard to the performance of state-of-the-art computers. For estimation of the false alarm probability F we shall use the approximation of probability density w( u ) and numerical characteristics of statistic u. Since statistic u = | Y | ³ 0, probability density w( u ) is equal to zero at the negative values of argument. In this case the approximation of probability density w( u ) by the Laguerre series is the most expedient [10]. A sufficiently good approximation provides the first term of series [11] coinciding with gamma distribution, the parameters of which are the first two distribution cumulants k1 and k 2 . Having simulated sampling {u s } of input values of the threshold device, for calculating the estimates of sampling cumulants k$ 1 and k$ 2 in the MathCAD system we shall use the following in-built functions: RADIOELECTRONICS
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D
10–5
0.75
477
2 1
0.50
2
10–6
1 0.25
10–7
0 10
15
20
25
–5
u0
0
Fig. 2.
5
q, dB
Fig. 3.
k$ 1 = mean({u s }) and k$ 2 = var({u s }), where function mean({u s }) determines the average value of elements of sampling {u s }, while function var({u s }) determines the dispersion of elements of sampling {u s }. The false alarm probability F based on the gamma distribution and sampling cumulants k$ 1 and k$ 2 is determined by expression ¥
F = ò w( u )du = u0
¥
1 G( k$ 12
¥
$2 $
-u k /k ò e u 1 2 du, 1
(12)
/ k$ 2 ) u0
$2 $
where G( k$ 12 / k$ 2 ) = ò e - t t k1 / k2 -1dt is the gamma function. 0 The calculation of expression (12) is performed by using the in-built function of gamma distribution 2 pgamma(u 0 , k$ 1 / k$ 2 ). This function determines the probability of the event that random quantity u will have the value less or equal to the specified value u0. Then for the probability of statistic u exceeding threshold u0 we get: F = 1 - pgamma( u 0 , k$ 12 / k$ 2 ). The relationships of the false alarm probability F as a function of detection threshold level u0 obtained by above method at N = 21 and S = 104 are presented in Fig. 2. Line 1 corresponds to detection algorithm (1) and line 2 corresponds to algorithm (6). The values of detection threshold levels u0 for the specified algorithms at F = 10–6 are equal to 24.78 and 15.88, respectively. Detection characteristics are determined by simulation of a signal burst against the background of noises in accordance with algorithms (11) by varying the signal-to-noise ratio q for each test run. The correct detection probability D is determined by the classical Monte Carlo method in accordance with its statistical definition as detection frequency: D = p$ = S 0 / S . Figure 3 displays the smoothed detection characteristics obtained at N = 21, df sT = 0.01, F = 10 -6 , and S = 103. Curve 1 corresponds to detection algorithm (1) and curve 2 corresponds to algorithm (6). As can be seen, the proposed detection algorithm of signal burst with linear adjustment of parameters at D = 0.5…0.8 yields a gain of no less than 2 dB as compared to the detection algorithm of signal burst with alternating repetition periods. In determining estimate dispersion Dj$ characterizing the accuracy of unambiguous measurement of the radial velocity of target we must initially determine the mathematical expectation of estimate m Dj$ by averaging the cyclic samples of phase sampling {Dj$ s }, s =1, S 0 . However, the direct averaging of phase samples results in essential errors, since for the cyclic quantity (phase samples) the difference of order 2p is as small as the difference of the order of zero. For elimination of these errors trigonometric functions of estimates should be averaged similar to [12], in particular quantities exp(iDj$ s ) = cos Dj$ s + i sin Dj$ s . Now the unknown mathematical expectation can be determined as follows: m Dj$ = argZ = arctan(Im Z / Re Z ), where Z = ås=01(cos Dj$ s + i sin Dj$ s ). S
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POPOV sDj$ , rad 1 0.4 2 0.2
0
5
10
q, dB
Fig. 4.
Dispersion of the estimate is determined in accordance with expression 2
s Dj$ =
1 S0 1 S0 2 2 2 ( Dj$ s - m Dj$ ) @ Dj$ s - m Dj$ . å å S 0 - 1 s=1 S 0 s=1
The relationships of estimation root-mean-square error s Dj$ as a function of the signal-to-noise ratio q are presented in Fig. 4. Line 1 corresponds to algorithm (2) and line 2 corresponds to algorithm (7). At q = 5…10 dB the estimation error of algorithm (7) is less by 21…26% as compared to that of algorithm (2) that corresponds to an enhanced accuracy of estimation. From functional relationship (8) between estimates of radial velocity v$ r and Doppler phase shift Dj$ it follows that the root-mean-square error of radial velocity measurement is equal to s v$r = bs Dj$ and maintains the specified gains. CONCLUSIONS The synthesized algorithms of joint detection-and-measurement of the sequence of coherent-pulse signals with linear adjustment of their parameters make it possible to unequivocally measure the radial velocity of target in the specified range and maintain the single-valued measurement of the range. The proposed functional block diagram of detecting-and-measuring device performs single-channel coherent processing of digital samples of complex envelope of arriving signals against the background of intrinsic noise of the receiving device and can be implemented by the hardware and software means of digital computer technology. As a result of computer statistical simulation, it was established that the synthesized algorithms of detection-and-measurement possess advantages as compared to the known algorithms in terms of the threshold signal-to-noise ratio and the measurement accuracy of the target radial velocity. REFERENCES 1. M. I. Skolnik (ed.), Radar Handbook, 3rd ed. (McGraw–Hill, 2008), 1352 p. 2. D. I. Popov, S. V. Gerasimov, E. N. Mataev, RF Patent No. 2017167, Byull. Izobret., No. 14 (1994). 3. D. I. Popov, “Synthesis of detecting-and-measuring devices of Doppler signals,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 42(4), 11 (1999) [Radioelectron. Commun. Syst. 42(4), 10 (1999)]. 4. D. I. Popov, “Adaptive threshold devices,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 49(3), 30 (2006), http://radio.kpi.ua/article/view/S0021347006030058 [Radioelectron. Commun. Syst. 49(3), 26 (2006)]. 5. D. I. Popov, RF Patent No. 2513656, Byull. Izobret., No. 11 (2014). 6. D. I. Popov, RF Patent No. 2507536, Byull. Izobret., No. 5 (2014). 7. H. Cramer, Mathematical Methods of Statistics (Stockholm, 1946). 8. V. A. Likharev and D. I. Popov, “Efficiency analysis of moving target selection systems by computer simulation,” Radiotekhnika, No. 4, 99 (1970). 9. N. P. Buslenko, D. I. Golenko, I. M. Sobol’, V. G. Sragovich, Yu. A. Shreider, The Method of Statistical Tests (Monte-Carlo Method) (Fizmatgiz, Moscow, 1962) [in Russian, ed. by Yu. A. Shreider]. 10. R. Deutsch, Nonlinear Transformations of Random Processes (Prentice-Hall, Englewood Cliffs, N. J., 1962). 11. D. I. Popov, “Analysis of digital systems of signal detection by computer simulation,” Radiotekhnika, No. 12, 66 (1980). 12. D. I. Popov, “Estimation of passive interference parameters,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 46(3), 71 (2003) [Radioelectron. Commun. Syst. 46(3), 62 (2003)]. RADIOELECTRONICS
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