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I. NOMENCLATURE

Monopulse-Radar Angle Tracking in Noise or Noise Jamming

ARNOLD D. SEIFER, Senior Member, IEEE Raytheon

In m n y monopulse radars, feedback in the angle-tracking servo system is taken to be directly proportional to the monopulse ratio. In those radars, monopulse measurements are conditioned on simultaneous occurrences of receiver sumchannel video exceed& a detection threshold if a detection fails to occur, the measurement is ignored, and the angle-tracking servo is made to “coast.” Such conditioning is shown here to be necessary in order that the noise power be finite in the servo feedback. As an extension of I. Kanter’s work, the conditional mean value and conditional variance of the monopulse ratio are therefore derived and quantified in terns of threshoid level as well as signal-to-noise ratio. Furthermore, the formulation here permits the noise covariance between receiver difference and sum channels to be complex rather than only real valued, as assumed by Kanter, thereby no longer restricting the sources of noise j a d n g to be positioned in the receiving-antenna mainlobe and to be copolarized with the antenna response there. Nonfluctuating and

Rayleigh-fluctuating target cases are considered and compared, and fluctuation loss is quantified.

Manuscript received November 11,1989; revised July 26,1991.

IEEE Log No. 9107179.

Author‘s current address Lockheed Sanders, Inc, PO. Box 0868, Nashua, NH 03061-0868.

0018-9251/9263.00 @ 1992 IEEE

622 IEEE TRANSACTIONS ON AEROSPP

2a2 2b2 F

Gi

P

A

P

X ai

Variance of D Variance of S Sum-channel, baseband, signal voltage of the fluctuating target Sum-channel, baseband, signal voltage of the j th jammer ( j = 1,2, ...) Threshold-to-noise voltage ratio Monopulse ratio for the true target position Dj/G,, where D, denotes the difference-channel, baseband, signal voltage from the j th jammer SUm-Channel, baseband, signal voltage of the nonfluctuating target Event of the detection threshold being exceeded by IS1 Imaginary part of the coefficient of correlation between D and S Sum-channel, baseband voltage of jamming plus receiver thermal noise Voltage level of detection threshold (a2/b2)(1 - p2 - C2) 3 - pa/b Difference-channel baseband voltage of jamming noise plus receiver thermal noise Real part of the coefficient of correlation between D and S Sum-channel signal-to-noise ratio Sum-channel jamming-to-noise ratio

II. INTRODUCTION

In monopulse radar, angle-tracking error is often measured by a digital computation of the monopulse ratio, which is defined as

R E Re{D/S} (1)

where D and S are simultaneous samples of signal plus noise taken at baseband from the monopuke-receiver difference and sum channels, respectively. Because a sample from each channel consists of an I (in-phase) and a Q (quadrature) component, D and S in (1) are “phasors” and as such are both complex valued: S’ denotes the complex conjugate of S; IS1 denotes the absolute value of S; and Re{. . . } denotes “real part.” Quantity R in (1) is a measure of angle-tracking error, which provides the feedback in the angle-tracking servo system.

In the presence of receiver thermal noise or noise jamming, I. Kanter had shown that R has infinite variance regardless of how large the signal-to-noise ratio happens to be [l-31. That result is indeed correct, for division by zero in (1) is always an event of non-zero probability density when the real and imaginary parts of S are jointly Gaussian distributed. However, one detail in radar processing had not been

LCE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

considered. In particular, monopulse measurements of R are often conditioned on the event of a threshold detection occurring from the receiver sumchannel video. In the absence of such an occurrence, the measurement is ignored, and the angle-tracking servo is made to "coast." Under such conditioning, the denominator in (1) is bounded away from zero by A, where X represents the detection-threshold voltage level. Therefore, the existence of all moments of R is assured, including its conditional variance, for the following inequality follows from (1):

from which it can be shown that for n = 1,2,. ..,

IW"1I I E[IRl"l< X-"E[lwl (conditioned on IS] > A)

where E[. . .] denotes mathematical expectation. Since the real and imaginary parts of D are jointly Gaussian distributed, the conditional expectation of IDJ" must exist. This implies, by the above chain of inequalities, the existence of E[R"], conditioned on IS( > A. In this way, noise power in the conditioned feedback to the angle-tracking servo system will always be finite when the servo error is taken proportional to R.

Conditioning monopulse measurements by threshold detection is not the only way to deal with the infinite variance of the monopulse ratio [4], [5, sec. 11.7. All values of R, from -cm to 00, have a one-to-one correspondence with all angular positions between the two nulls bounding the principal lobe of the monopulse, receiving-antenna sum pattern in the elevation or traverse plane. Those angular positions, as radar-measured angle-tracking errors, can be used instead of R itself for feedback in the angle-tracking servo only if the nonlinear function representing the one-to-one correspondence is always known. The angular position corresponding to each random outcome of R is also random but, unlike R, must be finite in variance because the antenna mainlobe is bounded in its angular extent. In this way, feedback data to the angle-tracking servo will always be finite in variance.

Only within a local interval about the principal null of the monopulse receiving-antenna difference pattern, R will be nearly proportional to angular displacement of target position from that null, where the constant of proportionality is the monopulse slope, k, [6, sec. 9.11. Except in certain geometries of hostile jamming, as identified later in Section V of this paper, conditioning measurements of R by threshold detection will usually limit the random outcomes of R to a sufficiently small variance that the proportional relationship between R and angle can apply. In this way, only the monopulse slope need be specified, instead of the entire nonlinear function that would otherwise be required, in order

to determine the feedback to the angle-tracking servo. Only this approach is examined here.

The target signal is assumed to be either constant in strength from echo to ech+A or to fluctuate randomly with a Rayleighdistributed amplitude. As in Kanter's studies, noise from jamming is taken here to be a Gaussian process. Since receiver thermal noise is also Gaussian, the real and imaginary parts of D and S taken altogether are jointly Gaussian distributed in the presence of either target-fluctuation case. Therefore, the joint probability distribution of D and S is completely specified in terms of their mean values and their covariance matrix.

Kanter assumed a real-valued covariance between D and S, which was consistent with his consideration of noise-jamming sources restricted in their location to the principal lobe of the receiving-antenna sum pattern, and restricted in their polarization to match the antenna response there. A more general covariance is assumed here, being taken to be complex valued, thereby permitting the sources of noise jamming to be unrestricted in their positions and polarizations [7]. The need to consider a complex-valued covariance is also anticipated in [5, secs. 11.2, 11.31.

Fluctuation loss is the decibel difference in signal-to-noise ratio between a fluctuating and a nonf luctuating target which are both tracked equalIy well, in angle, by the radar in terms of some measure of angle-tracking performance, such as mean-square error, error variance, or bias error. In each target-fluctuation model, the performance of the angle-tracking servo system is affected by signal-to-noise ratio in two ways. First, feedback in the servo system is provided by the radar-measured monopulse ratio, whose error statistics depend directly on the target signal-to-noise ratio. Second, servo feedback is not sampled uniformly in time but is sampled randomly at only those moments during which a threshold detection happens to occur in the receiver sum channel. Here, the servo will coast more frequently and for longer periods of time when the target signal-to-noise ratio is lowered, thereby lowering the probability of detection. During a coasting period, feedback is therefore suspended in sensing servo error during a target maneuver, for example, causing the servo to lag further behind the target motion. Furthermore, the more often the servo coasts, the fewer will be the number of feedback samples that will be statistically smoothed by the servo during its integration time, thereby worsening the influence of the random errors in these samples.

A complete accounting of fluctuation loss must therefore include the effects of target signal-to-noise ratio not only on measurement error in the monopulse ratio, but also on the probability of threshold detection and the impact of coasting on servo performance. However, the effect of coasting is not considered here and is deferred to a future study. AS a first attempt to

SEIF'ER MONOPULSE-RADAR ANGLE TRACKING IN NOISE OR NOISE JAMMING

1 ' _ _

623

quantify the effect of target fluctuation on monopulse angle tracking, only the error in the monopulse ratio is considered here.

conditional variance of R are formulated for both target-fluctuation cases. Their fundamental properties are established in Sections IV and V, where the more general results are identified in Section IV. In order to illustrate the application of the theory and establish further principles, two examples are considered in Section V. In Section VI, the major issues and their conclusions are summarized. In this context, topics are identified for further study.

In Section 111, the conditional mean and

Ill. ANALYSIS

For convenience, both the sum-channel envelope IS(, and the detection-threshold voltage level A, are now normalized with respect to the rms (root-mean-squared) sum-channel noise voltage, thus defining L and eo:

L = lSl/JE[Iql2]; eo = X / V q l q l 2 ] (2)

where q denotes the baseband noise process from the receiver sum channel. In terms of (2), the event of a threshold detection is denoted by A:

A = {L IL >eo}. (3) The conditional moments of R are then given by

m

E[R" I A] = l _ r " p ~ l ~ ( r ; e o ) d r (n = 1,2, ...).

(4a)

Here, P R ~ A ( r ; ! ~ ) denotes the conditional probability density function for R, given the occurrence of event A, and is obtained from

P R ~ A (r; eo) = - P D JmPLn(e , r )de to

1:

(4b)

where P L R ( ~ , r) denotes the joint probability density function for L and R, and where PO denotes the probability of detection:

PO Pr[A] = P ~ ( t ) d t . (4c)

In (k), p ~ ( l ) represents the marginal probability density function for L. In terms of (k), the conditional variance of R is given by

a2[R I A] = E[R2 I A] - (E[R I A])2. (5)

In the following, p ~ ~ ( e , r ) in (4b) is derived in terms of the mean values and covariance matrix of the real and imaginary parts of D and S, which are all jointly Gaussian distributed. Since the problem lends itself well to complex notation, it is used wherever

possible. Here, some or all of the arguments of a probability density function are expressed as complex variables; however, the function itself is always determined real valued and nonnegative [S , 91.

The variances and covariances of D and S are more easily discussed in terms of DO and SO as defined

DO D - E[D]; SO S - E[S] (6a) by

from which U and b denote

. = d e ; b = d s . (6b)

The real and imaginary parts of the coefficient of correlation between D and S are denoted by p and <:

p + is = E[S;Do]/(2ab)

p' + c* 5 1.

(6c)

which are bounded by

(7)

In (6c), i denotes the imaginary unit.

regarded [ to be zero. However, 5 can be expected to depart from zero in a more general ECM (electronic countermeasure) environment. For example, < can differ from zero whenever jamming sources happen to be positioned where the polarization of the receiving-antenna sum response differs from that of the antenna difference response [7]. Such positions can occur in the far sidelobe and backlobe regions of the antenna pattern, where polarization responses are unpredictable. Although both regions are quite weak in receiving cross section, jamming sources there can nevertheless dominate over receiver thermal noise because only one-way loss in propagation is suffered. Here, no restriction on [ is imposed.

As mentioned in Section 11, Kanter always

Denote w 5 (a/b)(p + is). (8)

In terms of D and S, C is defined orthogonal to S:

(93) C 2b'D - 2b2wS

so that C and S are uncorrelated

E[S*C] - E[S*]E[C] = 0. (9b)

Therefore, C and S are jointly Gaussian distributed, since the transformation in (Sa) is linear, so that their joint probability density function is given by

where p denotes

624 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

and where E[C] is obtained from (sa):

E[C] = 2b2E[D] - 2b2wE[S]. (104

Denoting the ratio D/S by Q and using (Sa), observe that

Q = - = - - S 2 6 2 s c + w . (11a)

The Jacobian of the transformation from C to Q can be expressed in terms of conjugate coordinates [lo] as

a(C,C*)/a(Q,Q*) = 4b4)SI2 (1lb)

from which the joint probability density function for Q and S can be represented in terms of that for C and S:

PQS (4, s) = 4b4 1 s I2pCS ( 2b2s[ 4 - U ] s). (1lc)

Denoting

U _= Re{S}; V _= Im{S}; @ arg{S}

where Im{. . . } denotes “imaginary part” and arg{. . . } denotes “argument,” or phase angle, observe from (2) that

v = (d*)Lsin@. U = ( J = ) ~ c o s ~ ; (12b)

The Jacobian of the transformation in (12b) is

w , v ) l w , q = E[tql2F (12c)

from which the joint probability density function for Q, L, and 9 can be represented in terms of that for Q and S:

pQL@ (4,[,$) = E[l~121~pQs(~,~e’”JE[lr l121) . (12d)

In terms of its real and imaginary parts, argument q in (12d) can be represented by r + i t , from which the joint probability density function for L and R can be expressed as

PLR(-%r) = 1; [ I p Q L @ ( r + i t , f , @ ) d t d @ .

(13)

Equations (k), (4b), (44, (9 (loa), (lob), ( W , (llc), (12d), and (13) yield the conditional variance and the conditional mean value [for n = 1 in (4a)l of R in terms of the four parameters a, b, p, and 5 of the covariance matrix of D and S, and in terms of the mean values E[D] and E[S] required in (loa) and (1Oc). The formulation of these parameters depends upon each of the two target-fluctuation cases, which are now examined separately. The noise power E[1qI2] in (12d) can be determined from b alone or from both b and the signal-to-noise ratio, depending upon the fluctuation case, as is observed shortly in (16) and (30) below.

A. The Nonfluctuating-Target Case

In this case, the phase of the target echoes need not be constant even though the amplitude is. For example, at sufficiently short wavelengths, phase is much more sensitive than amplitude to changes in optical path length due to fluctuations in the propagating medium or to changes in position of the target or radar. Because of the uncertainty in the behavior of phase of a constant-amplitude target signal, a joint Gaussian probability distribution for the real and imaginary parts of D and S cannot be ensured when noise is added.

However, the signal phasor in the receiver difference channel must always be parallel to the signal phasor in the sum channel, assuming that the phase shifts are balanced between receiver channels. Therefore, using the target signal in the sum channel as a phase reference, the ratio D / S can be reduced to a numerator and denominator that are both jointly Gaussian distributed, when noise is added, whose mean values are real valued. Redefining D and S to now represent the numerator and denominator, respectively, of that reduced ratio, their mean values can be shown to be

E[D] = Pp; E[S] = p > 0 (14)

where /3 denotes the amplitude of the target signal in the sum channel and is therefore positive valued, and where

i _=The monopulse ratio of the target signal alone, for which noise and jamming are imagined to be absent

(15)

which is always real valued by (1). The mean values of D and S are observed in (14)

to be only the target signal in the difference and sum channels. By subtracting those mean values from D and S in (6a), Do and SO are therefore samples of noise alone, with noise jamming included if present. For this reason, U , b, p, and C are properties of only the noise, upon which the nonfluctuating-target signal exerts no effect. In particular,

E[1qI2] = 2b2 (Nonfluctuating target) (16)

while the target signal-to-noise ratio in the sum channel is given by

x G p2/E[lq12] = p2/(2b2) (Nonfluctuating target).

(17)

Denoting

SEIFER MONOPULSE-RADAR ANGLE TRACKING I N NOISE OR NOISE JAMMING

- T

v E F - p U / b

and using ( S ) , (loa), (lOc), (llc), (14), (16), and (17), it can be shown that (126) yields

P Q L I ( ~ +it94+)

23 = 22 exp (- { p i n 2 + + (e - f i m + > 2

= P

(19)

which is the joint probability density function for R, T, L, and a, where

T G Im{ D / S } . (20)

From (k), (4b), and (13),

E[R" I A] = - J"J_:J_,J_: PD Lo x P ~ Q L Q (r + it,!,+) dt dr d+ d! (21)

where ~ Q L Q ( ~ + it,[,@) is given by (19). For n equal to unity or two, (21) is evaluated in Appendix A by taking the order of integration as indicated. First integrate with respect to variable f; then with respect to r; then with +; and finally with respect to t.

Defining the functions

~ ~ ( t ~ , & = e-('O-mz [ e - ' ~ ~ ~ o ( ~ o f i l (22a)

2Al A +A2 02[R I A] = p - + x v 2 L PD PD

where PD is given by the Marcum Q-function [U, p. 40, eq. (3-18)]:

It is helpful to observe that for large positive values of the argument of the modified Bessel functions of the

first kind, the following asymptotic formula holds [12, sec. 9.7:

as z-+oo. (25) 1 e-"I,(z) - - rn This combination of exponential and Bessel function has therefore been grouped within the brackets in (22) and is helpful in numerical evaluations and numerical integrations.

The difference Al - A2 can never be negative in the third term on the right-hand side of (23b). This observation follows from a recurrence relation on the order of the modified Bessel functions [12, sec. 9.61, yielding

Therefore, the last term in (23b) is the only term that is never positive.

B. The Rayleigh-Fluctuating Case

In the following, the target signal is now assumed to fluctuate randomly in strength with a Rayleigh-distributed amplitude; therefore, the real and imaginary parts of the target signal at baseband are uncorrelated, Gaussian-distributed random variables that are equal in variance and zero in mean value [13, chap. 71. In this respect, a fluctuating-target signal sampled at one moment in time is as random as noise from jamming. For this reason, results derived below for the monopulse ratio of a fluctuating target can be applied instead to the monopulse ratio of a source of noise jamming which is being tracked by the radar in the presence of receiver thermal noise and noise from the other jamming sources. Kanter included this alternative in his work [2]. Therefore, in the presence of both a fluctuating target and Gaussian noise,

E[D] = E[S] = 0 (Rayleigh-fluctuating target).

(27)

Equations (22), (B), and (24), which quantify the conditional mean value and conditional variance of R for a nonfluctuating target, were derived on the basis that D and S are jointly Gaussian distributed. Because D and S are likewise distributed in the present case of a Rayleigh-fluctuating target, these equations remain correct in their form for application here. Only the parameters in these equations require appropriate transformation to the Rayleigh-f luctuation case in order to provide the proper representation for the mean values of D and S and for their covariance matrix, all of which completely define the Gaussian distribution.


-~

When sampled at some moment in time, the sum-channel target signal is denoted here by the complex-valued random variable F, so that

S = F + q . (28)

The signal-to-noise ratio for the fluctuating-target case is then given by

x = E[IF12]/E[lq12] (Fluctuating target). (29)

In terms of x as given by (29), observe that from @a), (a), (271, (2% and (291,

2b2 = E[lSI2] = o( + 1)E[lq12] (Fluctuating target)

(30)

which when combined with (2) can be shown to yield

XI^ = e o / ~ m Fluctuating target).

(31)

The significance of (31) becomes apparent shortly. Setting p equal to zero in order to reduce

(14) to (27), observe that parameter x of the nonfluctuating-target case also becomes zero by (17). For this reason, x must now be equated to zero in (22), (a), and (24) before proceeding further.

nonf luctuating-target case really represents the left-hand side of (31) in the current case. Therefore, the next step in modifying (22), (a), and (24) requires that parameter t o there be replaced everywhere by the right-hand side of (31).

Following the modifications just prescribed, Ao, A I , A2, and PO in (22a), (22b), (22c), and (24) become

By (2) and (16), parameter eo in the

A2=O (324

where El(z) denotes the exponential integral [12, sec. 5.11:

E ~ ( z ) f lw (l/x)e-” d x . (324

Because 12(z) vanishes for z = 0, the integrand in (22c) is observed to vanish when x is set to zero, thereby verifying (32). When x is set to zero in (22b), and parameter there is replaced by the right-hand side of (31), A1 is observed to reduce to (32b). Similarly, (22a) and (24) reduce to (32a).

The equality between A0 and PD in (3%) causes (23a) to reduce to

E [ R I A] = pa/b. (33a)

First setting x = 0 in (23b) and then applying (32a) and (32b) to the result yields

1 g2[R 1 A] = -,u2exp 2 (A) X + 1 E1 (A) X + 1 . (33b)

Denoting the total noise at baseband in the receiver difference channel by ( ( t ) , observe that

D = f F + < . (34)

From (b), (6b), (271, and (3%

2a2 = E[IDI2] = f2E[IFI2] + E[1<I2] (35)

and from (6c) and (Z),

2ab(p + ic) = E[S*D] = fE[IFl2] + E [ q * t ] .

Therefore, from (29), (30), (35), and (36), it can be shown that

(36)

The right-hand side of (33a) is observed to be the real part of (37b). From (lob), p2 in (33b) can be represented by

In this way, E [ R I A] and g2[R 1 A] for a Rayleigh-fluctuating target are completely formulated by (33a), (33b), (37a), (37b), and (37c) in terms of x, l o , i , and the covariance matrix of the noise, < and q, from the difference and sum channels.

IV. RESULTS

Conditioning imposed upon the probability distribution of R by the occurrence of event A is removed when the voltage level, A, of the detection threshold is not greater than zero. In that case, the threshold-to-noise ratio, l o , defined by (2), will fail to exceed zero in (3), and the probability of A, PO, will equal unity in (4c), (24), and (32a). Under these circumstances, the conditional moments of R reduce to their corresponding “nonconditioned” moments, which have already been derived by Kanter [l-31:

For eo = 0:

E [ R I A] = E[R];

c2[R I A ] = g2[R]; (38)

PD = 1.

Equation (38) provides a correspondence between the results derived here with those exhibited by Kanter.

621 SEIFER MONOPULSE-RADAR ANGLE TRACKING IN NOISE OR NOISE JAMMING

__ - -- r _ _ ~-

In particular, the infinite variance of R occurs when conditioning by A is removed:

(r2[R] = f l o ~ 2 [ R I A ] = (39)

Equation (39) follows from (22b), (23b), and (26) for the nonfluctuating-target case, and from (32d) and (33b) for a Rayleigh-fluctuating target.

While the conditional variance of R is observed in (lob), (23b), and (33b) to depend explicitly on parameter 5 in each target-fluctuation case, as does the conditional probability density function for R, the complete absence of C from the formulation of the conditional mean value of R in (22a), (23a), (24), and (33a) is remarkable. Recall that 5, defined by (6c) to be the imaginary part of the coefficient of correlation between D and S, had been tacitly taken to be zero by Kanter.

For a Rayleigh-fluctuating target, E[R I A ] is observed in (33a) to be independent also of parameter l o , which implies by (38) equality between E[R] and E[R I A ] , the latter having been evaluated in (33a):

E[R] = E [ R I A] = pa/b

(Rayleigh-fluctuating target). (9

Kanter derived the first equality in (40) in an unpublished memorandum [14]; he evaluated E[R] to be pa/b in an earlier publication [2].

For the nonfluctuating target, (22a) and (23a) yield by (38)

E[R] = lim E[R 1 A ] = i - (P - pa/b)e-x (41)

f0'0 . I

(Nonfluctuating target)

which agrees with Kanter's result [l, 31. Applying (25) to (22a), and using the result in

(23a), the conditional bias in R for the nonfluctuating case, being the difference between E[R I A ] and i , can be approximated by

for loa>> 1. (42)

Observe the difference between conditional bias in (42) and the nonconditioned bias, implied by (41), in regard to their exponential factors. For this reason conditional bias will be significantly larger in magnitude when the signal-to-noise ratio exceeds the threshold-to-noise ratio in decibels (i.e., when

> eo). Although the exponential factor in (42) is maximum when equals t o , conditional bias can be shown nevertheless to be a monotonically decreasing function of x, since the effect of PD in the denominator will tend to cancel the attenuation by the exponential when fi becomes less than eo.

Equation (23a), for the nonfluctuating target, can be rewritten as

where those terms linear in i have been combined. The coefficient of i in (43) is never negative. This observation, which is now demonstrated, ensures negative feedback in the angle-tracking servo mechanism, which is crucial to the servo stability.

Integrating (24) by parts, where h(z) is observed to be the derivative of lo(z) [12, sec. 9.61, it can be shown using (22a) that

PO - A0 = 2 f i e - L U + ' z ) I ~ ( ~ ~ d l > 0 ( X > 0). lorn Therefore, PD must exceed Ao, so that in (43)

0 < 1 - Ao/PD < 1 (for x > 0). (44) Substituting the real part of (37b) into (33a),

monopulse bias of a Rayleigh-fluctuating target is observed to be inversely proportional to x + 1:

In (49, E[q* t ] and E[1qI2] are properties of only the noise and are therefore independent of x, as is also P. In contrast to the exponential attenuation by signal-to-noise ratio observed in (42) for the nonfluctuating-target case, monopulse bias will therefore have a tendency to be more severe for a Rayleigh-f luctuating target.

decreases monotonically as the detection-threshold level is increased. This result will follow by demonstrating that

For a Rayleigh-fluctuating target, (T[R I A ] always

a -(r2[R I A ] 5 0. a t 0

Replacing l:/(x + 1) in (33b) by z , and observing that z is an increasing function of l o , it suffices to show that the derivative of e'El(z) is never positive. From (32d), observe that

(47) d 1

-[e"El(z)] = e'El(z) - - < 0 dz z -

thereby verifying (46), where the inequality in (47) is a known mathematical property 112, sec. 5.1, item 5.1.191.

For sufficiently large values of signal-to-noise ratio, it is shown in Appendix B that

exp (A) El (A) X + 1 X + 1


I I

where 7 denotes Euler's constant [12, sec 6.11:

7 = 0.5772... . ( a b )

Substituting (48a) into (33b), 02[R I A] for a Rayleigh-fluctuating target can be approximated by

for X B 1 and x >e: . (49)

V. EXAMPLES

Two examples are now considered. In the first example jamming is absent, and the target competes against only receiver thermal noise. In the second example a single, mainlobe, copolarized source of noise jamming is postulated, where noise from the jamming is assumed to dominate over receiver thermal noise. As examples, their study can lead to principles of only limited generality; however, their impact on our understanding is found significant.

A. The Covariance Matrix of D and S

Before proceeding to the first example, the covariance matrix of D and S must first be related to more direct parameters of noise and jamming. Among the four parameters, a, b, p, and <, of the covariance and the variances of D and S , only the three combinations a / b , pa/b, and < a / b are required in (lob), (18), (=a), (Db), (33a), and (33b) for E[R I A] and 02[R I A] for the nonfluctuating and Rayleigh-fluctuating targets. These combinations are now quantified in terms of receiver thermal noise and noise from jamming.

In the general case of multiple noise jammers, their superposition in the presence of receiver thermal noise yields the total noise in each receiver channel, which can be represented at baseband by

5 = M + c r j G j (Difference channel) (5Oa) i

7 = N + C G j (Sum channel) (50b) i

where M and N denote instantaneous baseband voltages of receiver thermal noise from the difference and sum channels, respectively, and where Gj ( j = 1,2,. . .) denotes the complex-valued random variable representing the noise-jamming signal from the j th jamming source in the receiver sum channel. Parameter rj denotes the ratio Dj/Gj, where Dj is the complex-valued contribution from the j th jamming signal to the difference-channel baseband.

%king the real part of rj , as prescribed by (1) in which D and S are replaced by Dj and Gj,

respectively, the monopulse ratio is obtained for the j th jamming signal, by itself, as it would appear if all other jamming signals, if the target signal, and if receiver thermal noise were imagined to be all absent. In general, rj is complex valued. However, as explained earlier, rj is real valued if the j th jamming source happens to be positioned within the principal lobe of the receiving-antenna sum pattern and also happens to be copolarized with the antenna response there. In that case, the signal from such a jamming source provides no contribution to the parameter < of the covariance between D and S.

While complex-valued G; quantifies the random amplitude and random phase of the j th jamming signal, parameter r; depends only on the position of the jammer within the receiving-antenna pattern; rj is independent of the characteristics of the jamming signal. Similarly, F depends only on the position of the target and not on its signal.

jamming on the monopulse ratio in as simple a context as possible, M and N are assumed in this section to be uncorrelated and equal in noise power:

In order to examine the effects of noise and noise

E[lNl21 = E[lMl21 (51a)

E[(N*MI] = 0. (51b)

For an amplitude-comparison monopulse system employing a typical assembly of feed horns to, say, a parabolic-reflector antenna, ( S a ) would be a consequence of the balance in gain between the difference and sum channels and their equivalence in noise figure. Furthermore, the internal noise in one channel will be independent of (and, hence, uncorrelated with) the internal noise in the other channel, thereby implying (51b).

However, for a phased-array antenna employing a corporate feed to active modules, each module terminated at its input to an array element, significant thermal noise is generated by the array components before being combined by the monopulse-comparator network. As a phase-comparison monopulse system, noise presented by one-half of the array, say the top half, to one comparator input will be equal in power and uncorrelated with noise presented by the bottom half to the other input. Noise from the difference output of the comparator (which, in this case, measures target elevation angle) can then be shown to be equal in power and uncorrelated with noise from the sum output, thereby demonstrating (5la) and (51b).

(6b), (6c), and (16), observe that 1) The Nonfluctuating-Target Case. From (6a),

2a2 = E[IDo~~] = E[1<12];

2b2 = E[1~1~]; (52)

2ab(p + i<) = E[q*<].

SEIFER: MONOPULSE-RADAR ANGLE TRACKING IN NOISE OR NOISE JAMMING

T

629

Therefore, from (5Oa), (50b), (%a), (51b), and (52),

where Rj denotes the sum-channel jamming-to-noise ratio:

(54)

Equations (53a), (53b), and (53c) relate the parameters of jamming to those of the covariance matrix of D and S for the nonfluctuating-target case.

square of the right-hand side of (53a) is observed from (52) to be the quotient E[1<12]/E[lq12]. Furthermore, the right-hand sides of (53b) and (53c) are observed from (52) to be the real and imaginary parts, respectively, of E[q*<]/E[lq12]. In these terms, (37a) and (37b) can be shown to yield

2) The Rayleigh-Fluctuating Target Case. The

In obtaining (55b) and (55c), P is observed to be real valued by its definition as a monopulse ratio as required by (1) and (15).

B. Example 1. The Nonhostile Environment

( 5 9 , so that In the absence of jamming, R j vanishes in (53) and

a / b = l ; p = < = O ; p = l ;

(Nonfluctuating target) (56)

and

p = d k ( P 2 + 1) + 1l/(x + 1)

(Rayleigh-fluctuating target) (57)

where p is given by (lob) or (37c).

is observed to be proportional to 3 in both target-fluctuation cases:

Using (23a), (33a), (42), (56), and (57), bias

(Nonfluctuating target) (58b) and

E[R I A] - P = -P/(x + 1) (59) (Rayleigh-fluctuating target).

Therefore, in the absence of jamming, conditional bias is not expected to be very significant when the principal null of the receiving-antenna difference pattern is maintained closely on the target position by the angle-tracking servo mechanism, causing P in (58) and (59) to be small in magnitude.

If the target, in its motion, is nonmaneuvering in the context of the dynamics and response of that servo mechanism, then by (58) or (59) the servo response can be expected to lag only negligibly behind the target position during the steady-state period, leaving P essentially zero:

P = O

(Thermal noise only; nonmaneuvering target).

In this case, the servo action appropriately drives the null of the antenna difference pattern to produce a zero-average response in the servo feedback. That average response is E[R I A], which can be shown from (58) or (59) to be directly proportional to P, thereby implying (60).

From (18), (22), (24), (56), and (57), the conditional variance of R, given by (23b) or (33b), is observed in this example to depend only upon the three parameters x, l o , and P for both target-fluctuation cases. In order to examine the effect of receiver thermal noise on that variance in as simple a context as possible, the target is considered to be nonmaneuvering. Therefore, during the steady-state period of the angle-tracking servo response, (60) will apply, and u[R I A] can then be plotted as a function of just the two remaining variables, x and l o . If desired, however, similar plots can be obtained with values of P other than (60) by using the appropriate equations derived above. In the following, the two target-fluctuation cases are examined separately for

(60)

U P I Al. 1) The Nonfluctuuting-Target Case. Applying (56)

and (60) to (18) and (Bb), the conditional standard deviation of R can be shown to reduce to

(for F = p = < = 0). u[R I A] = d m (61)


I

ABOVE NOISE POWER

x = SIGNAL-TO-NOISE RATIO IN WAlTS / W A T

0.0 5.0 10.0 15.0 20.0 25.0 3

SIGNAL-TO-NOISE RATIO IN DECIBELS

Fig. 1. Conditional standard deviation of monopulse ratio versus signal-to-noise ratio and threshold-to-noise ratio. Receiver thermal

noise only. Nonfluctuating target.

Equation (61) is plotted in Fig. 1 as the dashed curves for five different values of lo . For reference, the function 1/m is plotted there as the solid curve, which is a straight line sloped downward because the vertical axis is scaled logarithmically and the horizontal axis is also logarithmic by its decibel unit.

approaches the solid curve for sufficiently large values of signal-to-noise ratio:

c [ R I A] x l/m when > l o > 1.

Observe how each dashed curve in Fig. 1

(62)

The use of (62) has been a standard in the radar industry [6, sec. 9.21, though its appearance here has for the first time specified that the left side of (62) should be the conditional standard deviation of R. In this way, the nonconditioned, infinite variance of R , correctly established by Kanter's theory, can be reconciled with practices already in use in radar design but which have never before been rigorously justified. Furthermore, Fig. 1 quantifies exactly how large the signal-to-noise ratio must be for (62) to apply satisfactorily.

Equation (62) can be explained as follows. When x sufficiently exceeds to, Po in the denominator of the radicand in (61) becomes nearly unity and therefore remains essentially constant with further increase in x. The numerator, being AI , is represented by the integral in (22b). The dominant contribution to that integral will occur for values of l (the variable of integration) within a local neighborhood about a, where the argument of the exponential factor, -(e - m2, vanishes, causing that exponential to be maximum. This observation is identical in principle with the asymptotic analysis supporting Laplace's method [15, sec. 2.41. Using Laplace's method and applying (25) to the appropriate factors of the

0.075

o.loo 7 ' 15.0 0.125 0

- VOLTS / VOLT

5.0

0.0 0.0 5.0 10.0 15.0

I i U

0

I * t.0


Fig. 2. Threshold-to-noise ratio versus signal-to-noise ratio along contours of constant conditional standard deviation of monopulse

ratio. Receiver thermal noise only. Nonfluctuating target.

integrand in (22b), (62) can be rigorously shown to be an asymptotic formula which approximates (61) for sufficiently large values of x.

In Fig. 1, c[R I A] is observed to decrease monotonically with increasing l o for x fixed in value. This observation can be demonstrated in general by showing that a[R I A] in (61) always has a negative-valued partial derivative with respect to eo. In particular, observe from (22b), (24), and (61) that

&a(r2[R I A] = PDaAl/aeo - AlaPD/aeo aio -(G+ - - -e 0 x ) ~ o ( u o f i ~ ~ ~ e - ( 1 2 + x ) I o ( 2 t f i

2de x (e + eo)(e - eo)- < o eoe

from which the desired assertion can be shown to follow.

Other ways of exhibiting the data in Fig. 1 may be found useful. In Fig. 2, for example, contour plots of constant levels of a[R I A] are exhibited, and in Fig. 3 a [ R I A] is plotted against the probability of detection for several different values of the probability of false alarm, which is given by

PFA = e-':. (64)

2) The Rayleigh-Fluctuating Target Case. Applying (57) and (60) to (33b), c[R 1 A] can be shown to be given by

(P =O). (65)

Using (a), and approximating x + 1 by x when x is large, (65) can be approximated as follows for x

SEIFER MONOPULSE-RADAR ANGLE TRACKING IN NOISE OR NOISE JAMMING

7- - -- -~

631

0.35

J PROBABILITY OF FALSE ALARM

10-1

Z 0.3

0 . 0 3 ~ " " " " " " " " " ' " " " ' " " ' " " ~ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PROBABILITY OF DETECTION

Fig. 3. Conditional standard deviation of monopulse ratio versus probability of detection and probability of false alarm. Receiver

thermal noise only. Nonfluctuating target.

0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.00


Fig. 4. Conditional standard deviation of monopulse ratio versus signal-to-noise ratio and threshold-to-noise ratio. Receiver thermal

noise only. Fbyleigh-fluctuating target.

sufficiently large:

( x > > l ; x>>e; ; f = 0). (66)

Equation (65) is plotted in Fig. 4, and its difference from Fig. 1 in terms of the data exhibited shows the effect of target fluctuation. One way to quantify that difference is in terms of fluctuation loss, which is considered shortly. In Fig. 5, a[R I A] is plotted against PD for several different values of PFA, and Fig. 5 can be similarly compared with Fig. 3.

Denoting x by xo in (62), and denoting x by XI in (66), x o and will become functionally related when

632

I-

$ . v)

5 9

2 4

P

I;

E P

W

P

I

>

d 0

cn


0.20 -


0.20 -

n ncl I I I I 1 1 1 1 1 I I I 1 1 1 1 -.-- 0.01 0.1 1 .o

PROBABILITY OF DETECTION

Fig. 5. Conditional standard deviation of monopulse ratio versus probability of detection and probability of false alarm. Receiver

thermal noise only. Rayleigh-fluctuating target.

cr[R I A] in (62) is equated to that in (66):

x o = Xl/[ln(Xl/.ei) - 73 (XI >> t; > 1; i = 0) (67)

In particular, (67) estimates the signal-to-noise ratio xo, that would produce the same value of a[R I A] for a nonfluctuating target as produces for the Rayleigh-fluctuating target. The ratio X ~ / X O then defines the fluctuation loss, which is denoted by LF:

LF f xI/xO (a)

( a b ) M ln(xl/@ - y > 1 (XI >> > 1; P = 0).

Equation ( a b ) is an approximation of LF, which is valid only when XI, t o , and P satisfy the indicated conditions. For P equal to zero, LF can be determined more generally and more accurately by avoiding the approximations (62) and (66) and using instead (61) and (65). However, in that case, determination of LF is more difficult, requiring a numerical evaluation by machine computation, since xo and x1 are functionally related implicitly rather than explicitly as in (67).

Fluctuation loss as determined above has been defined only in terms of cr[R I A]. Bias never became an issue because of (58), (59), and (60) bias is zero in both target-fluctuation cases. Whenever bias is significant in either fluctuation case, fluctuation loss becomes a more complex matter to define. This issue is not considered any further here, and other issues on fluctuation loss were identified earlier in Section 11.

C. Example I t . Noise From Jamming Only

In order to examine the effect of noise jamming on the monopulse ratio in as simple a context as possible, three assumptions are invoked here.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

-

1) Noise jamming is assumed to originate from a single, point source.

2) That source is assumed to be positioned within the principal lobe of the receiving-antenna sum pattern, and is further assumed to be copolarized with the antenna response there.

3) Noise from jamming is assumed to dominate over receiver thermal noise. By Assumption 1, each summation over index j in (50), (53), and (55) consists of only a single term, for which subscript j is equal to unity. In order to emphasize in this case that parameters rl and Q1 are properties of that single jammer, their subscripts are changed to J: r1 and RI become rJ and Q J , respectively.

From Assumption 2, rJ is real valued, implying by (53c) or (5%) that parameter < of the covariance between D and S is equal to zero, as discussed earlier:

< =O; Im{rJ} =O;

(Mainlobe, copolarized jamming). (69)

In a realistic ECM environment, Assumption 3 can often be expected whenever 2 occurs, and RJ (defined in (54)) is large with respect to unity:

QJ >> 1 [by Assumption 31. (70)

By (17), (29), and (50b), x is the ratio of the target-signal power to the total noise power, where the noise power includes receiver thermal noise plus noise from jamming. Therefore, Assumption 3 or its equivalent in (70) will render x to be essentially the target signal-to-jamming ratio. By similar argument using (2) and (50b), Assumption 3 will also render to to become essentially the threshold-to jamming ratio.

In the limit as QJ increases without bound, (53) and (55) become


and

(Rayleigh-fluctuating target)

where p is obtained from (lob) and (69). Using (23a), (33a), (42), (71), and (72), bias is observed to be

proportional to rJ - P in both target-fluctuation cases:

E[R I A] - P = ( r j - P)Ao/PD (73a) e - ( t o - m2

2PD d m m for loa>> 1

x (rJ - P)

(Nonfluctuating target) (73b)

(74)

and E[R I A] - F = (r / - P)/(x + 1)

(Rayleigh-fluctuating target).

Applying (69) and (71) to (18) and (23b), and applying (72) to (33b), a[R I A] is observed to be proportional to the absolute value of rJ - P:

A1 +A2


and

(Rayleigh-fluctuating target) (76)

where the right-hand sides of (75) and (76) are independent of r~ and P. Therefore, bias and g[R 1 A] are both likely to be large when rJ is large in magnitude and not offset, in the difference rJ - P, by an F of similar magnitude and sign. In fact, P must usually remain small, so that the target signal has the benefit of sufficient directive gain of the receiving-antenna sum pattern. Otherwise, noise would likely obscure the target signal sufficiently to render A an event of too rare an occurrence to adequately permit tracking.

Jamming parameter rJ can become quite large in magnitude, for example, whenever the position of a stund-ofi jammer-which is carried by a vehicle separate from the target-curs near the null of the principal lobe of the receiving-antenna sum pattern. In this case, the jamming noise level in the receiver sum channel is considerably smaller than that in the difference channel, resulting in a large magnitude of r J .

In Fig. 6, the quotient a[R I A] / l r~ - P I is plotted (as the dashed curves) for the nonfluctuating-target case using (75), while (76) is similarly plotted (though with solid curves) in Fig. 7 for the Rayleigh-fluctuating target. As mentioned earlier, P must usually remain small in magnitude, not more than a couple of tenths of a Volt per Volt, in order that the target signal not suffer excessive loss in antenna directive gain. If,


-~ 1

633

P t I:

8

k

w

n

I

> 0

1

0.1

0.01

I THRESHOLD IN DECIBELS ABOVE NOISE POWER

I 0.0 5.0 10.0 15.0 20.0 25.0 30.0


Fig. 6. Conditional standard deviation of monopulse ratio, divided by IrJ - PI, versus signal-to-noise ratio and threshold-to-noise ratio.

Receiver thermal noise dominated by noise from a single, mainlobe, copolarized jammer. Nonfluctuating target.

0.35 L 1 - <..

I -7 0.30 - >

f 0.25 P

P 0.20

s w I: 0.15 3

B g 0.10

k

E & 0.05

- A THRESHOLD IN DECIBELS

ABOVE NOISE POWER I 3.0

0 . O o r ~ " " " " " " ' ~ " " ~ " " ~ ' ~ ~ ~ I 0.0 5.0 10.0 15.0 20.0 25.0 30.0


Fig. 7. Conditional standard deviation of monopulse ratio, divided by lr, - PI, versus signal-to-noise ratio and threshold-to-noise ratio.

Receiver thermal noise dominated by noise from a single, mainlobe, copolarized jammer. Fbyleigh-fluctuating target.

for example, Ir~l were 20 dB, because the jammer happened to be positioned near the null of the antenna mainlobe, then I ~ J - PI would be essentially 10 Volts per Volt (V/V), by which the values of the vertical axis in Fig. 6 or 7 are multiplied in order to obtain a[R I A]. In this case, a[R I A] is observed to be as large as 2 V/V in Fig. 6, and 21 V/V in Fig. 7, when the signal-to-noise ratio is 10 dB and the threshold is set 6 dB above noise power (6 dB = 20 loglolo).

However, under the same circumstances, the magnitude of the bias in R can be shown, using (73b) and (74) to be only 0.3 V/V for the nonfluctuating target and 0.9 V/V for a Rayleigh-fluctuating target. Bias of the nonfluctuating target would have been much larger and more comparable to a[R I A] had the difference between !O and J?s not caused a reduction

by the factor of 0.26 from the exponential in (73b). Not having the benefit of attenuation by such an exponential factor, bias of the Rayleigh-fluctuating target therefore exceeds that of the nonfluctuating target by a factor of 3 in this case.

The monopulse ratio is essentially proportional to the angular displacement between target position and the principal null of the receiving-antenna difference pattern only when that displacement is not more than a few tenths of an antenna (sum-pattern) beamwidth, where the monopulse ratio is not more than a few tenths of a Volt per Volt [6, Fig. 9.61. Therefore, feedback in an angle-tracking servo which assumes that proportional relationship will report servo angle errors that are much larger in magnitude than their actual values when o[R I A] is excessive, thereby potentially driving the servo beyond its designed limitations. Even if that proportional relationship could be replaced by the actual function which relates monopulse ratio to angle-tracking error, the nonlinearity of that function in the presence of an excessively large a[R I A] could be expected to cause additional bias in the estimated target position, which would be suffered by the angle-tracking servo.

which, as in Fig. 1, appears straight and sloped downward. Observe how each dashed curve in the figure approaches the solid curve as x becomes sufficiently large, thereby implying for the nonfluctuating-target case

The solid curve in Fig. 6 is a plot of 1/m,

(Nonfluctuating target).

Applying (48a) to (76), and approximating x + 1 by x, observe that a[R I A] for a Rayleigh-fluctuating target can be approximated, when x is sufficiently large, by

(x >> 1; x >>e;) (78) (Rayleigh-fluctuating target).

Comparing (62) with (77), and comparing (66) with (78), the jamming-only case is observed to differ from the thermal-noise-only case by the presence of the factor I ~ J - PI in (77) and (78) for both target-fluctuation models. However, this comparison is not completely balanced, since the restriction P = 0 by (60) on o[R I A] for the thermal-noise-only case is not imposed upon the jamming-only case.

Unlike the thermal-noise-only case in Figs. 1 and 4, a[R I A] in the jamming-only case is observed in Figs. 6 and 7, for both target-fluctuation models, not to be a monotonically decreasing function of x but to possess a relative maximum with respect to x when

634 IEEE TRANSACTIONS ON AEROSPA LCE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

l o , rJ, and 3 are all fixed in value. In other words, an increase in signal-to-noise ratio toward the point of relative maximum appears to worsen the noisiness in the monopulse ratio. In this regard, Figs. 6 and 7 should be interpreted more carefully. In particular, the process of tracking is more complex than can be indicated from these figures. The radar-tracking servo mechanism usually tends to position the radar-antenna beam to point where E[R 1 A] is either small or zero. This action causes all three parameters, rr, P, and x, to vary, where the monopulse-antenna patterns are directly involved. Therefore, a proper interpretation of the maxima in Figs. 6 and 7 has yet to be found.

VI. CONCLUSIONS

This work is one step toward achieving a more comprehensive design philosophy of monopulse radar. In particular, the proper level of the detection threshold for optimal tracking is an issue of fundamental importance, which has never before been answered, and part of that question is addressed here. While a higher threshold reduces the variance of each monopulse measurement, as observed in the present study, fewer measurements are smoothed by the angle-tracking servo during its integration time, thereby somewhat offsetting any improvement in angle-tracking accuracy. Here, the higher threshold lowers the probability of detection and causes the servo to coast more often. Because the conditional variance of R decreases from infinity as the threshold is raised from zero, the existence of an optimal threshold setting seems to be implied on the issue of noise alone, regardless of whether the target maneuvers in its motion. The presence of a maneuver affects only the value of the optimal threshold, not its existence.

of the monopulse ratio by the simple expression 1/(2x), where x denotes signal-to-noise ratio, has been a standard of the radar industry, has never before been rigorously justified, and is in direct conflict with Kanter’s conclusion that the variance is infinite. However, by conditioning the variance on the event of a threshold detection occurring in the receiver sum channel, the 1/(2x) law is demonstrated here for certain cases and is shown not to be valid in other cases. Nevertheless, in all cases, the conditional variance is shown to be finite as long as the detection-threshold level exceeds zero in voltage.

conditions are met: 1) the target is nonfluctuating with a deterministic signal strength, 2) noise from jamming is absent, so that the target signal competes against only receiver thermal noise, 3) the principal null of the receiving-antenna difference pattern is maintained, by the angle-tracking servo mechanism, on the position of a nonmaneuvering target, 4) x sufficiently exceeds the square of the threshold-to-noise (voltage) ratio to, and

The traditional practice of estimating the variance

The 1/(2x) law holds whenever all of five

5) &J is sufficiently large. The 1/(2x) law is observed here to fail when noise jamming occurs or when x fails to exceed t; sufficiently.

are satisfied, a Rayleigh-fluctuating target requires a somewhat larger signal-to-noise ratio to achieve the same conditional variance in monopulse ratio as that of a nonfluctuating target. That excess in signal-to-noise ratio is due to fluctuation loss, which is shown to be given, in decibels, approximately by

When all but the first of the above five conditions

1Olog,,[~(x/tz) - 71 for x >> tg > 1

While the angle-tracking servo reduces the effect where 7 is Euler’s constant (y = 0.5772.. .).

of the conditional variance of the monopulse ratio by smoothing the random errors of monopulse measurement, it cannot reduce bias. Conditional bias in the monopulse ratio is significantly larger for a Rayleigh-fluctuating target than for a nonfluctuating target when x sufficiently exceeds ti. Here, bias is inversely proportional to x + 1 for a Rayleigh-f luctuating target but decreases exponentially with the square of the difference Jjs- l o for a nonfluctuating target.

For a Rayleigh-fluctuating target, conditional bias in the monopulse ratio is independent of the threshold parameter, l o , causing conditional bias and nonconditioned bias to be equivalent for that target-f luctuation model. Unlike conditional variance, conditional bias in both target-fluctuation cases is independent of the imaginary part of the covariance between D and S.

the monopulse ratio is quantified, where C denotes the imaginary part of the coefficient of correlation between D and S. Although 5 is known to be zero in the presence of mainlobe, copolarized noise jamming, the impact on 5 in other jamming geometries or from other jammer polarizations has yet to be examined. That effort would have to model in some detail the response of the monopulse receiving antenna and would have to determine the effect of that response on < in a more general noise-jamming environment.

randomly and simultaneously with the event of a threshold detection in the receiver sum channel. The probability of detection therefore directly affects servo accuracy and stability, as also does the correlation time of the target signal strength in the event of a fluctuating target. The impact of random sampling is another issue that requires further study.

In the present study, the effect of parameter C on

Feedback in the angle-tracking servo system occurs

APPENDIX A

For each of the two values of exponent n (n = 1,2), the four-fold integral in (21), in Section 111, is reduced in the following to an expression that is free


1 - - - ~-

635

of integrations in more than one variable at a time. We begin by integrating (19) with respect to variable t . Letting w denote

a

observe that (19) depends on t only through w, and w is manifested in (19) only as the square root of the last term in the argument of the exponential function. Noting that the derivative d w / d t is equal to !/,U, the integration of (19) with respect to t reduces essentially to the problem of integrating

Therefore,

where B denotes

B = l p - a +fi(vcos$-C-sin$). a (A2) b b

With the exception of the factor rn, the only other occurrence of r in (Al) above is in the last term of the argument of the exponential function. Letting w' denote the square root of that term,

1 P

w' E -(er - B)

whose derivative, dw' /dr , is equal to l / p , the integration of (Al) with respect to r reduces to the problem of integrating

In particular, for each of the two values for n (n = 1,2), it can be shown that

636

Therefore, from (Al) to (A4), it can be shown that

1: J_m_ rpQLa(r + i t ,e ,$)dtdr

I) cos2$ - 2 x 4 - a sin20 . b

(A6)

The third integration is performed on (As) and (A6) with respect to variable q$. Because the interval of integration, -n 5 $ 5 1r, is symmetric about zero, all terms in (As) and (A6) containing the odd function sin$ or sin2q$ will vanish upon integration. The remaining terms integrate into modified Bessel functions [12, sec. 9.61:

1: J_mlJ_ml rpQLa (r + it,e, @) dt dr d$

= e-(X+l*) [pi10(2efi)2t + 2vfi11(2fi)]

(A7)

r2pQLQ(r + i t , e ,$)d tdrd$

In the final integration, which is taken with respect to variable e over the interval l o 5 e < 03, observe from (24) in Section I11 that the first term on the right-hand side of each of (A7) and (A8) integrates into PO multiplied by some constant. Furthermore, those terms containing I l ( 2 e m can be integrated by parts, where Il (z) is observed to be the derivative of

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 28, NO. 3 JULY 1992

- _

I I

Io(z) [12, sec. 9.61:

= PO - e-(X+':)Io(2tofi)

where (24) has been recalled once again. With these observations, (23a) and (23b) in Section 111 follow from (5) and (21).

APPENDIX B

In this Appendix, (48a) in Section 111 is demonstrated. Using power-series representations of El(z ) [12, sec. 5.11 and e', observe that

- 1 n t - y + z - - + .. . 4

z In(l/z) - y for 0 < z g 1 (B1)

where Euler's constant y is exhibited in ( a b ) in Section IV. Letting

2 = @(x + 1) (B2)

observe that the conditions imposed upon x in (48a) imply the condition 0 < z << 1 in (Bl). Observe further that

l / z = (X/w+ 1 / x )

so that W / Z ) = Mx/G> + + 1 / x ) (B3)

where

x 0 for x>> 1. (B4)

Therefore, (Bl) to (B4) above yield (48a).

ACKNOWLEDGMENT

Among the comments on the original manuscript, the use of complex rather than real variables in the derivation of (19) in Section 111 was suggested by one reviewer and considerably improved the mathematical development. The author is also indebted to those at Raytheon Company who provided support and many useful technical discussions during the progress of this work.

JEENCES

Kanter, I. (1977) The probability density fundion of the monopulse ratio for N looks at a combination of constant and Rayleigh targets. IEEE Transactions on Information Theory, IT-23 (Sept. 1977), 643-648.

Multiple Gaussian targets: The track on jam problem. IEEE Transactions on Aerospace and Electronic Systems,

Kanter, I. (1977)

AES-13 (NOV. 1977), 620-623. Kanter, 1. (1978)

Three theorems on the moments of the monopulse ratio. IEEE Transactions on Information Theory, IT-24 (Mar. 1978), 272276.

Connolly, T. (1980) Statistical prediction of monopulse emrs for fluctuating targets. In Record of ihe IEEE I980 Zntermtional Radar Conference, Arlington, VA, Apr. 1980, 458463.

Monopulse Principals and Techniques. Dedham, MA: Artech House, 1984.

Radar System Analysis. Dedham, MA: Artech House, 1976.

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SEIEER MONOPULSE-RADAR ANGLE TRACKING IN NOISE OR NOISE JAMMING 637

Arnold D. Seifer (S’59-M674M79) received the B.S., M.S., and Ph.D. degrees in mathematics from Rensselaer Polytechnic Institute, Poy, NY, in 1%2, 1%4, and 1968, respectively.

He has been with Raytheon Company since 1980, as a principal engineer, and has been involved in systems analysis and systems design with many of Raytheon’s programs. From 1%7 to 1980, he worked at General Dynamics Corp., Groton, CT; at the Applied Physics Laboratory of The Johns Hopkins University, Laurel, MD; and at Emerson Electric Company, St. Louis, MO. He is an applied mathematician involved in interdisciplinary activities, which have included Ocean acoustics with application to problems in ASW, remote sensing of the Ocean surface by radar sea echoes, analysis and design of radar and automatic fire-control systems, and reliability modeling of fault-tolerant processors.

He has published papers in several journals. Dr. Seifer is a member of the Society for Industrial and Applied Mathematics.


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