Optimisation of Periodic Search Strategies for...

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LIST OF F IGURES 1 Coincidence of three periodic pulse trains........................ 28 2 Illustration of a triangle in the (α, ) plane in which intercept time is constant.... 28 3 Partitioning of the (α, ) plane into triangles of constant intercept time........ 29 4 Optimisation itineraries against three example threat emitters. ............ 30 LIST OF TABLES I The Farey series up to order five between 0 and 1................... 27 II Threat emitter list example................................ 27 III Period ratio and minimum tolerance parameters for threat emitters. ......... 27 IV Optimal parameters for search strategy example. ................... 29 V Expected intercept times for the optimal periodic search strategy vs. empirical average intercept times for the jittered search strategy . ................ 31

Transcript of Optimisation of Periodic Search Strategies for...

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LIST OF FIGURES

1 Coincidence of three periodic pulse trains. . . . . . . . . . . . . . . . . . . . . . . . 28

2 Illustration of a triangle in the (α, ε) plane in which intercept time is constant. . . . 28

3 Partitioning of the (α, ε) plane into triangles of constant intercept time. . . . . . . . 29

4 Optimisation itineraries against three example threat emitters. . . . . . . . . . . . . 30

LIST OF TABLES

I The Farey series up to order five between 0 and 1. . . . . . . . . . . . . . . . . . . 27

II Threat emitter list example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

III Period ratio and minimum tolerance parameters for threat emitters. . . . . . . . . . 27

IV Optimal parameters for search strategy example. . . . . . . . . . . . . . . . . . . . 29

V Expected intercept times for the optimal periodic search strategy vs. empirical

average intercept times for the jittered search strategy. . . . . . . . . . . . . . . . . 31

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Optimisation of Periodic Search Strategies for

Electronic Support

I. Vaughan L. Clarkson

School of Information Technology & Electrical Engineering

The University of Queensland

Queensland, 4072

AUSTRALIA

[email protected]

Abstract

Solutions to the sensor-scheduling problem in Electronic Support are central to successful receiver

operation. The scenario is examined in which a programmable, frequency-agile receiver is required to

monitor a wide bandwidth for threat emitters from a known list. A theory is developed to allow the

sweep time of the receiver, together with individual dwell times on each band, to be jointly optimised

in order to minimise the intercept time with threat emitters. In theoretical and simulation studies, it is

found that improvements of more than 10%, and sometimes much more, can be achieved on maximum

and expected intercept times with respect to other periodic and jittered search strategies.

Index Terms

Electronic support, superheterodyne receiver, emitter intercept, synchronisation, receiver search

strategy, Farey series, threat-emitter list, sensor scheduling, radar warning receiver, scan-on-scan.

This paper results from work that was performed under Contract 4500161275 for the Defence Science and Technology

Organisation (DSTO), Department of Defence, Australia.

Preliminary versions of this paper have been submitted for publication as a DSTO Research Report and presented as an invited

paper at the Defence Applications of Signal Processing (DASP ’05) workshop.

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I. INTRODUCTION

Electronic Warfare Support Measures (or Electronic Support or ES for short) can be defined

as:

‘. . . that division of electronic warfare involving actions taken under direct control

of an operational commander to search for, intercept, identify, and locate sources of

radiated electromagnetic energy for the purpose of immediate threat recognition.’ [1]

Apart from Electronic Warfare related to communications, i.e., in non-communications EW, the

emitter of interest in ES is usually a radar. The primary tools for gathering information are ES

receivers. The bandwidth over which threat emitters may operate is generally too wide to permit

the use of a receiver which is at once both highly sensitive and able to discriminate multiple

simultaneous emissions. A common compromise is to use a highly sensitive receiver of more

modest bandwidth which is swept (or stepped) through different centre frequencies in order to

maintain surveillance over the wider search bandwidth.

Radars face a similar problem. It is generally not practical to employ a radar which is both

sensitive enough for the required detection range and able to resolve simultaneous targets in

angle. Therefore, most radars use a directional antenna which is scanned in angle.

The tuning and re-tuning of the centre frequencies constitute a search strategy or sensor-

scheduling problem for the receiver. How should we decide whether one strategy is better than

another? One way to measure the performance of a strategy is to determine the intercept time

with emitters on a threat-emitter list.

The sweeping of the receiver and the scanning of the radar make the sensor-scheduling problem

somewhat peculiar to ES. In order to intercept energy, the receiver must be tuned to the same

band as the radar at the same time as the radar is pointed towards the receiver. The sweeping

of the receiver must coincide with the scanning of the radar.

In the simplest setting, both the emitter and receiver employ a periodic search strategy. That

is, the receiver tunes to a particular centre frequency at regular intervals, and the radar points in

a particular direction at regular intervals. The interval for the receiver and interval for the radar,

i.e., the periods, may be, and usually are, different. However, even in this simplest setting, the

analysis has proved difficult.

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In fact, the literature on the ES sensor-scheduling problem is extremely sparse. The only

relatively substantial body of work pertains to the closely related problem of calculating the

probability of intercept or intercept time between emitter and receiver.

For the coincidence of strictly periodic and deterministic events, it appears that RICHARDS [2]

was the first to make a rigorous study. He recognised some of the links to number theory and

derived some results on probability of intercept. His approach was extended and the connection

to number theory, particularly Diophantine approximation and the Farey series, was strengthened

in [3]–[5].

A slightly more constrained approach, in which the events’ periods, phases and durations

are commensurate, was undertaken by MILLER & SCHWARZ [6]. They also recognised some

of the connections with number theory, through the theory of linear congruences. A method

of calculating coincidence times for deterministic and random phases is developed, as well as

the fraction of time for which coincidences occur. Their work was simplified and extended in

several respects in [7], namely (insofar as they are relevant to this paper) that further results

from the theory of linear congruence are applied to simplify the calculation of coincidence time

and approximations are developed where exact solutions are unavailable or difficult to compute.

Independently, in [8], similar results are derived and are applied specifically to ES, along with

considerations for calculating approximate intercept times and probabilities when the parameters

are not all commensurate.

On the other hand, STEIN & JOHANSEN [9] proposed a model for random pulse trains for

which they were able to obtain some statistical information on coincidences. This was extended

and applied to ES in [10], where expressions were derived for probability of intercept. However,

these expressions are not applicable for strictly periodic events.

WILEY [11, ch. 3], in addition to providing an excellent summary and illustration of the

work on probability of intercept up to that point, appears to be the first to have considered the

implications of intercept-time calculations to the design of search strategies. KELLY et al. [3]

also attempt to quantify the amount of randomness required in receiver search strategies to

overcome synchronisation effects (a possible problem in deterministic search strategies in which

the receiver is synchronised with the emitter so that the receiver is never in the right band when

the emitter is illuminating the receiver).

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In an earlier work [12], the author considered how a periodic search strategy should be devised

in order to intercept emitters on a threat-emitter list. The proposed criterion for deciding between

two candidate sweep periods was the maximum intercept time with any emitter in the list.

According to this criterion, the optimum sweep period is that which minimises the maximum

intercept time. Using a novel generalisation of the Farey series, the author derived a procedure

for calculating the optimal sweep period.

However, the author’s previous work assumed that the receiver was required to dwell for an

equal amount of time in each band. In this paper, we will relax this assumption. Hence, we will

use a slightly more sophisticated model for our receiver: one in which not only is the overall

sweep period programmable, but the individual dwell times on each band too. Again, the min-

max intercept-time criterion will be applied to select the optimal strategy. By way of a case

study, we will show that considerable reductions in the min-max intercept time can be achieved.

In Section II, we review some of the mathematical foundations for intercept-time calculation

and the ES sensor-scheduling problem. In Section III, a new interpretation of maximum and

expected intercept time is presented. This new interpretation leads directly to a method for opti-

mising individual dwell times, which we present in Section IV. Numerical results are presented

in Section V which demonstrate that a reduction of over 15% in min-max intercept time is

achieved by the new optimisation techniques when individual dwell times can also be varied.

In Section VI, we discuss the applicability of the theory to more general classes of emitters,

receivers and interception models.

II. MATHEMATICAL PRELIMINARIES

A. Receiver and Emitter Models

Throughout this paper, except where stated (notably Section VI), we assume that our ES

receiver is a frequency-swept receiver (FSR) with an omni-directional antenna. The receiver

has a number of bands through which it sweeps. The sweeping is not continuous: it does not

continuously adjust its centre frequency. Instead, it dwells at a particular centre frequency for a

certain length of time before moving on to the next. A frequency-swept superheterodyne receiver

is an example from the class of receivers that we call FSRs.

We restrict ourselves to periodic search strategies. Therefore, we assume that the sequence of

dwells repeats exactly after each pass through the search bandwidth. However, we assume that

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the dwell time in each band is adjustable individually and can be reset to new values from time

to time to meet changing environments throughout a mission.

We also allow for some dead time between each dwell: a period of time during which the

receiver is re-tuning between different centre frequencies and is not receiving or processing

intercepts.

To render the problem mathematically tractable without, it is hoped, sacrificing too much of

the relevance, in the current paper we make the following simplifying assumptions:

• For the radar emitter, we also assume a periodic search strategy is in operation and that

this period, like that of the receiver, is stable.

• The emitter has a narrow main beam and a fixed centre (or radio) frequency (RF).

• The beamwidth, RF and scan period are all known to good accuracy and are recorded in a

threat-emitter list.

What is not known a priori by the receiver is whether the radar is switched on or within range.

B. Interception Model

In order to receive energy from the emitter, the receiver must be tuned to the right band when

the emitter is pointing in the right direction. Mathematically, we can describe the receiver being

tuned to the right band as an event. Whether or not the event is occurring at any particular time

can be modelled by a function whose value is 1 or 0, respectively. We call such a function a

window function or pulse train. The time interval over which an event occurs is called a window

or pulse. Therefore, we can construct one pulse train to represent whether the receiver is tuned

to the right band, and another to represent whether the emitter is pointing in the right direction.

Because we are assuming that both the receiver and the emitter are employing periodic search

strategies, it follows that the corresponding pulse trains are both periodic. For the emitter pulse

train, the period is the scan period and the pulse width is the illumination time, the amount of

time for which any particular point in the radar’s search path is illuminated as the main beam

passes over it. For the receiver pulse train, the period is the sweep period and the pulse width is

the dwell time on that particular band. To each pulse train, we can associate a phase: the time

offset between the centre of a designated pulse and the time origin. Phase is not unique, since

the addition of the period to the phase gives another equally valid value for phase.

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For the receiver to intercept energy from the emitter, the two pulse trains must coincide, i.e.,

they must both have a value of 1. More generally, we can consider simultaneous coincidence of

multiple pulse trains. This is illustrated for three pulse trains in Figure 1. We label the period,

pulse width and phase of the ith pulse train Ti, τi and φi, respectively.

Let us now number the pulses from each pulse train. Let the 0th pulse be the pulse which

defines the phase, the 1st pulse to be the next after that, and so on. The kthi pulse from Pulse

Train i occurs when

|t− kiTi − φi| 6 12τi (1)

Clearly, a coincidence occurs with the kthj pulse from Pulse Train j if and only if

|t− kjTj − φj| 6 12τj (2)

is simultaneously satisfied with (1) at some time t.

We now state the following lemma about coincidence of pulse trains. The proof of the lemma

and of all other theorems are to be found in Appendix A.

Lemma 1: A necessary and sufficient condition for the existence of some t that simultaneously

satisfies (1) and (2) is that

|kiTi + φi − kjTj − φj| 6 12(τi + τj).

If a coincidence of minimum duration d is required then we must have

|kiTi + φi − kjTj − φj| 6 12(τi + τj − 2d). (3)

We return to the case of specific interest to us, where there are only two pulse trains repre-

senting the emitter scan and receiver sweep. Let Pulse Train 1 be the emitter pulse train and

Pulse Train 2 be the receiver pulse train. That is, the emitter scan period is T1, the illumination

time is τ1 and its phase is φ1. Similarly, the receiver sweep period is T2, the dwell time is τ2

and its phase is φ2. Interception takes place whenever

|qα− p+ β| 6 12ε, (4)

where, following [12], [13]:

• p and q represent pulse numbers from the emitter and receiver pulse train, respectively (in

place of k1 and k2),

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• α = T2/T1 is the period ratio,

• β = (φ2 − φ1)/T1 is the (normalised) relative phase and

• the quantity

ε = (τ1 + τ2 − 2d)/T1 (5)

is the tolerance (when d = 0 this quantity is the normalised sum of pulse widths).

C. The Farey Series

The Farey series of order n, Fn, is the sequence or series of fractions, written in lowest terms

and in ascending order, with denominator less than or equal to n. The sequence is usually defined

so that it consists only of those fractions between 0 and 1. With this definition, the first five

orders are listed in Table I. However, it is also sometimes defined between 0 and ∞, or between

−∞ to ∞. For our purposes, it is convenient to use the limits 0 and ∞.

In writing a fraction h/k in lowest terms, we mean that h and k have no common prime

factors and we specify that the denominator k is always positive. A basic theorem about the

Farey series is the following.

Theorem 1: The fractions h/k < h′/k′ are adjacent in a Farey series if and only if h′k−hk′ =

1.

In examining intercept-time problems, the following theorem is of more direct relevance.

Theorem 2: Suppose h/k < h′/k′ are adjacent in a Farey series. If h/k 6 α 6 h′/k′ and, for

some h′′/k′′,

kα− h > k′′α− h′′ > k′α− h′

then k′′ > k + k′.

The fraction (h+ h′)/(k + k′) is known as the mediant of the adjacent fractions h/k and

h′/k′. The mediant is itself adjacent to both h/k and h′/k′ in a Farey series of higher order.

III. MAXIMUM AND EXPECTED INTERCEPT TIME

A. Maximum Intercept Time

For two pulse trains, Pulse Train 1 and Pulse Train 2, we define the (maximum) intercept

time (with respect to Pulse Train 2) as the maximum number of consecutive pulses from Pulse

Train 2 required for coincidence with Pulse Train 1, where the maximum is taken over the

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relative phase. Therefore, the intercept time is the number of consecutive ‘looks’ required by

the receiver in the radar’s operating band in order to be certain of intercepting it.

Note that the intercept time, according to this definition, can be infinite. This happens if the

periods are commensurate and the sum of pulse widths is small, in a sense that can be made

precise [5]. In this case, the two pulse trains are said to be synchronised and, for certain relative

phases, the pulse trains are always out of step with each other. In ES, this is a highly undesirable

situation, since the radar of interest may never be intercepted.

Although intercept time has been studied in the past, especially by the author [4], [5], [12], [13],

some remarkable simplifications of the theory are possible, leading to a geometric construction

that will be useful in optimising the receiver search strategy. We begin by recalling the intercept

inequality (4). Note that if the intercept time is ` looks then, for any relative phase β, there must

exist some integers p and q such that the intercept inequality is satisfied with 0 6 q < `.

This leads us to our first theorem regarding intercept time.

Theorem 3: Consider a pair of pulse trains defined by period ratio α and tolerance ε. Suppose

that h/k < h′/k′ are fractions such that h/k 6 α 6 h′/k′. If

kα− h 6 ε and h′ − k′α 6 ε

then the intercept time is not greater than k + k′.

Now we state an important converse to this theorem.

Theorem 4: Consider a pair of pulse trains defined by period ratio α and tolerance ε. Suppose

that h/k < h′/k′ are adjacent fractions in a Farey series such that h/k 6 α 6 h′/k′. If

(k − k′)α− (h− h′) > ε (6)

then the intercept time is not less than k + k′.

Consider any two adjacent Farey fractions, h/k < h′/k′. When h/k 6 α 6 h′/k′, we know that

the intercept time is not greater than k+k′ when kα−h 6 ε and h′−k′α 6 ε, from Theorem 3.

The boundaries of these regions intersect at α = (h+ h′)/(k + k′) and ε = 1/(k + k′), with the

numerator in the latter expression being unity because h′k− hk′ = 1. At α = h/k, we find that

the inequality h′ − k′α 6 ε is the applicable one and this reduces to ε > 1/k at that point. At

α = h′/k′, the inequality kα− h 6 ε is applicable and this reduces to ε > 1/k′.

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On the other hand, we know that the intercept time is not less than k + k′ when (k − k′)α−

(h− h′) > ε, from Theorem 4. At α = h/k, this inequality reduces to ε < 1/k and, at α = h′/k′,

it reduces to ε < 1/k′.

Therefore, Theorems 3 and 4 together imply that there is a triangle with vertices{(h

k,

1

k

),

(h′

k′,

1

k′

),

(h+ h′

k + k′,

1

k + k′

)}(7)

in the (α, ε) plane inside which the intercept time is k + k′. This is illustrated in Figure 2. A

dotted line is drawn across the top of the triangle to indicate that this boundary is open whereas

the other two, in solid lines, are closed. In the centre of the triangle is written k+ k′: this is the

intercept time everywhere within the triangle.

We can now describe a recursive partitioning of the (α, ε) plane. We will show that this plane

can be recursively divided into triangles, and in each triangle the intercept time is constant.

To see how this partitioning is achieved, we first note that if the tolerance ε is greater than

or equal to 1 then intercept is always immediate, so the intercept time in this case is 1. Noting

that the Farey series F1 is0

1,1

1,2

1,3

1, . . . ,

we see that we can take h/k and h′/k′ as adjacent integers. According to our triangle construction

above, this yields triangles with vertices at (n, 1), (n+1, 1) and (n+ 12, 1

2) in which the intercept

time is 2.

Now, given any triangle with vertices as per (7), we can construct two new triangles adjoining

the lower left and lower right edges of the first, respectively. This is because h/k and h′/k′

are adjacent in a Farey series, so h/k and their mediant are adjacent also (in a series of higher

order), as are the mediant and h′/k′.

By adjoining these triangles recursively, the whole plane can be partitioned, apart from a set

of measure zero. This set is the set for which the intercept time is infinite, i.e., for which the

pulse trains are synchronised. A detailed justification of this claim is contained in Appendix B.

The partitioning of the (α, ε) plane is illustrated, for intercept times up to 7, in Figure 3.

B. Expected Intercept Time

In [4], [5], expressions were derived for the expected intercept time. The expected intercept

time is defined here as the expected number of looks required to intercept the emitter, assuming

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that the relative phase is uniformly distributed modulo 1.

We can now interpret these expressions in terms of the triangles that partition the (α, ε) plane.

From the previous section, we learned that every point in that plane outside the synchronisation

set with ε < 1 belongs to one and only one triangle with vertices as set out in (7), for appropriate

h, h′, k and k′. Furthermore, the maximum intercept time within the triangle is constant at k+k′

looks. For ε > 1, intercept is always immediate and the intercept time is always 1 look. For

ε > 1, it therefore follows that the expected intercept time is also always 1 look.

It is possible to show that, within one of the triangles, the expected intercept time, E[N ], can

be written as

E[N ] = 12[1 + k + k′ + kk′(kα + k′α− h− h′ − 2ε)]. (8)

This equation can be shown to be correct by appealing to the previously published results [4],

[5]. However, a translation between the notations is required and this would be tedious here.

Therefore, the details are left to Appendix B.

Notice that (8) is linear in α and ε. Therefore, we can define the expected intercept time

everywhere in the triangle by linear interpolation of its values at the vertices. By substitution

into (8), it is easily verified that:

• E[N ] = 12

+ 12k at the vertex α = h/k, ε = 1/k,

• E[N ] = 12

+ 12k′ at the vertex α = h′/k′, ε = 1/k′,

• E[N ] = 12

+ 12(k + k′) at the vertex α = (h+ h′)/(k + k′), ε = 1/(k + k′).

Thus, it is clear that the expected intercept time is a continuous, piecewise-linear function over

and between the partitioning triangles in the (α, ε) plane.

IV. COMPUTING THE OPTIMAL SEARCH STRATEGY

A. Optimisation Criterion and Constraints

We now return to the central problem of this paper, which is designing an optimal search

strategy for a FSR. We define a candidate search strategy as a particular set of dwell times for

each of the bands the FSR visits.

Importantly, we assume that accurate intelligence is available to us from which optimisation

criteria can be formulated. This intelligence takes the form of a threat-emitter list. The threat-

emitter list enumerates each of the radars that may be encountered during the mission. We

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assume that the list consists only of radars which employ periodic search strategies, or at least

that the search strategy will only be optimised against the subset of radars that do.

Each entry in the threat-emitter list is a different make of radar, or a different operating

mode of a particular make. For each make or mode, the RF, beamwidth, scan period and pulse

repetition interval (PRI) are recorded.

Our criterion for optimisation, as in [12], will be to minimise the maximum intercept time.

That is, over all feasible candidate search strategies, we deem that strategy to be optimum for

which the maximum intercept time over all emitters in the threat-emitter list is minimised. In

contrast to previous work in min-max intercept time [12], the current work allows dwell times

on each band to be individually adjusted in the optimisation process.

B. Optimisation for a Fixed Sweep Period

1) The Algorithm: Let us tackle the simplest scenario first. We assume that the sweep period

is fixed but the individual dwell times on each band can be varied. Of course, the constraint

here is that the sum of the dwell times, including dead times, cannot exceed the sweep period.

If the sum of dwell times is less than the sweep period, we assume the receiver ‘sleeps’ for the

remainder of the time in each period (in practice, however, we always allocate all the available

time in each period).

The principle underlying optimisation is simple. If we attribute more dwell time to a particular

band then the intercept times with the emitters in that band cannot increase. Conversely, if we

reduce dwell time then the intercept times cannot decrease. That is, as a function of dwell time,

intercept times with emitters in that band are monotonically non-increasing.

Therefore, we begin the optimisation process by setting the dwell times on each band to their

minimum allowable values. The minimum allowable value is set to be some multiple of the PRI

of an emitter in that band. Clearly, at least one RF pulse must be received, which requires a

dwell time of at least one PRI. In some receiver designs, two or more consecutive RF pulses

are collected to accurately measure the PRI and positively identify (detect) the emitter. Hence,

a coincidence with minimum duration is required, cf. (3). This minimum duration must be equal

to a number of PRIs according to the number of consecutive pulses required for detection. If the

PRIs are jittered then some additional allowance must be made to set a duration long enough to

ensure the required number of consecutive pulses will be intercepted.

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With each of the bands allocated the minimum allowable dwell times, we compute the relative

pulse-train parameters, α and ε, for each emitter with respect to the sweep period of the receiver.

Hence, for each emitter, it is possible to compute the intercept time, for example, by using the

partitioning of the (α, ε) plane as discussed in Section III and illustrated in Figure 3 or, as

discussed elsewhere, by using Euclid’s algorithm [4], [5].

It is possible that, even with a minimum amount of dwell time allocated to each band, the

sum of dwell times may exceed the sweep period. In this case, no feasible search strategy is

possible.

Note also that the intercept time with any one of the emitters may be infinite. This occurs

if, for any emitter, the ratio of periods, α, is rational, i.e., if α = h/k, and the tolerance, ε,

is less than 1/k. In each such case, extra dwell time must be allocated to that band in order

that ε = 1/k. Again, as a result of this step, the sum of the dwell times may exceed the sweep

period, in which case it is not possible to have a search strategy in which intercept times are

finite with all emitters in the threat-emitter list.

However, assuming a feasible search strategy can be found with finite intercept times, the

optimisation process continues by doling out the remaining available time in each sweep period.

We call this remaining available time ‘the kitty’.

In each band, we compute the maximum intercept time over all emitters (potentially) operating

in that band. In the band which has the maximum of the maximums (or in any one of them if

there is a tie), determine the emitter which incurs the maximum intercept time (or again choose

any one if there is a tie). Determine how much additional dwell time must be allocated to this

band in order to raise the value of ε so that the intercept time reduces. If there is enough time

left in the kitty, allocate that time to the band and repeat the procedure. If not, the min-max

intercept time has been found and an optimum search strategy has been computed.

2) An Example: Suppose we have a threat-emitter list as given in Table II. The threat-emitter

list consists of three threat emitters, labelled 1–3, operating in three separate bands, labelled

A–C. Suppose our receiver is to sweep through bands A–C periodically with a sweep period of

1 second and is required (for the purposes of ensuring detection) to dwell long enough in each

band to be able to intercept five consecutive RF pulses.

In order to calculate the minimum tolerance with a particular emitter, we observe that the

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pulse width of Pulse Train 1, i.e., the illumination time, is

τ1 =beamwidth

360◦× T1

and that the minimum pulse width of Pulse Train 2, i.e. the minimum dwell time, and also the

minimum duration of overlap, is

τ2 > d = required # pulses× PRI.

It follows from (5) that

ε >beamwidth

360◦− required # pulses× PRI

T1

.

For each emitter, we can now calculate α and minimum ε parameters, and these are listed in

Table III.

The minimum dwell required on each band is five times the PRI of the emitter in that band.

Although the sum of the minimum dwells is much less than the sweep period, the intercept time

is infinite with Emitters 1 and 3. This is because the tolerance with Emitter 1 needs to be at

least 1/42 = 2.381× 10−2 and, with Emitter 3, it needs to be at least 1/21 = 4.762× 10−2. In

order to satisfy these requirements, (5) implies that

τ2 > εT1 − τ1 + 2d.

Hence, the dwell time on Band A needs to be increased to 193.5 ms and, on Band C, to 438.8 ms.

Note that it is now clear that, if no dwell-time optimisation were applied and the dwell time

were made equal on each band at 333.3 ms, the intercept time with Emitter 3 would be infinite.

With the minimum dwell times allocated as described in Section IV-B1 to ensure that the

intercept time with each emitter is finite, the intercept times are 42 looks with Emitter 1, 199

with Emitter 2 and 21 with Emitter 3. By allocating all of the remaining kitty to Band B,

the intercept time with Emitter 2 can be reduced to 65. This therefore represents the min-max

intercept time and an optimal search strategy.

An optimal strategy with a sweep period of 1 second is to have the receiver dwell for 193.5 ms

in Band A, 367.7 ms in Band B and 438.8 ms in Band C. Any of the emitters in the threat-emitter

list will be detected (five consecutive RF pulses intercepted) within 65 seconds of becoming

operational or coming within range.

We observe that the order in which the bands are searched is not important, so long as the

order remains the same on each sweep.

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C. Optimisation Over a Range of Sweep Periods

1) The Algorithm: In addition to being able to optimise the dwell times for a given sweep

period, we expect to further reduce the intercept time if we can also optimise with respect to

the sweep period.

A first observation is that the optimum search strategy, when both dwell times and sweep

period can be varied, is the best of the optimum search strategies for every fixed sweep period

in the range.

We now describe an algorithm which optimises both on sweep period and on dwell times.

We begin at one end of the interval over which sweep period is to be optimised, say, at the

minimum allowable sweep period. We proceed to traverse the interval, looking for strategies

with successively lower min-max intercept times. Whenever a new best strategy is discovered,

the intercept time, sweep period and set of dwell times are recorded, discarding the previous

best. In contrast to Section IV-B, it is important that we calculate intercept time in seconds,

rather than in looks, by multiplying the number of looks by the sweep period.

With the sweep period set to the minimum allowable, we optimise the dwell times according

to the procedure of Section IV-B to obtain an initial best strategy. If no such initial strategy

exists with a finite intercept time, we define the initial best strategy to be the strategy in which

all dwell times are set to the minimum allowable values.

Each time we discover a new best strategy (including the initial best strategy), we create a

new candidate strategy with a reduced intercept time by increasing dwell times.1 That is, we

determine which emitters require the maximum number of looks and add just enough extra dwell

time in each corresponding band to cause the number of looks to be reduced. This causes the

sum of the dwell times to exceed the sweep period. Thus, the new candidate strategy is (at first)

infeasible.

Having created a new candidate strategy, we increase the sweep period until it becomes

feasible. The partitioning of the (α, ε) plane into triangles now comes into full effect. As we

increase the sweep period, α increases proportionately with each emitter. For each emitter, as

we increase α, we trace along the edge of the triangle with the greatest intercept time less

1We may also need to do this if, as sweep period increases, a candidate strategy with a certain maximum number of looks

has an intercept time, measured in seconds, that becomes longer than the current best strategy.

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than the current best intercept time. We keep track of the sum of the required dwell times in

each band. Since the triangle edges are line segments, the sum of the required dwell times is a

piecewise-linear function.

If, as we increase the sweep period, this piecewise-linear sum-of-dwell-times becomes less

than the sweep period then the candidate strategy becomes feasible and, hence, a new best

strategy. We take note of the new best strategy, create a new candidate strategy and continue, as

described above. If, instead, we reach the end of the permissible sweep period interval before

the candidate strategy becomes feasible, then the most recently discovered best strategy is the

optimal strategy.

We observe that it is possible—for instance, when the range of allowable sweep periods is

very narrow—that no feasible strategy with finite min-max intercept time exists. In that case,

synchronisation with one or more of the emitters remains a possibility regardless of dwell settings.

2) An Example: Let us return to the threat-emitter list of Table II. We now seek an optimal

search strategy over a range of sweep periods between 1 s and 1.008 s. The process of optimisation

is illustrated in Figure 4.

The illustration is a plot of the ‘itineraries’ on the (α, ε) plane of each emitter which are

followed while optimising the search strategy. In each itinerary, and as in Figures 2 and 3, the

horizontal axis is the α axis and the vertical axis is the ε axis, although the axis markings

themselves have been omitted for clarity. The top itinerary belongs to Emitter 1 in Band A, the

middle itinerary to Emitter 2 in Band B and the bottom to Emitter 3 in Band C. Each itinerary

shows how the optimisation process assigns a value of ε as a function of α as α increases to

cross the interval corresponding to the range of sweep periods.

We observe that the itineraries, shown in solid lines, trace the edges of triangles in each (α, ε)

plane, apart from occasional vertical jumps. The vertical jumps occur where the solution at last

becomes feasible, a new best intercept time is found, and more dwell time is added to create a

new infeasible solution. The additional dwell is at once ‘borrowed’ on the band which has the

longest intercept time. The optimal search strategy corresponds to the point at which the last

jump occurs in any of the itineraries.

The variable-sweep-period optimisation begins where the fixed-sweep-period optimisation

ends: a feasible solution with a sweep period of 1 s and intercept times of 42 looks with Emitter 1,

65 looks with Emitter 2 and 52 looks with Emitter 3. With a sweep period of 1 s, one look is equal

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to one second. We immediately add more dwell time to Band B, in which Emitter 2 operates, in

order to reduce its intercept time to 62 looks. This makes the solution infeasible since the sum

of the dwell times now exceeds the sweep period. The sweep period is then allowed to increase.

In order to maintain an intercept time of 42 looks with Emitter 1, we observe in the top itinerary

that the dwell time (through ε) which must be allocated to Band A needs to increase. On the

other hand, the amount of dwell time required to maintain an intercept time of 62 looks with

Emitter 2 and 52 looks with Emitter 3 decreases. The decrease with Emitters 2 and 3 dominates

the increase required for Emitter 1 so that when the sweep period reaches 1.00033 s, this solution

becomes feasible and represents a new best strategy. More dwell is immediately added to Band B

so that the intercept time with Emitter 2 is reduced to 59 looks, making the solution infeasible

until the sweep period is increased, and so on.

The last feasible solution which is encountered in the optimisation process occurs when the

sweep period reaches 1.00763 s. The parameters at this point are listed in Table IV. Therefore,

this represents the overall optimum search strategy for sweep periods between 1 s and 1.008 s.

3) Running Time: The running time is dominated by the number of triangles in each (α, ε)

plane which must be examined. Unfortunately, this is difficult to determine a priori. However,

once an initial feasible solution has been found, it becomes possible to estimate the remaining

running time. This is because the triangles to be examined all have an intercept time (measured

in looks) less than the initial feasible intercept time. The partitioning of the (α, ε) plane into

triangles is periodic in α with period 1, as illustrated in Figure 3. If the initial feasible intercept

time is ` looks, the maximum number of triangles to be examined per unit interval on the α axis

is the number of Farey fractions per unit interval in F`. This number is the value of Φ(`) [14,

p. 268], sometimes known as the summatory totient function. A simple upper bound, which is

good enough for our purposes, is that Φ(`) 6 `2. Hence, for each emitter, as α increases by one,

a maximum of `2 triangles must be examined in order to determine whether a better feasible

strategy exists. Given an initial feasible search strategy, we can conclude that the remaining

running time is (at worst) proportional to the square of the initial feasible intercept time, the

number of emitters in the threat-emitter list and the length of the sweep period interval over

which optimisation is to be performed. In practice, the author has found that, for specific, realistic

scenarios, involving scores of emitters, the optimisation process rarely takes more than a few

tens of seconds on a modern desktop computer, and never more than a few minutes.

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D. Search Strategies in the Limit

It is an interesting question to consider what happens if the sweep period is allowed to approach

zero or infinity. Are there certain strategies or intercept times that become obvious under these

conditions?

As the sweep time approaches zero, it becomes clear that the receiver will intercept any emitter

the first time it scans past the receiver. Hence, although the intercept time as measured by the

number of looks required by the receiver approaches infinity, the intercept time as measured in

elapsed time will be finite with any emitter in the threat-emitter list (since intercept will always

occurs on the first scan). That is, the min-max intercept time will be equal to the maximum scan

period of the emitters on the list, with some small adjustment for illumination time and PRI.

This is the best min-max intercept time that can be hoped for.

Consider the threat-emitter list example in Table II that has been used throughout this section.

We see that, if we were allowed to sweep as quickly as we liked, we should be able to achieve

a min-max intercept time of (slightly) better than 10.5 s, that being the longest scan period in

the list.

Of course, it is impossible to achieve this lower limit with a practical receiver. This is because

there is always some dead time when the receiver switches between bands, and because of the

requirement to dwell long enough on a band to receive at least one pulse from the emitter.

On the other hand, if the sweep period setting of our receiver has no upper limit, another

strategy becomes obvious. We could choose to dwell on each band for an amount of time equal

to the longest scan period of any emitter on our list that operates in that band. In this way, we

would be certain of intercepting any emitter in that band during the dwell. In effect, we would

be certain that the maximum intercept time would be just one look. The sweep period required

for this is the sum of the dwells, i.e., the sum of the maximum scan periods in each band.

In terms of our threat-emitter list example in Table II, we see that, with a sweep period of

8.4 s + 2.97 s + 10.5 s = 21.87 s, we achieve a maximum intercept time of just one look. This is

better than any of the intercept times achieved earlier in this section or in the next.

Why not simply use this strategy in preference to the other search strategies where sweep

periods have an upper limit? Indeed, if there are no other constraints under which the receiver

must operate, then this strategy may well be the best (or nearly the best), and the optimisation

procedure proposed in this paper will surely find it. However, a maximum sweep period is

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usually imposed because of the implicit assumption that low-PRI, very-high-threat emitters or

modes need to be detected as soon as possible after they come within range. Since they can be

detected on the first look, the intercept time with these emitters is determined by sweep period.

Limiting the maximum sweep period is therefore highly desirable.2

V. NUMERICAL RESULTS

Let us now continue with the simple numerical example that was begun in Section IV as a

case study that illustrates the benefits of the new optimisation procedures. Suppose we again use

the threat-emitter list given in Table II.

If no optimisation of sweep or dwell settings were performed and the sweep period were

arbitrarily set to 1 s and the dwell periods in each band were made equal, then the intercept time

would be infinite with Emitter 3. As seen in Section IV-B, this is because, for a sweep period

of 1 s, the period ratio with the scan period of Emitter 3 is rational, α = 1/21, and so we need

to make ε > 1/21 also to avoid being in synchronisation. This implies that we need to make

the dwell time in Band C at least equal to 438.8 ms.

If we allow optimisation of dwell times as described in Section IV-B while keeping the sweep

period fixed at 1 s then we can preferentially allocate dwell time to Band C in order to bring

it out of synchronisation. By following the optimisation procedure, we find that the optimal

settings are that the receiver dwells for 193.5 ms in Band A, 367.7 ms in Band B and 438.8 ms

in Band C. With these settings, any of the emitters in the threat-emitter list will be detected

within 65 seconds of becoming operational or coming within range.

Using only sweep period optimisation between 1 s and 2 s, but keeping dwell times equal

between bands, as described in [12], the min-max intercept time is found to be 31.41 s. This

occurs at a sweep period of 1.57 s.

Finally, if we allow the optimisation to be performed jointly in both the dwell times and

the sweep period, as described in Section IV-C, with the sweep period being allowed to vary

between 1 s and 2 s then the min-max intercept time is significantly lower again. The optimal

2Rather than imposing a maximum sweep period, an alternative would be to explicitly record high-threat modes in the threat-

emitter list and to impose constraints on intercept time with those modes. However, if the modes are ‘staring’ modes, i.e., zero

scan period, then the two constraints are effectively identical.

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search strategy in this range occurs for a sweep period of 1.905 s, with a maximum intercept

time of 26.67 s.

We now compare this with a jittered search strategy, inspired by the findings of [3]. In this

strategy, the sweep duration is varied randomly from sweep to sweep with a mean value of

1.5 s and has a uniform distribution over a range equal to half the mean value, i.e., it has a jitter

amplitude of 50%. The dwells on each band are equal, i.e., one third of the sweep duration. Being

random, it becomes impossible to guarantee (even theoretically) any finite maximum intercept

time. In 1,000 simulation trials, it was observed that the maximum intercept time was 157.6 s,

quite significantly higher than the 26.67 s achievable with the optimal periodic strategy. Further,

as detailed in Table V, it can be seen that in two out of three cases, average intercept times with

individual emitters are also significantly longer.

Although this numerical example is by necessity of a small scale, in order that all the details

may be given, many other scenarios have been tested. It is difficult to quantify the average

improvement achieved by optimising dwell times as well as sweep periods or vice versa, since

the very idea of an ‘average’ implies some distribution of the parameters in the threat-emitter

list. However, empirically, the author has found that reductions of between 10% and 20% are

common when dwells are optimised in addition to sweep periods.

VI. EXTENSIONS TO THE MODEL

In Section II, the receiver, emitter and interception models were proposed in the simplest

form that gives a concrete basis for analysis while retaining a reasonable correspondence with

our empirical understanding of the real-world environment. Nevertheless, it is reasonable to

inquire about the applicability of the theory and the performance of resulting sensor-scheduling

algorithms in cases where it is not obvious that the model applies. We now consider some

extensions to the receiver and emitter models and to the interception model.

A. Receiver and Emitter Models

Increasingly, modern radars are able to operate in a number of modes and are agile between

these modes to achieve better performance. For instance, pulse-repetition-frequency (PRF) jit-

tering, switching and staggering is used to resolve range ambiguities. RF agility is useful in

evading detection. Furthermore, the scanning strategy of the radar need not be circular, but may

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be concentrated in sectors using, for example, raster, spiral or lobe-switching scan strategies. How

do these characteristics of a modern radar affect a receiver sensor-scheduling strategy based on

the min-max intercept-time principle? Alternatively, how can the sensor-scheduling strategy be

adapted to take account of these characteristics?

In the case of PRF agility, this is readily accounted for in the model. In order to achieve

detection, it is usual that a certain number of consecutive pulses must be intercepted. So long

as the maximum time interval is known over which this number of pulses is emitted, it can be

accounted for in the min-max strategy through the minimum duration (d) parameter.

Non-circular scanning strategies need not represent a difficulty either. The key requirement for

the applicability of the min-max intercept-time principle is that the scanning strategy is periodic

with known period. In many widely used scanning strategies, this is the case. For example, a

radar employing a spiral or unidirectional raster scan behaves, for the purposes of our model,

in the same way as a circularly scanning radar: the emitter is illuminated once each period for

a certain illumination time. So long as the period and minimum illumination time are known,

the min-max intercept-time principle can be applied to design the sensor-scheduling strategy.

As for the scanning strategy, RF agility is not incompatible with the min-max principle so

long as the pattern of visits to any particular RF band is periodic with known period. In order

for the algorithm to devise a strategy with an upper bound on the intercept time, rather than

have a single entry in the threat-emitter table, RF agile emitters require multiple entries, each

entry identical except for the RF band. One entry is required for each RF band that the emitter

is known to visit. The scan period in this case is to be interpreted as the illumination revisit

period on that band.

On the other hand, the receiver model may be extended to allow for directional search. In

order to increase detection sensitivity, as well as to gain intelligence on the direction of arrival

of radar signals, many ES receivers employ directional antennas and search not only in RF but

also in angle. The penalty to be paid is a longer intercept or detection time, although this may be

more than made up for by the extra sensitivity and directional information that this configuration

affords. Consider a receiver in which angular sectors are searched in a periodic fashion, in each

sector performing a search over RF, so that the whole search strategy is periodic. In this case,

the min-max intercept-time principle can be applied: the sweep period is now to be interpreted

as the period between visits to any particular RF band and angular sector.

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B. Interception Model

The practical utility of the theory of min-max intercept time is founded upon the assumption

that the most important single factor in interception or detection is that the receiver be tuned to the

emitter’s RF band when the emitter is illuminating it. Until now, we have made the simplifying

assumption that, when this condition is met, interception can occur reliably whenever the emitter

is within range, i.e., whenever the received signal is sufficiently powerful.

Let us now assign a probability p that the emitter is detected even when the coincidence

conditions are met (including any minimum required duration of coincidence). We will assume

that detection from separate coincidences are independent events.

If we are to consider p < 1 then the notion of a maximum intercept time must be abandoned.

Instead, we must perform a stochastic analysis of intercept time. Here, we will derive an upper

bound on the expected intercept time, showing that min-max intercept time calculations are still

relevant in minimising the expected intercept time in this case.

Let N represent the number of looks required for detection, now a random variable. We seek

an upper bound on E[N ]. We observe that, provided the expected intercept time exists,

E[N ] =∞∑

n=1

nPr {N = n} =∞∑

n=1

n(Pr {N > n} − Pr {N > n+ 1}) =∞∑

n=1

Pr {N > n}. (9)

Let C(n) represent the number of coincidences between the receiver and emitter pulse trains in

n consecutive looks, also a random variable, depending on a number of factors including the

distribution of the relative phases. Clearly, C(n) 6 n. The probability that c coincidences all

fail to result in a detection at the receiver is (1− p)c. It follows that

Pr {N > n} =n−1∑c=1

Pr {C(n) = c}(1− p)c

According to the results of Section III-A, we know that there must be at least one coincidence

of the two pulse trains in any k+ k′ consecutive looks, where k and k′ are the denominators of

adjacent fractions in the Farey series as determined by the parameters α and ε. We can conclude

that Pr {C(n) = c} = 0 if c(k + k′) < n. It follows that

Pr {N > n} 6 (1− p)b(n−1)/(k+k′)c.

Hence, from (9),

E[N ] 6∞∑

n=1

(1− p)b(n−1)/(k+k′)c = (k + k′)∞∑

j=0

(1− p)j =k + k′

p.

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Thus, the expected intercept time in this stochastic analysis can be bounded above by a simple

expression involving the maximum intercept time derived from the foregoing deterministic

analysis.

C. Unlisted Emitters or Modes

The threat-emitter list is compiled from intelligence sources. It may be that there exist other

emitters or modes of interest that are not listed. Yet the search strategy proposed here is optimised

against a list which we now acknowledge is possibly incomplete. What effect would this have

on intercept time? Performance would not necessarily be poor but upper limits on intercept time

can no longer be calculated. However, in a certain sense, no search strategy, periodic or not, can

guarantee finite intercept times without accurate intelligence [15].

VII. CONCLUSIONS

In this paper, we have extended the results of [12] to devise an optimisation procedure for

periodic search strategies in frequency-swept ES receivers. The optimisation procedure minimises

the maximum intercept time against any periodic emitters in a threat-emitter list. In contrast

to [12], the optimisation is not only over sweep periods but dwell times on each band also. We

have demonstrated in a numerical example that this additional optimisation step yields further

substantial reductions in intercept time and, in simulations, that it is superior to a jittered search

strategy.

ACKNOWLEDGEMENT

I would like to thank Dr. Greg Noone for his advice, interest, encouragement, enthusiasm,

friendship and guidance, without which this research would have foundered.

APPENDIX A

PROOFS OF LEMMAS AND THEOREMS

Proof of Lemma 1: To prove necessity, we observe, using the triangle inequality, that

|kiTi + φi − kjTj − φj| = |(t− kiTi − φi)− (t− kj − Tj − φj)|

6 |t− kiTi − φi|+ |t− kjTj − φj| 6 12(τi + τj).

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To prove sufficiency, set

t =τj(kiTi + φi) + τi(kjTj + φj)

τi + τj.

Then,

|t− kiTi − φi| =τi

τi + τj|kiTi + φi − kjTj − φj| 6 1

2τi

and, similarly,

|t− kjTj − φj| =τj

τi + τj|kiTi + φi − kjTj − φj| 6 1

2τj.

Proof of Theorem 1: The proof of necessity is often given in introductory texts on number

theory: see, for example, [14]. However, the proof of sufficiency is less common: see, for

example, [16].

Proof of Theorem 2: We write h′′ and k′′ as linear combinations of h, h′, k and k′ so that

h′′ = ah+ bh′ and k′′ = ak + bk′. (10)

Solving for a and b, we find that

a =h′k′′ − h′′k′

h′k − hk′and b =

h′′k − hk′′

h′k − hk′.

Note that, in both cases, the denominator is 1, because h/k and h′/k′ are adjacent in a Farey

series. The numerators are integers because h and k are integers and so a and b are both integers,

and not both zero.

Now, let η = kα− h > 0 and η′ = k′α− h′ 6 0. Then

η′′ = k′′α− h′′ = (ak + bk′)α− (ah+ bh′) = aη + bη′.

If ab < 0 then either η′′ > η or η′′ 6 η′. If ab = 0 then, because we assume k′′ > 0 as the

denominator of a fraction, (10) implies that either a > 0 or b > 0, so again either η′′ > η or

η′′ 6 η′. Therefore, ab > 0. This implies that a > 0 and b > 0, and so k′′ > k + k′.

Proof of Theorem 3: We will prove the theorem by contradiction. Suppose, contrary to the

theorem statement, that the intercept time is greater than k + k′. Then there must exist some

relative phase β such that the intercept inequality (4) is not satisfied for any q with 0 6 q < k+k′.

Let u∗ and v∗ be the pair of integers which minimise |vα− u+ β| for 0 6 v < k+ k′. Without

loss of generality, we will assume that v∗α− u∗ + β < −12ε.

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First, we prove a contradiction in the special cases where α = h/k or α = h′/k′. Suppose

α = h′/k′. Observe that (k∗ − k′)α− (h∗ − h′) + β has the same value as k∗α− h∗ + β so we

choose k∗ such that k∗ < k′. If α = h′/k′ then 0 < kα− h 6 ε and so

|(v∗ + k)α− (u∗ + h) + β| < |v∗α− u∗ + β|

Therefore, v∗ + k > k + k′ or, equivalently, v∗ > k′, a contradiction. We can generate a similar

contradiction for α = h/k.

Having excluded α = h/k and α = h′/k′, suppose h/k < α < h′/k′. Since 0 < h′− k′α 6 ε,

|(v∗ − k′)α− (u∗ − h′) + β| < |v∗α− u∗ + β|

and so v∗ < k′. On the other hand, because 0 < kα− h 6 ε,

|(v∗ + k)α− (u∗ + h) + β| < |v∗α− u∗ + β|.

Therefore, v∗ > k′ and we have generated a contradiction.

Proof of Theorem 4: Consider the relative phase

β = 12(k + k′)α− 1

2(h+ h′).

In order for the intercept inequality (4) to be satisfied, we must have∣∣qα− p− 12(k + k′)α + 1

2(h+ h′)

∣∣ 6 12ε.

This can be rewritten as

−12ε+ 1

2(k + k′)α− 1

2(h+ h′) 6 qα− p 6 1

2ε+ 1

2(k + k′)α− 1

2(h+ h′).

Observing that, because of (6),

12ε+ 1

2(k + k′)α− 1

2(h+ h′) < 1

2(k − k′)α− 1

2(h− h′) + 1

2(k + k′)α− 1

2(h+ h′)

= kα− h

and, similarly, that

−12ε+ 1

2(k + k′)α− 1

2(h+ h′) > k′α− h′,

we find that we must satisfy the inequality

kα− h > qα− p > k′α− h′.

From Theorem 2, we see that either q < 0 or q > k + k′. Hence, the intercept time is not less

than k + k′.

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APPENDIX B

PROOF OF CERTAIN CLAIMS REGARDING MAXIMUM AND EXPECTED INTERCEPT TIME

In Section III, we made certain claims about maximum and expected intercept time that require

further justification. We do this by appealing to the previous results in [4], [5], and in particular

their exposition in the latter. To do this, we need to translate between notations.

First, we note that the definition of ε used in this paper is twice that used in [5].

Now, consider a point on the (α, ε) plane with ε < 1. First of all, referring to Chapters 2

and 5 of [5] and Equation 3.8 of the latter chapter, we observe that the probability of intercept

never reaches unity if α is rational and ε < 1/k when we write α = h/k. Thus, the intercept

time in this case is infinite, a situation which we call synchronisation.

Outside of this synchronisation set, we can find integers p/q and p′/q′ which [5, Section 3,

Chapter 2, and Section 2, Chapter 5]:

1) satisfy a unimodularity condition in that |pq′ − qp′| = 1 and,

2) with η = qα− p and η′ = q′α− p′, satisfy |η| 6 ε, |η′| 6 ε and |η − η′| > ε.

Therefore, by setting h/k to the lesser of p/q and p′/q′ and h′/k′ to the greater, we have adjacent

elements in a Farey series and we obey all the conditions of Theorems 1, 3 and 4. The intercept

time is therefore k + k′ = q + q′. In this way, we have shown that every point in the (α, ε)

plane outside of the synchronisation set and with ε < 1 lies in one of the partitioning triangles.

Moreover, we can now easily verify that (8) corresponds precisely to Equation 4.1 in Chapter 5

of [5].

REFERENCES

[1] American National Standards Institute, American National Standard for Telecommunications: Telecom Glossary 2000.

ANSI, Feb. 2001. American National Standard T1.523–2001.

[2] P. I. Richards, “Probability of coincidence for two periodically recurring events,” Ann. Math. Stat., vol. 19, no. 1, pp. 16–29,

Mar. 1948.

[3] S. W. Kelly, G. P. Noone and J. E. Perkins, “The effects of synchronisation on the probability of pulse train interception,”

IEEE Trans. Aerospace Electron. Syst., vol. 32, no. 1, pp. 213–220, Jan. 1996.

[4] I. V. L. Clarkson, J. E. Perkins and I. M. Y. Mareels, “Number theoretic solutions to intercept time problems,” IEEE Trans.

Inform. Theory, vol. 42, no. 3, pp. 959–971, May 1996.

[5] I. V. L. Clarkson, Approximation of Linear Forms by Lattice Points with Applications to Signal Processing. PhD thesis,

The Australian National University, 1997.

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26

[6] K. S. Miller and R. J. Schwarz, “On the interference of pulse trains,” J. Appl. Phys., vol. 24, no. 8, pp. 1032–1036, Aug.

1953.

[7] H. D. Friedman, “Coincidence of pulse trains,” J. Appl. Phys., vol. 25, no. 8, pp. 1001–1005, Aug. 1954.

[8] R. M. Hawkes, “The analysis of interception,” unpublished research report, Defence Science and Technology Organisation,

P.O. Box 1500, Edinburgh, 5111, South Australia, Nov. 1983.

[9] S. Stein and D. Johansen, “A statistical description of coincidences among random pulse trains,” Proc. IRE, vol. 46, no. 5,

pp. 827–830, May 1958.

[10] A. G. Self and B. G. Smith, “Intercept time and its prediction,” IEE Proc., vol. 132F, no. 4, pp. 215–222, July 1985.

[11] R. G. Wiley, Electronic Intelligence: The Interception of Radar Signals. Norwood, Massachusetts: Artech House, 1985.

[12] I. V. L. Clarkson, “The arithmetic of receiver scheduling for Electronic Support,” in Proc. Aerospace Conf., vol. 5,

pp. 2049–2064, Mar. 2003.

[13] I. V. L. Clarkson, “The Farey series in synchronisation and intercept-time analysis for Electronic Support,” Trans. AOC,

vol. 1, no. 1, pp. 7–28, Oct. 2004.

[14] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Oxford University Press, 5th ed., 1979.

[15] I. V. L. Clarkson and A. D. Pollington, “Performance limits of sensor-scheduling strategies in Electronic Support,” IEEE

Trans. Aerospace Electron. Syst., vol. 43, no. 2, pp. 645–650, Apr. 2007.

[16] A. Hurwitz, “Uber die angenaherte Darstellung der Zahlen durch rationale Bruche,” Math. Ann., vol. 44, pp. 417–436,

1894.

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TABLE I

THE FAREY SERIES UP TO ORDER FIVE BETWEEN 0 AND 1.

0

1

1

1

0

1

1

2

1

1

0

1

1

3

1

2

2

3

1

1

0

1

1

4

1

3

1

2

2

3

3

4

1

1

0

1

1

5

1

4

1

3

2

5

1

2

3

5

2

3

3

4

4

5

1

1

TABLE II

THREAT EMITTER LIST EXAMPLE.

Emitter Scan

number Band period (µs) PRI (µs) Beamwidth (◦)

1 A 8.4× 106 2.38633× 103 1.3

2 B 2.97× 106 1.37792× 103 2.6

3 C 10.5× 106 9.38 2.1

TABLE III

PERIOD RATIO AND MINIMUM TOLERANCE PARAMETERS FOR THREAT EMITTERS.

Emitter number α ε

1 5/42 2.191× 10−3

2 100/297 4.902× 10−3

3 2/21 5.829× 10−3

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��������������������

t = 0 //t

oo �//φ1

oo �//φ2

oo �//φ3

�oo �//T1

�oo �//T2

�oo �//T3

oo //τ1

oo //τ2

oo //τ3

Pulse train 1

Pulse train 2

Pulse train 3

Coincidences

Fig. 1. Coincidence of three periodic pulse trains.

//α

OOε

��������������������������������

///////////////

(h

k,1k

)

(h′

k′ ,1k′

)

(h+ h′

k + k′ ,1

k + k′

)

k + k′

Fig. 2. Illustration of a triangle in the (α, ε) plane in which intercept time is constant.

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29

1//α

_1

OOε

0

1

??????????????????????????????????????????? �������������������������������������������

?????????

2

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3 3

************************************************

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********************

4 4

''''''''''''''''''''''''''''''''''''''''''''''''''

����� /////////

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5 5 5 5

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6 6

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

��� *******

������������ ////

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���� ''''''''''''

������� ???

�����������������������������������������������������

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

7 7 7 7 7 7

Fig. 3. Partitioning of the (α, ε) plane into triangles of constant intercept time.

TABLE IV

OPTIMAL PARAMETERS FOR SEARCH STRATEGY EXAMPLE.

Emitter Intercept time

number Band (looks) (s) Dwell(µs)

1 A 25 25.191 332.5

2 B 41 41.313 312.5

3 C 31 31.236 362.6

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(217 ,

117

)

(867 ,

167

)

(325 ,

125

)(

542 ,

142

)jjjjjjjjjjjjjjjjjjj

ZZ

42

67

25

(13 ,

13

)

(2368 ,

168

) (1235 ,

135

)68

OO

65

'''SS

62

///WW

59

888[[

56

@@@@__

53

HHHcc

50

HHHcc

47

HHHcc

44

HHHcc

41

HHHcc

38

HHHcc

BB DD FF GGHHHHHH

JJJJKKKK

MMMMMM NN

KKKKKK

YYYYYYYYYYY(

221 ,

121

)

(773 ,

173

) (552 ,

152

)(

331 ,

131

)73

52

31

Fig. 4. Optimisation itineraries against three example threat emitters.

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TABLE V

EXPECTED INTERCEPT TIMES FOR THE OPTIMAL PERIODIC SEARCH STRATEGY vs. EMPIRICAL AVERAGE INTERCEPT TIMES

FOR THE JITTERED SEARCH STRATEGY.

Emitter Optimal periodic (s) Jittered (s)

1 11.56 20.16

2 13.65 11.49

3 11.31 23.76

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