1. Introduction
With the increase of electrical and electronic equipment in recent years, analysis of the coexistence between them has gathered great importance. Interference is often analyzed in the frequency domain, power domain, and signal processing, and some researchers have analyzed coexistence between two pieces of equipment operating at close frequency [
1,
2,
3]. Takuya and Shigeru [
4] propose an estimation method for the signal environment in ARNS (aeronautical Radio Navigation Service: 960∼1215 MHz) band to examine whether equipment sharing a frequency band interfere with each other. However, the analysis of coexistence in the frequency or power domain provides the worst case results because the involved systems are assumed to transmit continuously [
5], the result is consistent with the actual value only when baseband signal is analog, but the result is different from the actual value when the signal is pulsed with small duty cycle, since the interfering and desired signal do not overlap with each other most of the time, i.e., interference does not occur all the time. Some researchers use signal processing to analyze the coexistence between two pieces of equipment, for example Khodr A et al. [
6] studied the cancellation of DME interference for aeronautical communications using signal processing. Miguel A. [
7] assessed the impact of L-band digital aeronautical communications system (LDACS) on JTIDS (a military radio system known as Joint Tactical Information Distribution System) using signal processing based on some assumptions for JTIDS. Nevertheless, there are some limitations when using signal processing to analyze interference, since the detailed characteristics of signals are not open, especially for military equipment (e.g., JTIDS).
Both DME and ATCRBS are types of aeronautical radio equipment, and operate in the L-band at the same time, so coexistence between the two pieces of equipment must be analyzed to ensure that they can work properly. To avoid the interference between DME and ATCRBS when they are located in the same airplane, they operate in time division multiplexing access (TDMA). However, it is difficult for using TDMA if pieces of equipment are located in the different airplanes, and if two airplanes are near enough, interfering power from an airplane can impact on sensitive equipment fitted on the other airplane, hence it is important to analyze interference in the time domain to evaluate the performance degradation of the interfered equipment and discover the reason for interference. Najett et al. [
8] have verified the impact of the interference using average capacity based on the Shannon–Hartley theorem through analyzing the influence of channel occupation, and thought that time-domain approach seems to be more accurate. Generally, the analysis of coexistence in the time domain can be achieved by calculating the overlap probability between the interfering and desired signals.
There are two methods on analyzing the overlap between two or more pulse signals; one is known as Poisson Distribution Method (PDM), and the other is called Monte Carlo Method (MCM). The assumption is adopted that the pulses are randomly distributed with respect to the time forming Poisson process in PDM. Vassilios [
3] explained the mechanisms of JTIDS interference on DME in the time domain based on PDM. However, the interfering and desired signals are periodic in most cases, the pulse-stream does not correspond with the Poisson distribution if the number of signals is not large enough. The pulse-stream can be thought to be corresponding with the Poisson distribution only when there are multi-path periodic signals, and then PDM can be used to analyze the interference. Najett et al. [
9] have analyzed the interference of LDACS on DME in the time domain based on a Monte Carlo process, and obtained the result that their coexistence is possible, however, It takes a lot of time for MCM to analyze interference to ensure the accuracy. Many researchers have studied interference between two or more periodic pulse signals using the mean probability of pulse collision [
10,
11], but they have not studied the impact of PRF and initial time on the probability of pulse collision. This paper presents a PPOM to calculate the overlap probability between two or more periodic pulse signals. Moreover, we study the parameters affecting overlap probability such as PRF, pulse duration, and the difference of initial time between interfering and desired signals. Compared with PDM, PPOM can be used to analyze single-path or multi-path interference source. PPOM is different from MCM, since PPOM is an analytic method and needs less time to analyze.
In this paper, we apply PPOM to analyze the interference of DME on ATCRBS in the time domain, and then provide corresponding measures to reduce or avoid interference impact from DME. The remainder of the paper is organized as follows. In
Section 2, we revisit the principles of DME and ATCRBS briefly, and focus on the technical characteristics of the two systems that are pertinent to the analysis of interference such as signal waveform, pulse duration, and PRF. In
Section 3, we build a model to analyze the interference of DME on ATCRBS, expound and prove the periodicity of pulse overlap, and study the characteristics of pulse overlap. In
Section 4, we use PDM and MCM and PPOM to analyze the interference of DME on ATCRBS, respectively, and then simulate the variance of recognition probability with respect to difference of initial time or multiples of PRF between interfering and desired signals. Finally, we provide conclusions in
Section 5.
3. Analysis and Derivation of PPOM Formula
PPOM is an analytical method based on the periodicity of pulse signal, the overlap probability between two or more pulse signals is used to characterize the interference probability, so the key to PPOM is the calculation of overlap probability. Analysis and derivation of overlap probability are discussed in this section. Firstly, we build a model of pulse overlap based on the signal structure of DME and ATCRBS; secondly, we expound and prove the periodicity of pulse overlap; thirdly, we derive the formula of the probability of pulse overlap and study the variation of number in a cycle. Finally, we derive the formula of the mean overlap probability for DME and ATCRBS.
3.1. Modeling of Pulse Overlap for DME and ATCRBS
Pulse overlap is also known as pulse collision, which is defined as the rising edge time of the back pulse is earlier than the falling edge time of the front pulse in a pulse-stream. If there are two pulses with different duration, moreover, both rising and falling edge times of the narrow pulse are in the range of wide pulse duration, then the narrow pulse is considered to be overlapped by wide pulse fully. In fact, it can be named as overlap if the duration of one event covers that of the other event partly or fully. Overlap occurs not only between two pulses but also between pulse and idle duration.
According to the signal structure of DME and ATCRBS, the model for pulse overlap analysis in the time domain is built and illustrated in
Figure 4. There are two periodic pulse signals; one pulse signal is PL
representing ATCRBS signal, the other pulse signal PL
composed of two pulses in one cycle represents DME signal. The pulse duration of PL
is
, which denotes
in
Figure 2 or
in
Figure 3.
and
represent the period of PL
and PL
, respectively. The symbol
and
represent the initial time and
is the difference between
and
. The definitions of the other symbols in PL
are the same as that shown in
Figure 1.
There are two idle times in PL
represented by
and
. PL
will not overlap with PL
if
is overlapped by
or
completely. If
denotes the rising edge time of the
-
th pulse of PL
, the difference represented by
between
and the rising edge time of PL
located in the front adjacent to the
-
th of PL
is calculated to be:
where
is the remainder from dividing
by
.
will overlap with the
-
th pulse of PL
fully if the following inequality condition is satisfied:
Similarly,
will overlap with the
-
th pulse of PL
fully if the following inequality condition is satisfied:
If the -th pulse of PL overlaps with or fully, then overlap probability can be considered to be probability of PL pulse non-overlapped by PL pulse.
3.2. Periodicity of Pulse Overlap
3.2.1. Calculation for the Period of Pulse Overlap
Since PL
and PL
are periodicity signals, the overlap between PL
and PL
should have a cyclical characteristic, and the period of pulse overlap
can be calculated as follows:
where
g is the greatest common divisor of
and
.
g is equal to the ration of the greatest common divisor of numerators to the least common multiple of denominators ratio when
and
are fractions. Clearly, the number
of pulse PL
in one overlap cycle can be expressed as follows:
Let the data set
denote the difference of the rising edge time between PL
and the first pulse of PL
in one overlap cycle:
3.2.2. Calculation for the Period of When
If
, the congruence equation is true.
can be obtained as follows:
So, the period of is . When varies from 0 to , all the difference of rising edge time in one overlap cycle are equal. That is to say, the difference of rising edge time has nothing to do with the sequence number k, so it must be either all or none of the PL pulses overlap with PL.
3.2.3. Calculation for the Period of When
Let natural number set
,
is defined as Equation (
6), given any nature number
, we can solve the congruence equation:
where
k is unknown number. Since
which satisfies the necessary and sufficient conditions for solution to congruence equation, there must exist solutions to satisfy Equation (
9), if
denotes one of its solutions, then the general solution can be expressed as follows [
14]:
There exists a unique solution designated
in the integer set
k. So we can conduct that:
Therefore, the data set
can be expressed as follows:
where
, moreover, they are different integers. For any
, we can get:
Note that
, then
can be expressed as follows:
Hence
g is the period of
. For a nonnegative integer
and
, we write
. According to Equation (
10), the equation
is true, so
can be expressed as follows:
Note that the maximum value
, then:
3.3. Calculation for the Number of Pulse Overlap
Let
.
is the modulus from dividing
by
g,
is the modulus from dividing
by
g, and
is the modulus from dividing
by
g, and define
as the remainders from dividing
and
and
by
g, respectively. We can put:
The
-
th element of
in Equation (
16) is
. If
overlaps PL
fully, then by Equation (
3),
must be satisfied. Combining Equation (
17), we obtain:
Subject to
, the minimum value of
k represented by
is expressed as follows:
Subject to
and
, the maximum value of
k represented by
can be expressed as follows:
So the number
of
overlapping with
pulse fully in one overlap cycle is calculated to be:
Let
denote the maximum value of
,
denote the maximum numerical value of
, and
denote the cutoff value of
. For example, if
, then
and
]; if
, then
. The variation of
and
with respect to
is shown in
Figure 5.
Now we consider the calculation for
, for any
,
can be calculated as follows:
So
is determined by
and
. If
,
changes with
, the mean value of
in a period of
represented by
can be calculated as follows:
Similarly, if
or
, the expression of
is the same as Equation (
23). If
, according to Equation (
8), all the elements of
is equal to
and the period of
is
, so
can be expressed as follows:
So
can be calculated as follows:
Clearly, Equation (
25) is the same as Equation (
23), so
is determined by
and
g whether
or not. Similarly, the mean value of PL
pulses overlapped fully by
is
.
3.4. Calculation for the Probability of Pulse Overlap
The overlap probability in one cycle represented by
can be defined as follows:
Combining Equations (6) and (22), the variation of
is 1 in one overlap cycle, so the variation of
represented by
can be calculated as follows:
Let
and
denote the PRF of ATCRBS and DME, respectively, and suppose
(
a and
b are natural and coprime numbers). So the great common divisor
g of
and
is calculated to be:
Combining Equations (6), (23), and (26), the mean value of
denoted by
can be calculated as follows:
4. Results and Discussion
To verify the validity of PPOM, we calculate and simulate the impact of DME interference on ATCRBS using PPOM, PDM, and MCM, respectively, and then study the results based on different methods. Moreover, we analyze the impact of PRF and difference of initial time on recognition probability. In this paper, we focus on the interference impact of DME interference on ATCRBS in the time domain, some assumptions are made as follows:
Probability of recognition equals that of non-overlapping in the time domain
Signal propagation environment characteristics are neglected
Time of signal processing in the equipment is negligible
Aircraft is motionless
Interference comes from the DME only
Interrogation antenna is not rotational.
4.1. Calculation and Simulation Based on Different Methods
4.1.1. Calculation Based on PPOM
One-path DME interference is assumed to be coexisted with ATCRBS signal. Using Equation (
30), Mean Recognition Probability (MRP) for ATCRBS interfered by DME is calculated as in
Table 1. The interference name is shown in the first row of
Table 1, DMEInterX and DMERepX denote interrogation and reply signal operating in mode X, respectively, and DMEY denotes interrogation or reply signal operating in mode Y. ATCRBS signal is shown in the first column of
Table 1, SSR A and SSR C denote the interrogation signal operating in mode A and mode C, respectively. ATC denotes the reply signal without SPI and ATC_SPI denotes the reply signal with SPI. MRP is shown in the cell intersected by the row and column correspondingly, where
(in MHz) is PRF of DME signal.
For example, when interfering signal is DMEInterX and wangted signal is SSR A, from
Figure 2 duration of wanted signal is
= 8.8
s, from
Figure 1 we can derive that the intra idle time of DME pulse pair
s and the inter idle time DME pulse pair
, considering that PRF of DME signal
2800 Hz [
12], i.e.,
s, hence
. Therefore, MRP can be calculated as Equation (
30), combining that
, we get:
Similarity, the other MRP in
Table 1 can be calculated as mentioned above.
Table 1 shows that MRP is a linear decreasing function of
when the operation mode of DME and ATCRBS are selected. Moreover, the calculation formula of MRP for DME RepX interference on ATCRBS is the same as that for DME Y interference, because
in DME RepX or DME Y is larger than the ATCRBS signal duration. From
Table 1, it can be seen that the operation mode of DME or ATCRBS has effect on
. For example, when interference is DME Y and its PRF
is equal to the maximum value 2700 Hz, the reply signal is ATC_SPI, then
can be calculated as
when ATCRBS is operating in mode C, while
is calculated by
when ATCRBS is operating in mode A without SPI. Similarity, when interference is DME InterX and its PRF
is equal to the minimum value 10 Hz, the reply signal is ATC,
can be calculated as
when ATCRBS is operating in mode A.
4.1.2. Calculation for MRP Based on PDM
The coexistence of
paths DME signal and one-path ATCRBS signal is assumed to constitute a pulse-stream. Since DME signal is composed of two pulses in one cycle, the pulse density of the pulse-stream can be defined as follows:
where
is the total pulse density of pulse-stream,
is the pulse density of DME pulse-stream,
is the pulse density of ATCRBS pulse,
is PRF of ATCRBS signal, and
is PRF of the
-
th path DME signal.
If
n is sufficiently large, the distribution of the arrival time from pulse-stream to ATCRBS can be considered to be in accordance with the Poisson distribution. There are two pulses next to an ATCRBS signal in the pulse-stream; one is in front of the signal, the other is behind the signal. ATCRBS signal is interfered if one of the pulses next to it overlaps with it. The front pulse in the pulse-stream cannot overlap with ATCRBS signal if the pulse is also an ATCRBS signal, because the duty cycle of an ATCRBS signal is larger than 1.
and
are the probabilities that the front pulse is ATCRBS signal and DME signal, respectively. The probability of DME signal non-overlapped by the back pulse in the pulse-stream is
. Similarly,
is the probability of ATCRBS non-overlapped by the back pulse in the pulse-stream. Therefore, the probability of ATCRBS signal duration non-overlapped by a DME pulse in the pulse-stream
can be calculated as follows:
4.1.3. MRP and MRN vs PRF Based on Different Methods
The interference is one-path DME interrogation signal operating in mode X, the PRF of interference is 1000∼2800 Hz with 200 Hz interval. The desired signal is ATCRBS reply with SPI with 25.15
s pulse duration and 2000 Hz PRF. The simulation times of MCM are 8000, and the initial time of DME and ATCRBS are the uniform distribution of data among their respective periods. Using PPOM, the MRP of ATCRBS signal is calculated by
shown in
Table 1, and the mean recognition number (MRN) of ATCRBS can be calculated as Equation (
23). The results based on different methods are depicted in
Figure 6. MRP is plotted on the left Y axis, MRN is plotted on the right Y axis, and the X axis represents the PRF of DME.
Figure 6 shows that MRP based on PPOM agrees well with that based on MCM, while MRP based on PDM is less than MRP based on MCM, the reason is that the pulse-stream composed of one channel DME signal and ATCRBS signal does not in agreement with Poisson distribution at all. PDM cannot be used to calculate MRP if the number of DME interference source is a small amount. However, if the number of DME interference source is large enough and the interference sources are statistical independence, furthermore, the PRF of DME is the uniform distribution of data from 30 Hz to 150 Hz; consequently, this kind of pulse-stream can be considered to accord with Poisson distribution.
The total MRP of ATCRBS signal is equal to the product of MRP non-overlapped by the entire DME signal. The total MRP with respect to the paths of DME signal is depicted in
Figure 7. The number of interference source is (1∼626) with 25 interval. The number on the curve is pulse-stream density (in pulses per second), for instance, when the number of interference source is 101, the pulse-stream density is 20,518 pulses per second.
Figure 7 shows that the result based on MCM is almost the same as that based on PPOM, just as shown in
Figure 6. In addition, the difference between MRP based on PDM and MRP based on PPOM gets smaller and smaller as the number of interference sources increases. In other words, MCM can be substituted by PDM to calculate MRP when the number of DME interference sources is large enough.
4.2. Variation of Recognition Probability vs Multiples of PRF
As mentioned above, the variation of overlap probability between two periodic pulse signals is periodic. Moreover, according to Equation (
29), the maximum variation of recognition probability
is equal to
when initial time difference varies in a period, which means that
can be adjusted through changing the multiples between DME PRF (
) and ATCRBS PRF(
). If the interfering signal is DME RepX, the desired signal is ATC and
is 1996 Hz. When
initiates from 700 Hz to 2700 Hz with 0.1 Hz interval, the variation of
is depicted in
Figure 8. It can be seen that
ranges from 0 to 1. Moreover, when
is close to the integer multiple of
,
is close to zero, and the closer to the integer multiple, the closer to zero. However, once
is an integer multiple of
, the difference value equals to 1.
4.3. Analysis of Recognition Probability Varying with Difference of Initial Time
According to Equation (
22), the recognition probability varies with difference of initial time. If the interference is DME RepX and its PRF
is 2561 Hz, the desired signal is SSR C and its PRF
is 394 Hz.
Figure 9 illustrates the recognition probability varying with the difference of initial time in a cycle. The definitions of
and
are the same as
shown in
Figure 5, and the definitions of
and
are the same as
shown in
Figure 5, and
,
. There are two idle durations in DME RepX pulse signal whose durations are all larger than that of ATCRBS signal, so there are four jumping points at
,
, and
. From
Figure 9, it can be seen that the MRP is 0.87041 according to the formula
shown in
Table 1, the maximum value of recognition probability difference is 0.5 according to Equation (
29). It is indicated that the distribution of recognition probability with respect to difference of initial time can be adjusted through changing ATCRBS PRF under the condition of keeping the MRP unchanged.
5. Conclusions
From the above discussion, the conclusion can be reached that PPOM is an accurate and efficient method to analyze interference between two or more periodic pulse signals. Moreover, PPOM can not only be used to analyze EMC, but it can also be used to reduce or avoid interference from the enemy equipment in electronic warfare. Equally important is that using PPOM, we can reduce or avoid interference by adjusting PRF or difference of initial time. Moreover, the maximum difference value and distribution of overlap probability in one overlap cycle can also be changed.
The interference from DME to ATCRBS in the time domain is analyzed based on PPOM, and the results show that the mean reply efficiency ranges from 0.7298 to 0.9994, as calculated in
Section 4.1.1. It is worth noting that the mean reply efficiency can be adjusted by changing operation mode of DME or ATCRBS, for example, other things being equal, the reply efficiency interfered by DME operating in mode X is larger than that interfered by DME operating in mode Y, and the reply efficiency operating in mode A is larger than that operating in mode C.
The limitation of PPOM is that it is derived from the periodicity of the signal; the statistical value of overlap probability can be obtained only if the signal is not periodic. Moreover, in this paper, the time of signal processing in the equipment is neglected, in fact, after recognition of a proper signal, the equipment shall not reply to any other signals for a long time, referred to as dead time. For example, the dead time of a transponder shall end no later than 125 microseconds after the transmission of the last reply pulse of the group [
12], so the recognition probability in this paper is larger than the actual value. Furthermore, we did not take into account signal propagation environment characteristics. As a further step, to corroborate with the actual condition, some assumptions should be taken into account.