Radar Correspondence Group

Subject: Work programme on the feasibility of using statistical and
operational aspects in the protection criteria for radiodetermination
radar systems.

Contacts: Hugues de BAILLIENCOURT, +33 2 98 34 12 32,

<hde_bailliencourt@compuserve.com>

**1-Introduction**

Documents 8B/214 and 8B/259 from United Kingdom are a reaction to the
French input in 8B concerning the feasibility of using statistical and
operational aspects in the protection criteria for radiodetermination
radar systems and in particular concern the subject of the effect of
fluctuating targets (RCS) on the required protection criteria for radar
systems.

It seems that the French input on the issue of fluctuating target was
misunderstood and therefore the following contribution is aiming to
provide a comparison of several 8B documents as the new ITU-R report on
test measurements on maritime radars, documents 8B/214, 259, 72 and 60.
The RCG is requested to discuss this issue and provide its views.

**2-Generality on the integration improvement factor**

The figure below from Skolnik (2nd edition) provides the integration
improvement factor for Swerling I, II, III, IV and V versus the number
of pulses integrated and the Pd per pulses.

![](media/image1.png){width="5.44375in" height="3.65in"}

The important elements to be noticed from the above curves are the two
following principles:

-For integration of elements which are dependent, the integration
improvement factor (I(n)) is Pd per pulse independent. This is the case
of integration during one scan of several pulses with a Swerling I, III
or V as they are dependent from pulse to pulse. This is confirmed in the
two United Kingdom contributions where in figure 2 of document 8B/214
and in figure 7 of 8B/259 the integration of pulses of Swerling V does
not change the gradients of the curves Pd versus S/N but only translate
them to the left.

-For integration of elements which are independent, the integration
improvement factor (I(n)) is Pd per pulse dependent. This is the case of
integration during one scan of several pulses with a Swerling II or IV
type of fluctuation as they are independent from pulse to pulse. The
gradients of the curve Pd versus S/N are modify as the integration
improvement factor is higher for high Pd and lower for low Pd (see the
above Skolnik figure). This is one of the reasons to use several
frequencies for pulses of one scan as in this case we introduce
independency between pulses and therefore improve the integration
improvement factor for high Pd per pulses.

**This can be summarised as follow: the integration of dependent
elements does not modify the gradients of the Pd versus S/N curves
however the integration of independent elements modify them.**

In the case of Swerling I type of fluctuation, the power is dependent
from pulse to pulse during one scan but independent from scan to scan
(definition of Swerling I). Therefore the Pd versus S/N curve will be
first translated to the left due to the integration of several pulses
during one scan and then will be translated to the left and will
increase it gradient due to the integration of several scans. The figure
below shows how the curves Pd versus S/N are modified by the
integrations (integration of pulses and then integration of scans) for
Swerling I.

![](media/image2.wmf){width="6.285416666666666in"
height="3.4923611111111112in"}

**3-IMO criteria of 8 out 10 scans for the track initiation**

Using the Markov chain (seen annex 1) it is possible to have the link
between the probability of track initiation, the Pd per scan and the
number of scans. The result is presented in the following figure in the
case of 8 out of 10 scans (IMO criteria for track initiation):

![](media/image3.wmf){width="6.3875in" height="4.3069444444444445in"}

Example of interpretation of the above figure:

If we have a Pd per scan of 0.9 then the probability to initiate the
track after 8 scans is a around 0.5, around 0.8 after 9 scans and 0.95
after 10 scans.

**4-Aeronautical criteria for track initiation**

In the case of aeronautical radar there is no agreed criterion for the
track initiation however it is proposed to use the criteria from
document 8B/XXX which is 3 scans out of 4 scans. This provides the
following figure using the Markov chain. This should be discussed within
RCG and/or 8B to validate or not this criterion.

![](media/image4.wmf){width="6.3875in" height="4.3069444444444445in"}

**5-Maritime radars case (New ITU-R report on test measurements, 8B/214
and 8B/259)**

The Swerling I fluctuation model is used in this case as in document
8B/214 and 259.

***5-1 New ITU-R report on test measurements (Note: here the aim is not
to modify the report.)***

The way the targets were generated for the test was to generate non
fluctuating pulses and to adjust the power in order to have 0.9 of
probability of detection per scan. The following figure then presents
the budget link in the case of non fluctuating targets.

From the 8B/72 figure (see below), figure agreed by 8B, the Pd per scan
for Swerling I fluctuating targets and for an S/N+I(n) of around 13dB,
which correspond to an Pd per scan of 0.9 for non fluctuating targets,
is around 0.53.

![](media/image5.png){width="4.835416666666666in"
height="3.626388888888889in"}

From the figure of section 3, it can be seen that for a Pd per scan of
0.53 the probability of track initiation is around 0.1 (see figure
below).

![](media/image3.wmf){width="5.11875in" height="3.451388888888889in"}

Therefore this will provide the following budget link in the case of
fluctuating targets (real case).

It can be shown that if a real radar is designed to have a S/N of
13dB-I(n) then the probability of track initiation after 10 scans using
the IMO criteria is around 0.1 dB (from figure in section 3). Therefore
the IMO provides an attenuation instead of a gain. To be sure to
initiate the track (probability of track initiation equal to 1) it can
be seen from the figure in section 3 that we need a quasi infinite
number of scans and then it is like impossible. This is unacceptable for
maritime radars.

Therefore the maritime radars can not be designed to have a S/N of
13dB-I(n) but in the real case the S/N should be around 21 dB-I(n) in
order to have a track initiation of around 0.95 after 10 scans. The
figure below shows the budget link that real maritime radar should have.

In this case it can be seen, from the figure in section 3 that to be
sure to initiate the track (probability of track initiation very close
to 1) we need to wait at least around 16 scans.

***5-2 Document 8B/214 and 259 case***

If we take the case of 5 scans integration from document 8B/214 and 259
the S/N is around 9.5 dB --I(n) (see figure below from document 8B/214).
Then the Pd per scan is therefore 0.25 which does not correspond to the
way that the test measurements were performed in the new ITU-R report
(0.9 dB per scan and not 0.9 after 5 scans).

![](media/image6.png){width="4.247916666666667in" height="3.1375in"}

The budget link, based on the above figure and the IMO criteria of 8 out
of 10 scans, is therefore as follows:

![](media/image3.wmf){width="5.11875in" height="3.451388888888889in"}

It can be seen for the figure in section 3 that in this case the
probability of track initiation is 0 which is of course unacceptable for
maritime radars.

The real S/N should be again 21 dB-I(n) and not 9.5dB-I(n) in order to
have a probability of track initiation of 0.95 after 10 scans using the
IMO criterion. In this case the points that should be used in order to
determine the operational impact of an interference should be:

S/N=21 dB-I(n) and therefore a Pd for one scan of 0.9 for 2 scans
integrated of around 0.98 and for 3, 4 or 5 scans integrated around 1
(see figure below).

![](media/image6.png){width="4.247916666666667in" height="3.1375in"}

It can be seen that in this case the gradient of the curves at S/N at 21
dB-I(n) is much less for the integration of 2, 3, 4 or 5 scans.

***5-3 Conclusion for the maritime radars case***

For the maritime radars case the point where the figure 3 of document
8B/214 should be drawn is at the level of scan (Pd per scan) and not
after the integration of several scans as the input of the IMO criteria
(8 scans out of 10 scans) is the Pd per scan and not the Pd after
several scans. In order to have an acceptable level of track initiation
after 10 scans (around 0.95: to be determine by IMO) the S/N should be
around 21 dB-I(n) with I(n) the integration improvement factor due to
the integration of several pulses within one scan.

Therefore the curves that should be used to develop the figure 3 of
document 8B/214 (impact of the interference) are the one presented in
document 8B/72 but translated to the left with a value of I(n) as they
were for one pulse. Their gradients will not change.

The following table summaries the comparison between document 8B/214,
259, 72 and the New ITU-R report.

  ---------------- ---------- --------- ------- --------------------------
                   S/N+I(n)   Pd per    Pd per  Probability of track
                   in dB      pulse     scan    initiation: 8 out of 10
                                                scans

  Document 8B/214  9.5        Depend on 0.25    0
  and 259                     n                 

  New ITU-R report 13         Depend on 0.53    0.1
                              n                 

  Document 8B/72   21         Depend on 0.9     0.95
                              n                 
  ---------------- ---------- --------- ------- --------------------------

**6-Aeronautical radars case (8B/60)**

The budget link in document 8B/60 using figure in section 4 is:

which implies the following one for fluctuating targets (the real case)

Again a probability of track initiation of 0.4 after 4 scans (4\*60/rpm=
48 seconds if RMP is 5 rotations per minute) is unacceptable. From the
figure of section 4 the probability to be sure to initiate the track
corresponds to 30 scans (360 seconds) which is unacceptable also.
Therefore a real aeronautical radar can not also be designed to have an
S/N+I(n) of 13 dB but should have an S/N+I(n) of 21 dB in order to have
a probability of track initiation of around 0.94 after 4 scans.

The budget link should therefore be as follows:

The following table summaries the comparison between document 8B/60 and
72.

  ------------ ----------- -------- -------- ------------------------------
               S/N+I(n) in Pd per   Pd per   Probability of track
               dB          pulse    scan     initiation: 3 out of 4 scans

  Document     13                   0.53     0.4
  8B/60                                      

  Document     21                   0.9      0.94
  8B/72                                      
  ------------ ----------- -------- -------- ------------------------------

**7 Conclusion**

This document aims to compare the process presented in several ITU-R 8B
contributions in order to clarify what are the differences in the
assumptions.

It also outlines the necessary precision required when specifying radar
technical and operational features. It is then obvious that in order to
know the operational impact of an interference (as ask in the ITU-R
question) the automatic track initiation process is an important one:

-   a decrease of the probability of detection per scan due to
    interference will then decrease the probability of track initiation
    and therefore the operational impact will be a delay in the track
    initiation. The variation of the probability of track initiation due
    to interference depends on the target fluctuation type and the
    criteria for the track initiation as the IMO one (8 out of 10
    scans).

-   Therefore the detection criteria as in IMO document section 5.3.1
    should be specify by:

    -   *The track initiation delay (For IMO is 8 scans out of 10
        scans)*

> *and*

-   *The probability of track initiation at 10 scans or the probability
    of detection per scan. In the IMO document this is missing. This is
    needed as section 3 figure shows that we can still be sure to
    initiate the track even with a Pd per scan of 0.8 but in this case
    we have to wait 30 scans. Therefore, the criterion of 8 scans out of
    10 scans is not enough to specify the minimum signal that radars
    should received.*

Finally the Markov Chain provides a tool to have the link between the
level of interference at the radar antenna which can be a time variable
and it most important operational impact: the delay in track initiation.
Other operational impacts should be investigated as the impact on the
lost of track but still in this case the Markov chain should be a good
tool.

Annex 1

Example of Markov Chain

Example of track initiation criteria: 3 scans out of 4 consecutive
antenna scans (mainly for aeronautical radars);

The following figure represents the Markov chain of such a criterium:

![](media/image7.wmf){width="6.527777777777778in"
height="3.5319444444444446in"}

Where:

State 1 corresponds to 0000 (all the four scans do not have any
detection), 1000 (the first scan has a detection and the 3 last scans do
not have any detection), 0100 and 1100;

State 2 corresponds to 0001 and 1001;

State 3 corresponds to 0010 and 1010;

State 4 corresponds to 0011;

State 5 corresponds to 0101;

State 6 corresponds to 0110;

State 7 corresponds to the track initiation: 0111, 1011, 1101, 1110 and
1111;

This model provides the following Markov matrix M(t):

  ------------ ---------------- -------------- ------------ -------------- -------------- ------------ --------------
  **States**   **1**            **2**          **3**        **4**          **5**          **6**        **7**

  **1**        **1- Pd(t)~n~**  **Pd(t)~n~**   **0**        **0**          **0**          **0**        **0**

  **2**        **0**            **0**          **1-         **Pd(t)~n~**   **0**          **0**        **0**
                                               Pd(t)~n~**                                              

  **3**        **1-Pd(t)~n~**   **0**          **0**        **0**          **Pd(t)~n~**   **0**        **0**

  **4**        **0**            **0**          **0**        **0**          **0**          **1-         **Pd(t)~n~**
                                                                                          Pd(t)~n~**   

  **5**        **0**            **0**          **1-         **0**          **0**          **0**        **Pd(t)~n~**
                                               Pd(t)~n~**                                              

  **6**        **1- Pd(t)~n~**  **0**          **0**        **0**          **0**          **0**        **Pd(t)~n~**

  **7**        **0**            **0**          **0**        **0**          **0**          **0**        **1**
  ------------ ---------------- -------------- ------------ -------------- -------------- ------------ --------------

Where:

Pd~n~ is the probability of detection of the considered scan n.

Note: A Matlab program, which computes the Markov matrix in the general
cases (M scans out of N consecutive scans was used for the IMO criteria
(At least 8 out of 10 scans).

For each scan, the probability of track initiation is determined by the
seventh element of vector Vnscan defined as follow:

$$V_{\text{nscan}} = \text{Vo} \ast \prod_{\text{nscan} = 0}^{\text{nscna}}{M(\text{nscan} \ast \text{rpm})}$$

Where:

Vo is the initial probability vector (example for initial state 1
Vo=1000000)

nscan is the number of scans

rpm is the antenna rotation per minute

**For example (Case of at least 3 out of 4 scans)**

If the initial state is 1 and the probability of detection per scan is
constant versus time and equal to 90% therefore the probability of
having the track initiation before the fourth scan is V~4~(7)= 94.77%
(see below):

$$V_{\text{nscan}} = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \ast \begin{bmatrix}
0\text{.}1 & 0\text{.}9 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0\text{.}1 & 0\text{.}9 & 0 & 0 & 0 \\
0\text{.}1 & 0 & 0 & 0 & 0\text{.}9 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0\text{.}1 & 0\text{.}9 \\
0 & 0 & 0\text{.}1 & 0 & 0 & 0 & 0\text{.}9 \\
0\text{.}1 & 0 & 0 & 0 & 0 & 0 & 0\text{.}9 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}^{4} = \lbrack 0\text{.}\text{01  0}\text{.}\text{009  0}\text{.}\text{009  0}\text{.}\text{0081  0}\text{.}\text{0081  0}\text{.}\text{0081  0}\text{.}\text{9477}\rbrack$$

The seventh element of Vscan is Vnscan(7)=0.9477 which corresponds to a
probability of 94.77% to be in state 7 (track initiation) before the
fourth scan (P(4)).