RCG-44

MATHEMATICAL MODEL FOR RADIATION PATTERNS FOR RADAR ANTENNAS\
FOR USE IN INTERFERENCE ASSESSMENT

Rec. ITU-R F.1245-1

The ITU Radiocommunication Assembly,

considering

a)  that there is no defined radiation pattern for radar system antennas
    stated, for use in interference assessments, in ITU Recommendations

b)  that a mathematical model is required for generalized radiation
    patterns of antennas for statistical interference analyses involving
    interference entries such as from other radar and communication
    systems,

recommends

1.  that, in the absence of particular information concerning the
    radiation pattern and peak sidelobe level of the radar system
    antenna involved, the mathematical reference antenna model of
    Annex-1, equation (3), should be used for interference analysis.

2.  that, if real antenna patterns are available, then those should be
    used.

ANNEX 1

MATHEMATICAL MODEL FOR RADIATION PATTERNS\
FOR RADAR ANTENNAS\
FOR USE IN INTERFERENCE ASSESSMENT

Two mathematical models for radar antenna patterns are currently used in
interference analysis with radars. These are ITU-R F.1245-1 and ITU
M.1652. Descriptions of these two patterns are provided in annex-2 and
annex-3 for reference.

In this annex a comparison is made between recommendations ITU-R
F.1245-1 and ITU M.1652 and the calculation of the cosine and the
proposed capped cosine patterns.

The uniform illuminated linear or rectangular aperture voltage gain is
approximated by:

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------
  $$f(\theta) = \frac{\text{sin}\left\lbrack 2\text{.}\text{7831}\left( \frac{\theta}{\theta_{3}} \right) \right\rbrack}{2\text{.}\text{7831}\left( \frac{\theta}{\theta_{3}} \right)}$$   \(1\)

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------

The cosine illuminated aperture voltage is approximated by:

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------
  $$$g(\theta) = \frac{\text{cos}\left\lbrack 3\text{.}\text{7352}\left( \frac{\theta}{\theta_{3}} \right) \right\rbrack}{1 - 5\text{.}\text{6544}\left( \frac{\theta}{\theta_{3}} \right)^{2}}$   \(2\)

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -----------

The peak sidelobe for the $\text{sin}(x)/x$ pattern is -13 dB and for
the cosine pattern is -23 dB. Typical ITU recommendations provide data
for the antenna gain and antenna vertical (elevation) and horizontal
(azimuth) beamwidths, θ~3~. Given these parameters it is possible to
construct the antenna pattern in dBi by either
$\text{20}\text{log}\left( \left| f(\theta) \right| \right)$ or
$\text{20}\text{log}\left( \left| g(\theta) \right| \right)$, where
$f(\theta)$ and $g(\theta)$ are the voltage patterns.

Using equation 2, a reference antenna pattern using a mask that follows
the antenna peak sidelobe levels of the cosine pattern, capped cosine
pattern, is developed in equation 3. The proposed mask of the capped
cosine pattern, with a -40 dBi floor, is given by the FINAL_MASK(θ)
equation as:

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------- ------------------------------------- -------
  $$\text{MASK}(\theta) = \text{20}\text{*log}\left\lbrack \frac{\text{cos}\left\lbrack 3\text{.}\text{7352}\left( \frac{\theta}{\theta_{3}} \right) \right\rbrack}{1 - 5\text{.}\text{6544}\left( \frac{\theta}{\theta_{3}} \right)^{2}} \right\rbrack$$   *if*    $$0 \leq |\theta| \leq \theta_{3}$$   \(3\)

  $$\text{MASK}(\theta) = \left\lbrack \frac{\text{ln}\left( 2\text{.}\text{27} \ast \frac{\theta}{\theta_{3}} \right)}{- 0\text{.}\text{0563}} \right\rbrack$$                                                                                             *If*    $$|\theta| \geq \theta_{3}$$          

  $$\text{FINAL}_{\text{MASK}}(\theta) = \text{Maximum}\left\lbrack - \text{40},\text{MASK}(\theta) \right\rbrack$$                                                                                                                                                                                       
  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------- ------------------------------------- -------

Since the peak sidelobe level is not provided and since the most common
antenna pattern used in practical radar systems is the cosine pattern,
it is then safe to recommend that the pattern that should be used, for
all cases except cosecant-squared pattern, for interference analysis is
that of the capped cosine pattern mask given in equation 3. Equation 4
should be used for cosecant-squared antenna patterns.

The cosecant squared pattern may be calculated as a sin(x)/x pattern for
angles up to the beamwidth angle. Beyond that point the pattern is
approximated by a cosecant-squared pattern. The formula for this pattern
becomes:

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------
  $$\begin{array}{r}                                                                                                                                                                      \(4\)
  h(\theta) = \frac{\text{sin}\left\lbrack 2\text{.}\text{7831}\left( \frac{\theta}{\theta_{3}} \right) \right\rbrack}{2\text{.}\text{7831}\left( \frac{\theta}{\theta_{3}} \right)} \\   
                                                                                                                                                                                          
  \end{array}$$                                                                                                                                                                           

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------

Where θ~Max~ is the csc^2^ pattern maximum angle.

The cosine patterns and the proposed capped cosine pattern along with
ITU-R F.1245-1 (for D/λ \< 100) and ITU M.1652 (22\<Gain\<48) are
compared to the figures below. Contour plots and a 3-D plot are also
provided as examples. Data from the contour plots may be used in
simulation analysis tools.

![](media/image1.wmf){width="3.997916666666667in"
height="3.3319444444444444in"}

Figure 1: Antenna Pattern Comparison beamwidth of 1.0 degree

![](media/image2.wmf){width="3.9993055555555554in"
height="3.295138888888889in"}

Figure 2: Antenna Pattern Comparison beamwidth of 1.0 degree

![](media/image3.wmf){width="3.997916666666667in"
height="3.3319444444444444in"}

Figure 3: Antenna Pattern Comparison beamwidth of 2.0 degrees

![](media/image4.wmf){width="3.9993055555555554in"
height="3.295138888888889in"}

Figure 4: Antenna Pattern Comparison beamwidth of 2.0 degrees

![](media/image5.wmf){width="3.9993055555555554in"
height="3.295138888888889in"}

Figure 5: Antenna Pattern Comparison beamwidth of 6.0 degrees

![](media/image6.wmf){width="3.9993055555555554in"
height="3.295138888888889in"}

Figure 6: Antenna Pattern Comparison beamwidth of 6.0 degrees

![](media/image7.wmf){width="3.5in" height="3.6875in"}

Figure 7: Antenna Contour Pattern for BW_H=1.2 deg & BW_V=6 deg

![](media/image8.wmf){width="3.5in" height="3.6875in"}

Figure 8: Antenna Contour Pattern for BW_H= BW_V=1.7 deg

The 3-dimentional antenna pattern may be generated by multiplying the
horizontal and vertical principal plane voltage cuts. In this case
vertical pattern is a matrix with the vertical voltage pattern in the
center column and all other columns and rows set to zero. The Horizontal
pattern is a matrix with the horizontal voltage placed in the centre row
and all other rows and columns set to zero. The equation for calculating
the 3-dimentional pattern is given by:

  ----------------------------------------------------------------------------------------------------------- -----------
  $$$P_{i,h} = \text{20}\text{log}\left\lbrack \sum_{k = 0}^{N}\left| H_{i,k}V_{k,h} \right| \right\rbrack$   \(5\)

  ----------------------------------------------------------------------------------------------------------- -----------

Where the elevation and azimuth matrices are defined in equation 6 and
7.

  ----------------------------------------------------------- -----------
  $$$\text{Elevation}_{\text{Matrix}} = \begin{matrix}        \(6\)
  0 & \text{el} & 0 \\                                        
  0 & \text{el} & 0 \\                                        
  0 & \text{el} & 0                                           
  \end{matrix}$                                               

  ----------------------------------------------------------- -----------

  ----------------------------------------------------------- -----------
  $$$\text{Azimuth}_{\text{Matrix}} = \begin{matrix}          \(7\)
  0 & 0 & 0 \\                                                
  \text{az} & \text{az} & \text{az} \\                        
  0 & 0 & 0                                                   
  \end{matrix}$                                               

  ----------------------------------------------------------- -----------

Figure 9 shows an example of a 3-dimentional pattern.

![](media/image9.wmf){width="5.114583333333333in"
height="4.052083333333333in"}

Figure 9: Example 3-Dimentional Antenna Plot

The following are examples of radar antenna pattern test data, in the 9
GHz band, that show the approximate peak sidelobe level and the pattern
floor.

![](media/image10.png){width="3.9965277777777777in"
height="3.3256944444444443in"}

Figure 10: Example Measured Antenna Plot

![](media/image11.png){width="4.002083333333333in"
height="3.339583333333333in"}

Figure 11: Example Measured Antenna Plot

# Results:

The following are the results of the analysis:

1.  The capped cosine pattern mask is a good approximation for
    interference analysis where multiple interferes are present.

2.  ITU-R M.1652 overestimates the antenna pattern, as compared to both
    the sin(x)/x and cosine patterns, in the mainlobe region for small
    beamwidth.

3.  ITU-R M.1652 underestimates the antenna pattern, as compared to both
    the sin(x)/x and cosine patterns, in the mainlobe region for large
    beamwidth.

4.  ITU-R F.1245 matches the 3-dB beamwidth but overestimates the
    antenna pattern, as compared to both the sin(x)/x and cosine
    patterns, in the mainlobe region.

5.  Both ITU-R M.1652 and ITU-R F.1245 overestimate the antenna pattern,
    as compared to both the sin(x)/x and cosine patterns, in the
    sidelobe region.

6.  The sin(x)/x pattern has high sidelobes and does not represent the
    majority of elevation and azimuth radar antenna patterns.

7.  The 3-dimentional pattern may be provided by multiplying the
    principal plane azimuth and elevation cuts to generate the needed
    values for simulation.

Annex-2\
Appendix-2 From ITU-R F.1245

# 1 Introduction

Recommendation ITU-R F.699 gives the reference radiation patterns of
point-to-point fixed service antennas, based on the peak envelope of
side-lobe levels. Therefore, the interference assessment using this
Recommendation will inevitably lead to overestimation of interference.

On the other hand, the main text of this Recommendation gives a
mathematical model for average radiation patterns of point-to-point
fixed service antennas, representing average side-lobe levels. However,
this can be applied only in the case of multiple interference entries or
time-varying interference entries.

A mathematical model is required for generalized radiation patterns of
antennas for use only in spatial statistical analysis such as deriving
the probability distribution function (pdf) of interference from a few
GSO satellite systems into a large number of interfered with fixed
service systems or stations.

# 2 Antennas with D/λ greater than 100

The reference radiation pattern of antennas with *D*/λ greater than 100
representing peak envelope side-lobe levels is given by *recommends* 2.1
of Recommendation ITU-R F.699. According to *recommends* 2.1 of this
Recommendation, the average side-lobe level is 3 dB below the peak
envelope side-lobe level. It seems reasonable to assume that the actual
side-lobe levels vary sinusoidally. Therefore, the actual radiation
pattern will be expressed as follows:

*G*(ϕ) = max \[*Ga*(ϕ), *Gb*(ϕ)\] for 0 ≤ ϕ \< ϕ*r* (1a)

$$ *G*(ϕ) = 32 -- 25 log ϕ* *+ *F*(ϕ) for ϕ*r* ≤ ϕ \< 48^o^ (1b)

*G*(ϕ)* *=* *--10 + *F*(ϕ) for 48^o^ ≤ ϕ ≤ 180^o^ (1c)

where:

$G_{a}(\phi)\  = \ G_{\text{max}}\  - \ 2\text{.}5\  \times \ \text{10}^{- 3}\left( \frac{D}{\lambda}\ \phi \right)^{2}$
(1d)

*Gb*(ϕ)* *=* G*1 + *F*(ϕ) (1e)

*G*1 = 2 + 15 log (*D*/λ) dB (2a)

ϕ*r* = 15.85 $\left( \frac{D}{\lambda} \right)^{- 0\text{.}6}$ degrees
(2b)

*F*(ϕ)* *=* *10 log
$\left( 0\text{.}9\ \text{sin}^{2}\left( \frac{3\ \pi\ \phi}{2\ \phi_{r}} \right)\  + \ 0\text{.}1 \right)$
dB (2c)

where ϕ*r* is assumed to correspond to the off-axis angle of the peak of
the first side-lobe and the phase at ϕ = ϕ*r* is assumed to be 1.5π. It
should be noted that the argument of sin function in equation (2c) is
expressed in radians and that the value of *F*(ϕ) is nearly zero or
negative. *F*(ϕ) = 0 corresponds to side-lobe peaks. The parameter 0.1
is introduced in equation (2c) in order to avoid the situation
that *F*(ϕ) falls below --10 dB.

# 3 Antennas with D/λ equal to or smaller than 100

In the case of antennas with *D*/λ equal to or smaller than 100, it will
be assumed that side-lobe peak levels are 3 dB higher than the average
side-lobe level given in the main text of this Recommendation.

Thus, the following pattern is provisionally presented as a generalized
radiation pattern of the antenna with *D*/λ equal to or smaller than
100:

*G*(ϕ) = max \[*Ga*(ϕ), *Gb*(ϕ)\] for 0 ≤ ϕ \< ϕ*r* (3a)

$$ *G*(ϕ) = 42 -- 5 log (*D*/λ) -- 25 log ϕ* *+ *F*(ϕ) for ϕ*r* ≤ ϕ \<
48^o^ (3b)

*G*(ϕ)* *=* *--5 log (*D*/λ) + *F*(ϕ) for 48^o^ ≤ ϕ ≤ 180^o^ (3c)

where:

$G_{a}(\phi)\  = \ G_{\text{max}}\  - \ 2,5\  \times \ \text{10}^{- 3}\left( \frac{D}{\lambda}\ \phi \right)^{2}$
(3d)

*Gb*(ϕ)* *=* G*1 + *F*(ϕ) (3e)

*G*1 = 2 + 15 log (*D*/λ) dB (4a)

ϕ*r* = 39.8$\left( \frac{D}{\lambda} \right)^{- 0\text{.}8}$ degrees
(4b)

*F*(ϕ)* *=* *10 log
$\left( 0\text{.}9\ \text{sin}^{2}\left( \frac{3\ \pi\ \phi}{2\ \phi_{r}} \right)\  + \ 0\text{.}1 \right)$
dB (4c)

Again, it should be noted that the argument of sin function in equation
(4c) is expressed in radians and that the value of *F*(ϕ) is nearly zero
or negative and *F*(ϕ) = 0 corresponds to side-lobe peaks. The reason
for introducing the parameter 0.1 in equation (4c) is the same as that
for equation (2c).

# 4 Conclusion

Equations (1a) to (1e) (together with (2a) to (2c)) and (3a) to (3e)
(together with (4a) to (4c)) are provisionally presented as mathematical
models of generalized radiation patterns of point-to-point fixed service
antennas for use only in spatial statistical interference assessment.

Annex-3\
Appendix-3 From ITU-R M.1652

There are no existing radar antenna reference patterns currently in ITU,
therefore the following is provided as a baseline. A statistical gain
antenna model is used to determine the radar antenna gain in the azimuth
and elevation orientations. The model gives the antenna gain as a
function of off-axis angle (θ) for a given main beam antenna gain (*G*).
The model includes separate algorithms for very high-gain, high-gain,
and medium-gain antennas, corresponding to antennas with gains greater
than 48 dBi, gains between 22 and 48 dBi, and gains between 10 and 22
dBi, respectively. Figure 1 illustrates the general form of the antenna
gain distribution. The equations for the angles θ*~M~* (first side-lobe
shelf), θ*~R~* (near side-lobe region), and θ*~B~* (far side-lobe
region) are given in Table 1. The antenna gains as a function of
off-axis angle, are given in Table 2 for very high-gain antennas,
in Table 3 for high-gain antennas, and in Table 4 for medium-gain
antennas. The angle θ is in degrees and all gain values are given in
terms of decibels relative to an isotropic antenna (dBi).

![](media/image12.wmf){width="3.857638888888889in"
height="2.736111111111111in"}

TABLE 1

Angle definitions

TABLE 2

Equations for very high-gain antennas (*G* \> 48 dBi)

TABLE 3

Equations for high-gain antennas (22 \< *G* \< 48 dBi)

TABLE 4

Equations for medium-gain antennas (10 \< *G* \< 22 dBi)