# Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses

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1 Recommendation ITU-R M (1/18) Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses M Series Mobile, radiodetermination, amateur and related satellite services

4 Rec. ITU-R M Annex 1 Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses 1 Introduction A generalized mathematical model for radiodetermination radar systems antenna patterns is required when these patterns are not defined in ITU-R Recommendations applicable to the radiodetermination radar system under analysis. Generalized antenna pattern models could be used in analyses involving single and multiple interferer entries, such as that from other radar and communication systems. This text describes proposed antenna patterns to be used. Given knowledge about beamwidth and the first peak side-lobe level, the proper set of equations for both azimuth and elevation patterns may be selected. The result of surveyed antenna parameter ranges from ITU-R Recommendations are recorded in Table 1. TABLE 1 Surveyed antenna parameter limits Antenna parameter Units Description Minimum value Maximum value Transmit and receive frequencies MHz Antenna polarization type Antenna type Beam type most common Horizontal, vertical, circular Omni, yagi element array, parabolic reflector, phased array Fan, pencil, cosecant squared Transmit and receive gain dbi Elevation beamwidth ( 3dB) degrees Pencil beam Cosecant squared (CSC ) (equation (1) and Table 4) 3.6 (θ 3) (θ Max) 3.6 (θ 3) 44 (θ Max) Azimuth beamwidth ( 3dB) degrees Pencil beam Elevation scan angle limit degrees 6 +9 Azimuth scan angle limit degrees 3 sector 36 First side-lobe level below main lobe peak db Table 1 was used to guide the development of the antenna types and patterns proposed.

5 Rec. ITU-R M Mathematical formulae.1 Radar antenna with a narrow beamwidth.1.1 Background In the absence of neither particular information concerning the 3 db beamwidth or the shape of the current distribution or illumination field on the antenna aperture, but in the presence of information given about the size of the antenna, the θ3 half power beam-width (degrees) is approximated by 7 λ/d, where λ the wavelength and D the antenna length are expressed in the same units; see Recommendation ITU-R F.699, recommends 4.1. If the information about the shape of the current distribution or illumination field on the antenna aperture is accessible, then a more accurate model of the antenna pattern can be used. To simplify the analysis, the antenna current distribution is considered as a function of either the elevation or azimuth coordinates. The directivity pattern, F(μ), of a given distribution is found from the finite Fourier transform as: jx F f ( x) e dx (1) where: f (x): relative shape of field distribution, see Table and Fig. 1 l μ: provided in the table below = sin l: overall length of aperture : wavelength : beam elevation or azimuth pointing (scan) angle relative to aperture normal θ: point of interest direction angle relative to aperture normal : point of interest direction angle relative to pointing angle direction (=θ-) x: normalized distance along aperture 1 x 1 j: complex number notation.

7 Rec. ITU-R M Theoretical antenna equations Equations of directivity patterns and associated parameters are given in Table for different shape of field distribution on the antenna aperture. TABLE Theoretical antenna directivity parameters Relative shape of field distribution f(x) where 1 x 1 Directivity pattern F(μ) 3 half power beamwidth (degrees) μ as a function of 3 First sidelobe level below main lobe peak (db) Equation No. Uniform value of 1 sin () 5.8 l 5.8sin ( ) () COS(*x/) cos ( ) 68.8 l 68.8sin ( ) 3 3 (3) COS (*x/) sin ( ) 83. l 83.sin ( ) 3 3 (4) COS 3 (*x/) COS 4 (*x/) 3 cos ( ) sin( ) l 95sin ( ) 16sin ( ) 16 l (5) 47 (6)

8 6 Rec. ITU-R M where θ3 is the 3 db antenna half-power beamwidth (degrees). The relative shapes of the field distribution functions f(x), as defined in Table, are plotted in Fig.. FIGURE Aperture distributions Aperture distributions Relative on axis gain (db) Normalized distance along aperture f(x) where 1 <= x <= 1 Uniform Cos 3 Cos Cos Cos 4 M Given that the half power beamwidth, θ3, is provided, the value of μ can be redefined as a function of l l the half-power antenna beamwidth. This is done by replacing the quantity in sin by a constant that depends on the relative shape of the field distribution; divided by the half-power beamwidth, θ3, as shown in Table. These constant values of 5.8, 68.8, 83., 95 and 16, shown in Table, can be derived by setting the equation for F(μ) equal to 3 db, and solving for the angle. Figure 3 shows various linear aperture antenna patterns for uniform, cosine (COS), cosine-squared (COS ) and cosine-cubed (COS 3 ) and cosine-to-the-fourth power (COS 4 ) field distribution functions. As the patterns are mathematically symmetric, they have been partially traced on the diagram. For comparison, all patterns are set to a same 3 db beamwidth of 6. degrees, meaning different ratios for λ/l.

10 8 Rec. ITU-R M θ3 is the 3 db antenna beamwidth (degrees); θ is the angle in either the elevation (vertical) or azimuth (horizontal) principal plane cuts (degrees); the average mask is the peak mask minus approximately 4 db. Note that the break points of the peak pattern are different from the average patterns. Table 3 shows the equations to be used in the calculations. Field distribution Uniform COS COS COS 3 COS 4 TABLE 3 Peak and average theoretical mask pattern equations Mask equation beyond pattern break point where mask departs from theoretical pattern (db) Peak pattern break point where mask departs from theoretical pattern (db) Average pattern break point where mask departs from theoretical pattern (db) Constant added to the peak pattern to convert it to average mask (db) Mask floor level (db) Equation No ln (7) ln (8) ln (9) ln (1) ln (11) The function ln() is the natural logarithm function. An example of the break point is shown in Fig. 4.

11 Rec. ITU-R M FIGURE 4 Break point example Antenna radiation pattern, beamwidth = 3 5 Break point (db) where the peak mask departs from the theoretical pattern Normalized antenna pattern (db) 1 15 Break point (db) where the average mask departs from the theoretical pattern Uniform Peak envelope Average envelope Angle measured from beam peak M Radar antenna with a cosecant-squared elevation pattern The cosecant-squared pattern is a special case. The power (not field-strength) is given by: where: G(θ): G(θ1): 1 CSC 1 CSC G G (1) cosecant squared pattern between angles of θ1 and θmax pattern gain at θ1 θ1: half power antenna beamwidth where cosecant-squared pattern starts = θ3 θmax: maximum angle where cosecant-squared pattern stops θ: elevation angle (degrees) θ3: half power antenna beamwidth (degrees). The average antenna pattern gain is not considered for the cosecant-squared pattern. It should be used for single and multiple interferers. The cosecant pattern is applied as shown in Table 4:

12 1 Rec. ITU-R M TABLE 4 Cosecant-squared antenna pattern equations Cosecant-squared equation Condition Equation No. sin ; 5.8 sin CSC 1 G Max CSC 1 (13) 3 (14) Cosecant floor level (example = 55 db) Max 9 (15) Note that G G sin sin G sin1 1 CSC 1 CSC 5.8 sin sin sin (16) refers to the amplitude of the power pattern, while sin refer to the Directivity pattern F(μ), field amplitude; which are square of sin power amplitude. The solution might be writing and G A graphical description of the patterns is shown in the Figures below sin 1 sin 3 5.8sin 1 3 and FIGURE 5 Cosecant squared beam coverage for search radar Height Maximum of CSC pattern Elevation of half power point on the main lobe Maximum height CSC beam Angle Fan beam Range Maximum range M

13 Rec. ITU-R M Theoretical diagrams and masks for different antenna radiation patterns FIGURE 6 Antenna pattern, peak sin in radians or sin 18 sin 57.3 uniform field distribution in degrees, and average envelope for Antenna radiation pattern, beamwidth = 3 1 Normalized antenna pattern (db) Uniform Peak envelope Angle measured from beam peak Average envelope M

14 1 Rec. ITU-R M FIGURE 7 Example polar plot antenna pattern, peak and average envelope for uniform field distribution 15 Antenna radiation pattern, beamwidth = db Uniform Peak envelope Average envelope M FIGURE 8 Antenna pattern, peak and average envelope for cosine field distribution Antenna radiation pattern, beamwidth = 3 1 Normalized antenna pattern (db) Angle measured from beam peak Cosine Peak envelope Average envelope M

15 Rec. ITU-R M FIGURE 9 Antenna pattern, peak and average envelope for a cosine-squared distribution Antenna radiation pattern, beamwidth = 3 1 Normalized antenna pattern (db) Cosine Peak envelope Angle measured from beam peak Average envelope M FIGURE 1 Antenna pattern, peak and average envelope for a cosine-cube distribution Antenna radiation pattern, beamwidth = 3 1 Normalized antenna pattern (db) Cosine 3 Peak envelope Angle measured from beam peak Average envelope M

16 14 Rec. ITU-R M FIGURE 11 Antenna pattern, peak and average envelope for a cosine-to-the-fourth power distribution Antenna radiation pattern, beamwidth = 3 1 Normalized antenna pattern (db) Cosine 4 Peak envelope Angle measured from beam peak Average envelope M FIGURE 1 CSC antenna pattern envelope Cosecant-squared pattern Example: 3 db beamwidth = 4, maximum CSC = 35 Start pattern using sin( x )/ x from negative 3 db Beamwidth devided by.88 to positive 3dB Beamwidth point Relative gain (db) 3 4 Cosecant-squared pattern Pattern floor level Angle (degrees) M Antenna pattern selection A suggestion for how the antenna pattern should be selected is based on information about half-power beamwidth and peak side-lobe level. This is provided in Table 5 given information about half power beamwidth.

17 Rec. ITU-R M Range of first side-lobe level below normalized main lobe peak (db) 13. to < to < 3 3 to < to <45 TABLE 5 Pattern approximation selection table Possible antenna distribution type and cosine raised to power n Theoretical pattern Equation number Mask equation number Uniform () (7) n = 1 (3) (8) n = (4) (9) n = 3 (5) (1) >= 45 n = 4 (6) (11) 4 Antenna pattern comparison One mathematical model for a radiodetermination radar antenna pattern that has been used in interference analysis is given in Recommendation ITU-R M.165. It includes equations for several patterns as a function of antenna gain. A comparison between the models developed in this Recommendation and Radar-C from Recommendation ITU-R M shows that the pattern in Recommendation ITU-R M.165 is not optimal. As shown in Fig. 13, the pattern from Recommendation ITU-R M.165 significantly overestimates the antenna gain off the antenna boresight ( ). It should also be noted that the equations defined in Recommendation ITU-R F.699 tends to overestimate the sidelobe levels of some radar systems, and it was not developed for radar systems.

18 16 Rec. ITU-R M FIGURE 13 Antenna pattern comparison Comparison of ITU-R M.165 to COS patterns (Example radar-c: gain = 44 dbi, El = Az beamwidth =.95, 1st SLL = 35 db) Normalized amplitude (db) ITU-R M.165 COS COS a verage pattern with floor ITU-R M.165 in pattern is not valid. This pattern overestimates interference Theoretical COS pattern Average COS pattern "mask + floor pattern" Angle (degrees) M Approximating three-dimensional (3-D) patterns Data from the contour plots may be used as simulation analysis tools. The three-dimensional (3-D) antenna pattern can easily be approximated. This is done by multiplying the horizontal and vertical principal plane voltage cuts. To do this, place the vertical principal plane pattern in the centre column of a square matrix, and set all the other elements to zero. Place the horizontal principal plane pattern in the centre row of a square matrix and set all the other elements to zero. Multiply the two matrices together and plot. Note that all patterns must be normalized. The equation for calculating the 3-D pattern is given by: N Pi, h log H k, ivh, k (17) k where the elevation and azimuth matrices, in units of volts, are defined in equations (18) and (19). The vertical pattern is given by:

19 Rec. ITU-R M El El El Vertical matrix (Vh,k) = (18) ElN ElN... The horizontal pattern is given by: Horizontal matrix (Hk,i) = Az1 Az AzN 1 AzN (19) Figures 14 and 15 show an example of a 3-D pattern. FIGURE 14 Contour Normalized (db) Pattern elevation (degrees) Elevation beamwidth = 7, azimuth beamwidth = Azimuth (degrees) Amplitude (db) M

20 18 Rec. ITU-R M FIGURE 15 Example of a 3-D relative azimuth and elevation antenna plot, uniform field distribution of a rectangular aperture, attenuation (db) versus off-axis angle; θ (elevation, degrees) and φ (azimuth, degrees) Elevation beamwidth = 7, azimuth beamwidth = 1 3 Amplitude (db) Amplitude (db) Elevation (degrees) Azimuth (degrees) M Measured pattern examples Figures 16 and 17 show examples of measured radar antenna patterns, in the 9 GHz band. The X axis represents the azimuth angle spanned on more than 36, and Y axis represents the power level received at each azimuth angle. This power pattern has to be normalized respectively to its maximum or an isotrope antenna to be considered as the antenna pattern or the directivity pattern. First analysis of such measured antenna patterns indicates that first side lobes appears near 3 dbc with a noticeable slope of side lobes leading to estimate roughly that a cos aperture illumination law was used. A theoretical mask floor at 6 dbc given by cos model would appear in this case a bit too low due to presence of backlobe and rear diffraction lobes in this antenna pattern, then if necessary, encouraging to use real antenna patterns instead of theoretical ones when possible.

21 Rec. ITU-R M FIGURE 16 Example measured antenna plot M FIGURE 17 Example measured antenna plot M Figures 18 and 19 show two other examples.

22 Rec. ITU-R M FIGURE 18 Measurement from AN/SPN-6 radar antenna at 3.6 GHz and 38 dbi 4 3 AN/SPN-6 frequency: 3 65 Mc Gain relative to isotropic (db) Degrees M Source: Statistical Characteristics of Gain and Mutual Gain of Radar Antennas, Project No. SF 1 4, Task 577, Department of the Navy 15 September, 1963 FIGURE 19 Doppler radar antenna pattern from meteorological radar with 5 db 1 st sidelobe and 6 db front-to-back ratio 1 Rlative magnitude (db) Azimuth (degrees) M

23 Rec. ITU-R M Source: USA Federal Meteorological Handbook No. 11, December 5 Part B, FCM-H11B-5 7 Patterns for phased array antennas The following equation could be used in the calculations for uniform linear array antenna normalized pattern: () where g: uniform linear array antenna normalized gain pattern f: Elementary radiating elements normalized gain pattern inserted in the uniform linear array antenna N: number of elementary radiating elements AF: uniform linear array antenna factor; ᴪ (radians): (1) with = (d/)(sin() - sin()) () where d: uniform elementary radiating element regular interspace λ: wavelength at the considered frequency : electronicaly beam steering angle off-axis angle. z FIGURE d x Radiating element No. 1 Radiating element No. N time d Radiating element No. 1 d Radiating element No. N M The specific nature of phased array antennas allows to steer electronically the mainlobe of the antenna pattern in a ±9 range from the mechanical antenna boresight. At large scan angles specific sidelobes effects in the antenna patterns should be taken into account as mainlobe significant

24 Rec. ITU-R M enlargement and desymmetrization (see Fig. ). In fact the main lobe maximum value decreases as cos() and further as the elementary radiating element pattern in the array. This result is a widened mainbeam, max gain losses, and consequently far sidelobes increase. For value of between ±6 and ±9 range from the mechanical antenna boresight the resulting pattern is so perturbed that it is no more usable (see Fig. 4). The practical values of are between and ±6 range from the mechanical antenna boresight. Furthermore, if the array lattice is bigger than / among the elementary radiating elements in the array, grating lobes of the mainlobe could appear for even less than ±6 range from the mechanical antenna boresight (see Fig. 3). And even if the array lattice is / among the elementary radiating elements in the array, sidelobes of the grating lobes of the mainlobe, situated at 9 and +9 from the mechanical antenna boresight, disturb the array pattern (see Fig. 4). FIGURE 1 Theoretical radiating pattern of an Uniform Linear Array of 3 radiating elements with a / lattice (blue curve) with a cosine radiating pattern (red curve) steered at boresight Normalized radiating pattern (db) M

25 Rec. ITU-R M FIGURE Theoretical radiating pattern of an Uniform Linear Array of 3 radiating elements with a / lattice (blue curve) with a cosine radiating pattern (red curve) steered at 6 Normalized radiating pattern (db) M FIGURE 3 Theoretical radiating pattern of an Uniform Linear Array of 3 radiating elements with a.6 lattice (blue curve) with a cosine radiating pattern (red curve) steered at 45 Normalized radiating pattern (db) M

26 4 Rec. ITU-R M FIGURE 4 Theoretical radiating pattern of an Uniform Linear Array of 3 radiating elements with a / lattice (blue curve) with a cosine radiating pattern (red curve) steered at 8 Normalized radiating pattern (db) M

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### Recommendation ITU-R SA (07/2017)

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### Frequency block arrangements for fixed wireless access systems in the range MHz

Recommendation ITU-R F.1488 (05/2000) Frequency block arrangements for fixed wireless access systems in the range 3 400-3 800 MHz F Series Fixed service ii Rec. ITU-R F.1488 Foreword The role of the Radiocommunication

### Water vapour: surface density and total columnar content

Recommendation ITU-R P.836-6 (12/2017) Water vapour: surface density and total columnar content P Series Radiowave propagation ii Rec. ITU-R P.836-6 Foreword The role of the Radiocommunication Sector is

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### , 16:9 progressively-captured image format for production and international programme exchange in the 50 Hz environment

Recommendation ITU-R BT.1847-1 (6/215) 1 28 72, 16:9 progressively-captured image format for production and international programme exchange in the 5 Hz environment BT Series Broadcasting service (television)

### Service requirements for digital sound broadcasting to vehicular, portable and fixed receivers using terrestrial transmitters in the VHF/UHF bands

Recommendation ITU-R BS.774-4 (06/2014) Service requirements for digital sound broadcasting to vehicular, portable and fixed receivers using terrestrial transmitters in the VHF/UHF bands BS Series Broadcasting

### Conversion of annual statistics to worst-month statistics

Recommendation ITU-R P.84-5 (09/206) Conversion of annual statistics to worst-month statistics P Series Radiowave propagation ii Rec. ITU-R P.84-5 Foreword The role of the Radiocommunication Sector is

### Recommendation ITU-R F (03/2012)

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### The use of diversity for voice-frequency telegraphy on HF radio circuits

Recommendation ITU-R F.106-2 (05/1999) The use of diversity for voice-frequency telegraphy on HF radio circuits F Series Fixed service ii Rec. ITU-R F.106-2 Foreword The role of the Radiocommunication

### Characteristics and protection criteria for radars operating in the aeronautical radionavigation service in the frequency band

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### Error performance and availability objectives and requirements for real point-to-point packet-based radio links

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### Interference criteria for meteorological aids operated in the MHz and MHz bands

Recommendation ITU-R RS.1263-1 (01/2010) Interference criteria for meteorological aids operated in the and 1 668.4-1 700 MHz bands RS Series Remote sensing systems ii Rec. ITU-R RS.1263-1 Foreword The

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### Interference mitigation techniques for use by high altitude platform stations in the GHz and GHz bands

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### The radio refractive index: its formula and refractivity data

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### Guide to the application of the propagation methods of Radiocommunication Study Group 3

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Recommendation ITU-R F.2005 (03/2012) Radio-frequency channel and block arrangements for fixed wireless systems operating in the 42 GHz (40.5 to 43.5 GHz) band F Series Fixed service ii Rec. ITU-R F.2005

### Bandwidths, signal-to-noise ratios and fading allowances in complete systems

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### User requirements for codecs for transmission of television signals through contribution, primary distribution, and SNG networks

Recommendation ITU-R BT.1868 (03/2010) User requirements for codecs for transmission of television signals through contribution, primary distribution, and SNG networks BT Series Broadcasting service (television)

### Broadcasting of multimedia and data applications for mobile reception by handheld receivers

Recommendation ITU-R BT.1833-3 (02/2014) Broadcasting of multimedia and data applications for mobile reception by handheld receivers BT Series Broadcasting service (television) ii Rec. ITU-R BT.1833-3

### Allowable short-term error performance for a satellite hypothetical reference digital path

Recommendation ITU-R S.2099-0 (12/2016) Allowable short-term error performance for a satellite hypothetical reference digital path S Series Fixed-satellite service ii Rec. ITU-R S.2099-0 Foreword The role

### International maritime VHF radiotelephone system with automatic facilities based on DSC signalling format

Recommendation ITU-R M.689-3 (03/2012) International maritime VHF radiotelephone system with automatic facilities based on DSC signalling format M Series Mobile, radiodetermination, amateur and related

### Recommendation ITU-R SF.1486 (05/2000)

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### Bandwidths, signal-to-noise ratios and fading allowances in HF fixed and land mobile radiocommunication systems

Recommendation ITU-R F.9-8 (02/2013) Bandwidths, signal-to-noise ratios and fading allowances in HF fixed and land mobile radiocommunication systems F Series Fixed service ii Rec. ITU-R F.9-8 Foreword

### Essential requirements for a spectrum monitoring system for developing countries

Recommendation ITU-R SM.1392-2 (02/2011) Essential requirements for a spectrum monitoring system for developing countries SM Series Spectrum management ii Rec. ITU-R SM.1392-2 Foreword The role of the

### Characteristics of systems operating in the amateur and amateur-satellite services for use in sharing studies

Recommendation ITU-R M.1732-2 (01/2017) Characteristics of systems operating in the amateur and amateur-satellite services for use in sharing studies M Series Mobile, radiodetermination, amateur and related

### Parameters for international exchange of multi-channel sound recordings with or without accompanying picture

Recommendation ITU-R BR.1384-2 (03/2011) Parameters for international exchange of multi-channel sound recordings with or without accompanying picture BR Series Recording for production, archival and play-out;

### Colour conversion from Recommendation ITU-R BT.709 to Recommendation ITU-R BT.2020

Recommendation ITU-R BT.2087-0 (10/2015) Colour conversion from Recommendation ITU-R BT.709 to Recommendation ITU-R BT.2020 BT Series Broadcasting service (television) ii Rec. ITU-R BT.2087-0 Foreword

### Technical characteristics for search and rescue radar transponders

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### Radio interface standards of vehicle-tovehicle and vehicle-to-infrastructure communications for Intelligent Transport System applications

Recommendation ITU-R M.2084-0 (09/2015) Radio interface standards of vehicle-tovehicle and vehicle-to-infrastructure communications for Intelligent Transport System applications M Series Mobile, radiodetermination,

### Recommendation ITU-R M (05/2011)

Recommendation ITU-R M.1652-1 (05/2011) Dynamic frequency selection in wireless access systems including radio local area networks for the purpose of protecting the radiodetermination service in the 5

### Guidelines for efficient use of the band GHz by the Earth explorationsatellite service (space-to-earth)

Recommendation ITU-R SA.1862 (01/2010) Guidelines for efficient use of the band 25.5-27.0 GHz by the Earth explorationsatellite service (space-to-earth) and space research service (space-to-earth) SA Series

### Morse telegraphy procedures in the maritime mobile service

Recommendation ITU-R M.1170-1 (03/2012) Morse telegraphy procedures in the maritime mobile service M Series Mobile, radiodetermination, amateur and related satellite services ii Rec. ITU-R M.1170-1 Foreword

### Preferred frequency bands for radio astronomical measurements

Recommendation ITU-R RA.314-10 (06/2003) Preferred frequency bands for radio astronomical measurements RA Series Radio astronomy ii Rec. ITU-R RA.314-10 Foreword The role of the Radiocommunication Sector

### Multi-dimensional signal mapping technique for satellite communications

Report ITU-R S.2306-0 (07/2014) Multi-dimensional signal mapping technique for satellite communications S Series Fixed satellite service ii Rep. ITU-R S.2306-0 Foreword The role of the Radiocommunication

### Recommendation ITU-R F.1571 (05/2002)

Recommendation ITU-R F.1571 (05/2002) Mitigation techniques for use in reducing the potential for interference between airborne stations in the radionavigation service and stations in the fixed service

### Power flux-density and e.i.r.p. levels potentially damaging to radio astronomy receivers

Report ITU-R RA.2188 (10/2010) Power flux-density and e.i.r.p. levels potentially damaging to radio astronomy receivers RA Series Radio astronomy ii Rep. ITU-R RA.2188 Foreword The role of the Radiocommunication

### Guidelines for narrow-band wireless home networking transceivers Specification of spectrum related components

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### Common application environment for interactive digital broadcasting services

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