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Professional Development Short Course On:
              Radar Systems Analysis & Design using MATLAB



                                         Instructor:

                               Dr. Andy Harrison



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Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 Consider a radar with an omni
  directional antenna. The peak power
  density (power per unit area) at any
                                                                                                 R
  point in space is

         Transmitted Power            watts 
    PD =                              2 
          Area of a Sphere            m 


 The power density at a given range
  away from the radar (assuming a
  lossless propagation medium) is

                   Pt                                      Pt = Peak Transmitted Power
             PD =
                  4π R 2                            4π R 2 = Area of a Sphere of Radius( R)

2
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 Radar systems utilize directional antennas in order to increase the
  power density in a certain direction.

 Directional antennas are usually characterized by the antenna gain and
  the antenna effective aperture, which are related by


                                                                               4π Ae
                                                                       G=
                                                                                  λ2
                                                θ                      G = Antenna Gain
                                                                       Ae = Effective Aperture (m 2 )
                                                                       λ = Wavelength (m)



3
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 The antenna gain is also related to the azimuth and elevation beam widths
  by
                                                   4π
                                        G=k                  k ≤1
                                                  θ eθ a
                          θ a = Azimuth Beamwidth (radians)
                          θ e = Elevation Beamwidth (radians)
 An accurate approximation for antenna gain is
                                    26000
                             G=                   → θ e θ a (degrees)
                                     θeθa
 The power density at a given range from the radar using a directive antenna
  is then given by

                                                  Pt G
                                            PD =
                                                 4π R 2

4
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Antenna Beamwidth



                                                                -3 dB from peak
                    Side Lobe Level
                                                                     θ = 8.75o




                                                                              Side Lobes




5
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 When the radiated energy from a radar impinges on a target, the
  induced surface currents on that target re-radiate or scatter
  electromagnetic energy in all directions.




                                                                           Transmitted Energy




              Scattered Energy



6
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 The amount of the scattered energy is proportional to the target size,
  orientation, physical shape, and material, which are all lumped
  together in one target-specific parameter called the Radar Cross
    Section (RCS) and is denoted by σ.

 The radar cross section is defined as the ratio of the power reflected
  back to the radar to the power density incident on the target.

                                            Pr
                                         σ=                 (m 2 )
                                            PD

                                   Pr = Reflected Power
                             PD = Incident Power Density


7
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Signature Modeling Techniques

 There are different ways to solve the EM scattering problem. The
  method used depends on variables such as
     Desired accuracy and appropriateness of the method
     Number of CPUs and amount of RAM available
     Amount of wall time available

 The different methods are often grouped together into what are
  called “low” and “high” frequency computational techniques.

 “Low frequency” methods solve the scattering problem in an
  “exact” sense, incorporating all electromagnetic effects. They are
  often limited by problem size or computer power.

 “High Frequency” methods approximate the scattering problem.
  They are generally less accurate, though often much faster and less
  demanding on CPU resources than a low frequency method.

8
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Finite Difference Time Domain (FDTD)

 FDTD solves the time-domain version of Maxwell’s Equations. FDTD
  is considered to be a “full wave” or “exact” method.
 For scattering problems, the target and some adjoining space must
  be discretized on a rectangular grid to support the electromagnetic
  fields.
 FDTD has high memory and CPU demands, and is not typically
  used for targets that are electrically large.




3D Maxwell Equations                          FDTD Grid                                  Time-Domain Field
9
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Method of Moments (MoM)

 The MoM is used in the scattering problem in the frequency-domain
  by solving the integral Maxwell’s Equations. It is considered to be a
  “full wave” or “exact” method.
 The MoM solves for the induced surface current on an object by
  discretizing the target surface into a number of subdomain “basis
  functions”. The number of basis functions (N) needed is
  proportional to the radar wavelength.
 The MoM creates a matrix equation of order N2. This ultimately limits
  its application to problems of small electrical size.




                                                                                  Surface Currents


10
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Method of Moments (MoM)

 The Fast Multipole Method (FMM/MLFMA) can be used to reduce the
  complexity of the MoM matrix system and allow the MoM to be used
  in problems previously unsolvable.
 Reliable, industrial-strength FMM software is slowly becoming
  commercially available. These codes still require supercomputer-
  class hardware to solve very large MoM problems.

                                                                                               81920 unknowns

                                                                                                Regular MoM:

                                                                                                 50 GB RAM

                                                                                                    FMM:

                                                                                                 1.2 GB RAM

                                                                                               A factor of 40/1 !
 8 Wavelength Sphere
11
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Approximate (High-Frequency) Methods

 “High Frequency” methods approach the scattering problem by assuming
  the target is very large compared to the radar wavelength.

 Approximations can then be made to simplify and expedite signature
  prediction by considering different scattering mechanisms separately.

 Physical Optics (PO) is a commonly used method that approximates the
  surface currents by assuming the surface is locally flat. This method does
  well at speculars but is fair to poor at off-specular angles.

 The Physical Theory of Diffraction (PTD) supplements PO by accounting for
  single diffractions from edges.

 Shooting and Bouncing Rays (SBR) supplements PO and PTD by using ray-
  optics to model multiple-bounce scattering.

 Other scattering mechanisms (traveling and creeping waves) can be
  considered, but are difficult to formulate analytically for general targets.



12
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Comparison of MoM and Approximate Methods

 Target: 16 x 8 inch “Top Hat”

 Monostatic RCS Calculated at 7
  GHz

 Target approximately 10 x 5
  wavelengths in electrical size.                                                         “Top Hat”




         PO Only                                   PO + PTD                                   PO + PTD + SBR

13
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Signature Modeling Software

 Finite Difference Time Domain (FDTD)
      Remcom xfdtd (packaging, machines)
      EMS+ EZ-FDTD (packaging, machines)

 Finite Element Method (FEM)
      Ansys, Inc. ANSYS (packaging, machines)

 Method of Moments (MoM)
      EM Software and Systems FEKO (wire and planar antennas)
      Tripoint Industries Serenity (Scattering by MoM/MLFMA)
      Tripoint Industries Galaxy (Scattering by MoM-BoR)

 Physical Optics / PTD / SBR
      Tripoint Industries lucernhammer MT (High-frequency scattering)
      SAIC xPatch (High-frequency scattering)



14
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 Typical RCS values of various objects at X-Band.

                    Object                                     RCS (m2)                   RCS (dBsm)
                 Pickup Truck                                       200                          23
                  Automobile                                        100                          20
                  Jumbo Jet                                         100                          20
                 Commercial                                          40                          16
              Cabin Cruiser Boat                                     10                          10
             Large Fighter Aircraft                                   6                          7.78
             Small Fighter Aircraft                                   2                           3
                  Adult Male                                          1                           0
          Conventional Winged Missile                                0.5                          -3
                       Bird                                         0.01                         -20
                     Insect                                       1x10-5                         -50
           Advanced Tactical Fighter                              1x10-6                         -60



15
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 The power scattered by the target is then
                                             Pt Gσ
                                 Ptarget   =                  (watts )
                                             4π R 2

 Recalling that the power density at a given range is

                                                      Pt
                                            PD =
                                                     4π R 2
 Substituting the target’s scattered power gives the total power
  density delivered back to the antenna.

                                                         Pt Gσ
                                      Pantenna =
                                                      (4π R )   2 2




16
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 Multiplying by the effective area of the antenna gives the total power
  received by the radar.

                                                     Pt Gσ
                                    Pradar =                        Ae
                                                  (4π R )   2 2



 Substituting for the effective aperture then gives


                                          Pt G 2 λ2σ
                              Pradar    =                        ( watts )
                                          (4π )3 R 4



17
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation


 Denote the minimum detectable signal power by Smin.

 It follows that

                                                     Pt G 2 λ2σ
                                           S min   =
                                                     (4π )3 R 4

 Rearranging the equation gives the maximum Range to the target.


                                                                14
                                     Pt G λ σ    2    2
                          Rmax     =
                                     (4 π )3 S 
                                                                        (meters)
                                              min 




18
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 In realistic situations, the returned signals received by the radar will
  be corrupted by noise.

 Noise is random in nature and may be described by its Power
  Spectral Density (PSD).


                                              PSD = k Ts
                                                                                      Joules 
           k = Boltzmann's Constant = 1.38 ×10 − 23                                          
                                                                                      Kelvin 
               Ts = System Noise Temperature                                    (Kelvin)


19
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 The input noise power for a radar of a given operating bandwidth is then
  given as

                              N i = PSD × B = k Ts × B
 It is always desirable that the minimum detectable signal power be greater
  than the noise power.

 The fidelity of a radar receiver is normally described by a figure of merit
  called the noise figure which is defined as

                                         SNRi   Si N i
                                      F=      =
                                         SNRo S o N o

                    SNRi = Input Signal to Noise Ratio
                    SNRo = Output Signal to Noise Ratio

20
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 The input signal power may be rewritten as

                                    Si = k Ts B F SNRo
 The minimum detectable signal is then

                             S min = k Ts B F (SNRo )min

 Setting the detection threshold to the minimum output SNR results
  in

                                                                                   14
                                      Pt G λ σ         
                                                        2    2
                  Rmax      =                          
                              (4 π ) k T B F (SNR ) 
                                     3
                                        s        o min 




21
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


The Radar Equation

 Rearranging gives the SNR at the output of the receiver.

                                         Pt G 2 λ2 σ
                              SNRo =
                                     (4 π )3 k Ts B F R 4

 In general, radar losses, L, reduce the overall SNR and thus


                                         Pt G 2 λ2 σ
                            SNRo =
                                   (4 π )3 k Ts B F L R 4




22
Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute


Radar Reference Range




     SNR vs Detection Range for Varying                             SNR vs Detection Range for Varying
                Peak Power                                                     Target RCS




23
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     Radar Systems Analysis & Design using MATLAB




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ATI's Radar Systems Analysis & Design using MATLAB Technical Training Short Course Sampler

  • 1. Professional Development Short Course On: Radar Systems Analysis & Design using MATLAB Instructor: Dr. Andy Harrison http://www.ATIcourses.com/schedule.htm ATI Course Schedule: http://www.aticourses.com/radar_systems_analysis.htm ATI's Radar Systems Analysis & Design:
  • 2. www.ATIcourses.com Boost Your Skills 349 Berkshire Drive Riva, Maryland 21140 with On-Site Courses Telephone 1-888-501-2100 / (410) 965-8805 Tailored to Your Needs Fax (410) 956-5785 Email: ATI@ATIcourses.com The Applied Technology Institute specializes in training programs for technical professionals. Our courses keep you current in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highly competitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presented on-site training at the major Navy, Air Force and NASA centers, and for a large number of contractors. Our training increases effectiveness and productivity. Learn from the proven best. For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.asp For Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm
  • 3. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Consider a radar with an omni directional antenna. The peak power density (power per unit area) at any R point in space is Transmitted Power  watts  PD =  2  Area of a Sphere  m   The power density at a given range away from the radar (assuming a lossless propagation medium) is Pt Pt = Peak Transmitted Power PD = 4π R 2 4π R 2 = Area of a Sphere of Radius( R) 2
  • 4. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Radar systems utilize directional antennas in order to increase the power density in a certain direction.  Directional antennas are usually characterized by the antenna gain and the antenna effective aperture, which are related by 4π Ae G= λ2 θ G = Antenna Gain Ae = Effective Aperture (m 2 ) λ = Wavelength (m) 3
  • 5. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  The antenna gain is also related to the azimuth and elevation beam widths by 4π G=k k ≤1 θ eθ a θ a = Azimuth Beamwidth (radians) θ e = Elevation Beamwidth (radians)  An accurate approximation for antenna gain is 26000 G= → θ e θ a (degrees) θeθa  The power density at a given range from the radar using a directive antenna is then given by Pt G PD = 4π R 2 4
  • 6. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Antenna Beamwidth -3 dB from peak Side Lobe Level θ = 8.75o Side Lobes 5
  • 7. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  When the radiated energy from a radar impinges on a target, the induced surface currents on that target re-radiate or scatter electromagnetic energy in all directions. Transmitted Energy Scattered Energy 6
  • 8. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  The amount of the scattered energy is proportional to the target size, orientation, physical shape, and material, which are all lumped together in one target-specific parameter called the Radar Cross Section (RCS) and is denoted by σ.  The radar cross section is defined as the ratio of the power reflected back to the radar to the power density incident on the target. Pr σ= (m 2 ) PD Pr = Reflected Power PD = Incident Power Density 7
  • 9. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Signature Modeling Techniques  There are different ways to solve the EM scattering problem. The method used depends on variables such as  Desired accuracy and appropriateness of the method  Number of CPUs and amount of RAM available  Amount of wall time available  The different methods are often grouped together into what are called “low” and “high” frequency computational techniques.  “Low frequency” methods solve the scattering problem in an “exact” sense, incorporating all electromagnetic effects. They are often limited by problem size or computer power.  “High Frequency” methods approximate the scattering problem. They are generally less accurate, though often much faster and less demanding on CPU resources than a low frequency method. 8
  • 10. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Finite Difference Time Domain (FDTD)  FDTD solves the time-domain version of Maxwell’s Equations. FDTD is considered to be a “full wave” or “exact” method.  For scattering problems, the target and some adjoining space must be discretized on a rectangular grid to support the electromagnetic fields.  FDTD has high memory and CPU demands, and is not typically used for targets that are electrically large. 3D Maxwell Equations FDTD Grid Time-Domain Field 9
  • 11. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Method of Moments (MoM)  The MoM is used in the scattering problem in the frequency-domain by solving the integral Maxwell’s Equations. It is considered to be a “full wave” or “exact” method.  The MoM solves for the induced surface current on an object by discretizing the target surface into a number of subdomain “basis functions”. The number of basis functions (N) needed is proportional to the radar wavelength.  The MoM creates a matrix equation of order N2. This ultimately limits its application to problems of small electrical size. Surface Currents 10
  • 12. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Method of Moments (MoM)  The Fast Multipole Method (FMM/MLFMA) can be used to reduce the complexity of the MoM matrix system and allow the MoM to be used in problems previously unsolvable.  Reliable, industrial-strength FMM software is slowly becoming commercially available. These codes still require supercomputer- class hardware to solve very large MoM problems. 81920 unknowns Regular MoM: 50 GB RAM FMM: 1.2 GB RAM A factor of 40/1 ! 8 Wavelength Sphere 11
  • 13. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Approximate (High-Frequency) Methods  “High Frequency” methods approach the scattering problem by assuming the target is very large compared to the radar wavelength.  Approximations can then be made to simplify and expedite signature prediction by considering different scattering mechanisms separately.  Physical Optics (PO) is a commonly used method that approximates the surface currents by assuming the surface is locally flat. This method does well at speculars but is fair to poor at off-specular angles.  The Physical Theory of Diffraction (PTD) supplements PO by accounting for single diffractions from edges.  Shooting and Bouncing Rays (SBR) supplements PO and PTD by using ray- optics to model multiple-bounce scattering.  Other scattering mechanisms (traveling and creeping waves) can be considered, but are difficult to formulate analytically for general targets. 12
  • 14. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Comparison of MoM and Approximate Methods  Target: 16 x 8 inch “Top Hat”  Monostatic RCS Calculated at 7 GHz  Target approximately 10 x 5 wavelengths in electrical size. “Top Hat” PO Only PO + PTD PO + PTD + SBR 13
  • 15. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Signature Modeling Software  Finite Difference Time Domain (FDTD)  Remcom xfdtd (packaging, machines)  EMS+ EZ-FDTD (packaging, machines)  Finite Element Method (FEM)  Ansys, Inc. ANSYS (packaging, machines)  Method of Moments (MoM)  EM Software and Systems FEKO (wire and planar antennas)  Tripoint Industries Serenity (Scattering by MoM/MLFMA)  Tripoint Industries Galaxy (Scattering by MoM-BoR)  Physical Optics / PTD / SBR  Tripoint Industries lucernhammer MT (High-frequency scattering)  SAIC xPatch (High-frequency scattering) 14
  • 16. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Typical RCS values of various objects at X-Band. Object RCS (m2) RCS (dBsm) Pickup Truck 200 23 Automobile 100 20 Jumbo Jet 100 20 Commercial 40 16 Cabin Cruiser Boat 10 10 Large Fighter Aircraft 6 7.78 Small Fighter Aircraft 2 3 Adult Male 1 0 Conventional Winged Missile 0.5 -3 Bird 0.01 -20 Insect 1x10-5 -50 Advanced Tactical Fighter 1x10-6 -60 15
  • 17. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  The power scattered by the target is then Pt Gσ Ptarget = (watts ) 4π R 2  Recalling that the power density at a given range is Pt PD = 4π R 2  Substituting the target’s scattered power gives the total power density delivered back to the antenna. Pt Gσ Pantenna = (4π R ) 2 2 16
  • 18. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Multiplying by the effective area of the antenna gives the total power received by the radar. Pt Gσ Pradar = Ae (4π R ) 2 2  Substituting for the effective aperture then gives Pt G 2 λ2σ Pradar = ( watts ) (4π )3 R 4 17
  • 19. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Denote the minimum detectable signal power by Smin.  It follows that Pt G 2 λ2σ S min = (4π )3 R 4  Rearranging the equation gives the maximum Range to the target. 14  Pt G λ σ  2 2 Rmax =  (4 π )3 S   (meters)  min  18
  • 20. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  In realistic situations, the returned signals received by the radar will be corrupted by noise.  Noise is random in nature and may be described by its Power Spectral Density (PSD). PSD = k Ts  Joules  k = Boltzmann's Constant = 1.38 ×10 − 23    Kelvin  Ts = System Noise Temperature (Kelvin) 19
  • 21. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  The input noise power for a radar of a given operating bandwidth is then given as N i = PSD × B = k Ts × B  It is always desirable that the minimum detectable signal power be greater than the noise power.  The fidelity of a radar receiver is normally described by a figure of merit called the noise figure which is defined as SNRi Si N i F= = SNRo S o N o SNRi = Input Signal to Noise Ratio SNRo = Output Signal to Noise Ratio 20
  • 22. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  The input signal power may be rewritten as Si = k Ts B F SNRo  The minimum detectable signal is then S min = k Ts B F (SNRo )min  Setting the detection threshold to the minimum output SNR results in 14  Pt G λ σ  2 2 Rmax =   (4 π ) k T B F (SNR )  3  s o min  21
  • 23. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute The Radar Equation  Rearranging gives the SNR at the output of the receiver. Pt G 2 λ2 σ SNRo = (4 π )3 k Ts B F R 4  In general, radar losses, L, reduce the overall SNR and thus Pt G 2 λ2 σ SNRo = (4 π )3 k Ts B F L R 4 22
  • 24. Radar Systems Analysis and Design Using MATLAB ♦ Applied Technology Institute Radar Reference Range SNR vs Detection Range for Varying SNR vs Detection Range for Varying Peak Power Target RCS 23
  • 25. To learn more please attend ATI course Radar Systems Analysis & Design using MATLAB Please post your comments and questions to our blog: http://www.aticourses.com/blog/ Sign-up for ATI's monthly Course Schedule Updates : http://www.aticourses.com/email_signup_page.html