AMTA Paper Archive


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Implementation of a Burst-Mode Technique and Variable Coherent Integration to Minimize Radar Data Collection Time
Christopher Fry,Charles Walters, John Raber, November 2013
Abstract— Compact ranges are ideal settings for collecting low-RCS measurement data at high pulse rates. However, until recently, two operating constraints have limited the efficiency of instrumentation radar systems in this setting: (1) system delays limiting Pulse Repetition Frequency (PRF) and (2) fixed integration across frequency resulting in more time spent on certain frequencies than required. In this paper, we demonstrate the capability to significantly increase data throughput by using a Burst-Mode to increase the usable PRF and a frequency table editing mode to vary integration levels across the frequency bandwidth. A major factor in the choice of PRF for a specific application is system hardware delays. We describe the use of a Burst-Mode of operation in the MkVe Radar to reduce delays caused by physical layout of the instrumentation hardware. Burst-Mode essentially removes setup time in the system, reducing the time between pulses to the roundtrip time of flight from the antenna to the target. Most pulsed-IF instrumentation radar users fix the coherent integration level for the entire measurement waveform, even though the set level of integration may not be required at all frequencies to achieve the desired sensitivity. We describe the use of a frequency table Parameter Editor Mode in the MkVe that allows the integration level to vary for each step in the waveform. We demonstrate the use of both methods to reduce data collection time by a factor of seven using a MkVe Radar installed in a compact range.
Radar Cross Sections in the Shorter Millimeter-wave Region:  Characterization and Calculation for Targets that include Rough Surfaces
Mark Patrick,Dane J. Phillips, Daniel L. Faircloth, Frank C. De Lucia, November 2013
Abstract— Measurements and strategies for the calculation of radar cross-sections in the shorter millimeter wave region, especially of objects that include rough surfaces, are discussed. Because of decreasing wavelength, roughness becomes more significant in this spectral region, but also more difficult to characterize. A tabletop radar cross-section measurement system was set up to measure scattering from canonical objects and rough objects with regular or random patterns using a swept frequency continuous wave system. Random, rough objects of different surface roughnesses were measured and fit to statistical distributions governed by optical speckle theory. In this paper we consider the inclusion of optical speckle theory in the electromagnetic codes to address both issues associated with the characterization of target surfaces and the time required for numerical calculations.
A 100 GHz Polarimetric Compact Radar Range for Scale-Model Radar Cross Section Measurements
Guy DeMartinis,Michael Coulombe, Thomas Horgan, Brian Soper, Jason Dickinson, Robert Giles, William Nixon, November 2013
Abstract— A fully polarimetric compact radar range operating at a center frequency of 100 GHz has been developed for obtaining radar cross section, inverse synthetic aperture radar imagery and high range resolution profiles on targets and structures of interest. The 100 GHz radar range provides scale-model RCS measurements for a variety of convenient scale factors including W-Band (1:1 scale), C-band (1:16 scale), and S-band (1:26 scale). An overview of the radar range is provided in this paper along with measurement examples of ISAR scale-model imaging, scale-model through-wall imaging, and preliminary kHz sweep-rate Doppler that demonstrate a few of the diverse and unique applications for this system. The 100 GHz transceiver consists of a fast-switching, stepped, CW microwave synthesizer driving dual-transmit and dual-receive frequency multiplier chains. The stepped resolution of the system’s frequency sweep is sufficient for unambiguous resolution of the entire chamber. The compact range reflector is a CNC machined aluminum reflector edge-treated with FIRAM™-160 absorber serrations and fed from the side to produce a clean quiet zone. This range is the latest addition to a suite of compact radar ranges developed by the Submillimeter-Wave Technology Laboratory providing scale-model radar measurements at nearly all of the common radar bands.
Transformation of Monostatic Near-Field Scattering Data By Fast Irregular Field Transformation Algorithms
Georg Schnattinger,Raimund Mauermayer, Thomas Eibert, November 2013
Abstract—It is well-known that a complete bistatic set of near-.eld scattering data is required to compute far-.eld radar cross section (RCS) quantities. In many practical applications, however, only monostatic scattering data is available. Almost all algorithms for the transformation of monostatic near-.eld data are based on the synthetic aperture radar (SAR) image representation.Since these algorithms are often acceleratedbythe fastFouriertransform(FFT),they usuallypose manylimitations on the measurement procedure such as regularly spaced grids and separate treatment of the different polarizations due to scalar processing. In this paper, a novel and .exible algorithm is presented which is not based on the FFT but on multi-level fast multipole method (MLFMM) principles. Therefore, it is similar to the fast irregular antenna .eld transformation algorithm (FIAFTA) which has been designed for the transformation of antenna .elds and measurements. Numerical results of different scenarios show that these principles can also be successfully applied to monostatic scattering data. In summary, this approach is superior to existing algorithms, because it provides more .exibility while it is still very ef.cient.
Detailed diagnostics of the BIOMASS Feed Array Prototype
Cecilia Cappellin,Sergey Pivnenko, Knud Pontoppidan, November 2013
Abstract—The 3D reconstruction algorithm of DIATOOL is applied to the prototype feed array of the BIOMASS synthetic aperture radar, recently measured at the DTU-ESA Spherical Near-Field Antenna Test Facility in Denmark. Careful analysis of the measured feed array data had shown that the test support frame of the array had a significant influence on the measured feed pattern. The 3D reconstruction and further post-processing is therefore applied both to the feed array measured data, and a set of simulated data generated by the GRASP software which replicate the series of measurements. The results of the diagnostics and the corresponding improvement of the feed array field obtained by removal of the undesired effect of the frame are presented and discussed.
Surface Electromagnetic Wave Characterization Using Non-invasive Photonic Electric Field Sensors
James Toney,Vincent E. Stenger, Peter Pontius, Andrea Pollick, Sri Sriram, Chi-Chih Chen, November 2013
Abstract— Electromagnetic properties of aircraft and missile skins have a large effect on radar cross sections and determine the level of stealth that is achieved over the various RF bands currently in use. RF absorption, reflection, and propagation along the skin surface all serve as important measures of the electromagnetic performance of the coated surfaces. Non-invasive probing of the electromagnetic field just above the propagating wave at multiple spots along the propagation direction can be used to determine and measure wave propagation parameters, including effective RF index, loss per length, wave impedance, and frequency dependent material properties of the coatings. Wide-band photonic electric field sensors have been demonstrated for probing of dielectric layers by measuring the traveling waves along the coated aircraft surface. The photonic E-field sensors are extremely linear and produce an exact real time analog RF representation of the electric field, including phase information. These ultra-wideband (UWB) photonic RF sensors are very small and contain negligible metal content, allowing them to be placed at close proximity without perturbing the RF surface waves. This is very important in accurately characterizing highly damped surface waves on absorber layers. This paper discusses the linearity, bandwidth, polarization, and sensitivity of the unique UWB photonic E-field sensor design. Experimental results are presented on surface-wave characterization measurements using these sensors.
Development of a Precision Model Positioning System for a Multi-Use Electromagnetic Test Facility at NASA Langley Research Center
Alex Deyhim,Eric Acome, Eric Van Every, Joe Kulesza, Richard Jane, November 2014
This paper describes the mechanical design, control instrumentation and software for a precision model positioning system developed for use in the Experimental Test Range (ETR) electromagnetic test facility at NASA Langley Research Center. ADC has a contract to design, build, and install major components for an updated indoor antenna characterization and scattering measurement range at NASA Langley Research Center. State-of-the-art electromagnetic systems are driving a demand to increase the precision and repeatability of electromagnetic test ranges. Sophisticated motion control systems can help meet these demands by providing electromagnetic test engineers with a level of positioning fidelity and testing speed not possible with previous generation technology. The positioning system designed for the Experimental Test Range at NASA Langley Reseach Center consists of a rail positioning system and four rail positioning carriages: an antenna measurement positioner, scattering and RCS measurement pylon, an azimuth rotator to support foam columns, and an electric personnel lift for test article access. A switching station allows for rail positioning carriages to be quickly moved on and off of the rail system. Within the test chamber there is also a string reel positioning system capable of positioning test articles within a 40’ x 40’ x 25’ volume. Total length of the rail system is 112’ with laser position encoding for the final section of the rail system. Linear guide rails are used to support the carriages and each carriage is position with a rack and pinion drive. Rails mount to steel weldments that are supported with 8” diameter feet. Capacity of the rail system is 7,300 lbs. A switching station allows for positioning components to be moved off of and onto the rail system independently and a place to dock positioning components when they are not in use. A curved linear guide rail supports the switching station so that the platform can be rotated manually. Hardened tapered pins are used to align the switching station with mating rail segments. The scattering and radar cross section (RCS) measurement pylon is a 4:1 ratio ogive shape and has a 3,000 lb load capacity. A pitch rotator tip or spline driven azimuth tip can be mounted to the pylon. The spline drive shaft can be removed to allow for the pitch tip to be mounted to the end of the pylon. Total height of the pylon is 18’ from the floor to the pitch positioner mounting plate. Keywords: RCS, Scattering, Pylon, Positioner, Antenna Design, Rotator, Instrumentation, electromagnetic, Radio Frequency, Radar
Advances in Instrumentation and Positioners for Millimeter-Wave Antenna Measurements
Bert Schluper,Patrick Pelland, November 2014
Applications using millimeter-wave antennas have taken off in recent years. Examples include wireless HDTV, automotive radar, imaging and space communications. NSI has delivered dozens of antenna measurement systems operating at mm-wave frequencies. These systems are capable of measuring a wide variety of antenna types, including antennas with waveguide inputs, coaxial inputs and wafer antennas that require a probing station. The NSI systems are all based on standard mm-wave modules from vendors such as OML, Rohde & Schwarz and Virginia Diodes. This paper will present considerations for implementation of these systems, including providing the correct RF and LO power levels, the impact of harmonics, and interoperability with coaxial solutions. It will also investigate mechanical aspects such as application of waveguide rotary joints, size and weight reduction, and scanner geometries for spherical near-field and far-field measurements. The paper will also compare the performance of the various mm-wave solutions. Radiation patterns acquired using some of these near-field test systems will be shared, along with some of the challenges encountered when performing mm-wave measurements in the near-field.
Nearfield RCS Measurements of Full ScaleTargets Using ISAR
Christer Larsson, November 2014
Near field Radar Cross Section (RCS) measurements and Inverse Synthetic Aperture Radar (ISAR) are used in this study to obtain geometrically correct images and far field RCS. The methods and the developed algorithms required for the imaging and the RCS extraction are described and evaluated in terms of performance in this paper. Most of the RCS measurements on full scale objects that are performed at our measurement ranges are set up at distances shorter than those given by the far field criterion. The reasons for this are e.g., constraints in terms of budget, available equipment and ranges but also technical considerations such as maximizing the signal to noise in the measurements. The calibrated near-field data can often be used as recorded for diagnostic measurements. However, in many cases the far field RCS is also required. Data processing is then needed to transform the near field data to far field RCS in those cases. A straightforward way to image the RCS data recorded in the near field is to use the backprojection algorithm. The amplitudes and locations for the scatterers are then determined in a pixel by pixel imaging process. The most complicated part of the processing is due to the near field geometry of the measurement. This is the correction that is required to give the correct incidence angles in all parts of the imaged area. This correction has to be applied on a pixel by pixel basis taking care to weigh the samples correctly. The images obtained show the geometrically correct locations of the target scatterers with exceptions for some target features e.g., when there is multiple or resonance scattering. Separate features in the images can be gated and an inverse processing step can be performed to obtain the far field RCS of the full target or selected parts of the target, as a function of angle and frequency. Examples of images and far field RCS extracted from measurements on full scale targets using the ISAR processing techniques described in this paper will be given.
Measurement of Operational Orientations Using Coordinate Transforms and Polarization Rotations
Douglas Morgan, November 2014
Antenna and Radar Cross Section (RCS) measurements are often required for orientation sets (cuts) that are difficult or impossible to produce with the positioning instrumentation available in a given lab.  This paper describes a general coordinate transform, combined with a general polarization rotation to correct for these orientation differences.  The technique is general, and three specific examples from actual test programs are provided.  The first is for an RCS measurement of a component mounted in a flat-top test fixture.  The component is designed to be mounted in a platform at an orientation not feasible for the flat-top fixture, and the test matrix calls for conic angle cuts of the platform.  The transforms result in a coordinated, simultaneous two-axis motion profile and corresponding polarization rotations yielding the same information as if the component had been mounted in the actual platform.  The second example is for a pattern measurement of an antenna suite mounted on a cylindrical platform (such as a projectile).  In this case, the test matrix calls for a roll-cut, but the range positioning system does not include a roll positioner.  The transforms again result in a coordinated, simultaneous two-axis motion profile and corresponding polarization rotations to provide the same information as the required roll-cut but without the use of a roll positioner.  Finally, the third example is for an antenna pattern measurement consisting of an extremely large number of cuts consisting of conic yaw cuts, roll cuts and pitch cuts.  The chosen method involves the use of the Boeing string suspension system to produce great-circle cuts at various pitch angles combined with the use of the coordinate and polarization transforms to emulate, off-line, any arbitrary cut over any axis or even multiple axes. Keywords:  Algorithm, Positioning, Polarization, Coordinates, RCS
Application of Huygens' Principle to a Dual Frequency Constant Beamwidth Reflector Operating in the Focused Near-Field
Herbert Aumann,Nuri Emanetoglu, November 2014
A technique is presented for determining the pattern of an antenna in the focused near-field from cylindrical near-field measurements. Although the same objective could be achieved by conventional near-field to far-field transformation followed by a back projection, the proposed technique has an intuitive appeal and is considerably simpler and faster. The focused near-field antenna pattern is obtained by applying Huygens’ principle, as embodied in the field equivalent principle, directly to near-field measurements and by including an “obliquity factor” to suppress backlobe radiation.  The technique was experimentally verified by comparison with far-field patterns obtained by conventional cylindrical near-field to far-field transformation and by EM simulations. Excellent agreement in sidelobe levels and beamwidth was achieved.  The technique was applied to the 25 in diameter reflector antenna of a harmonic radar operating at 5.8 GHz and 11.6 GHz. Since the operating range of this radar is less than 40 ft, the reflector is the near-field at both frequencies. By defocusing the reflector at the harmonic frequency the beamwidths and gains at both frequencies can be made the same. The defocusing is accomplished by exploiting the frequency dependent phase center displacement of a log-periodic feed.
Reflector Panel Gap Analysis for the U. S. Army’s Electronic Proving Ground Outdoor Range
Jeffrey Bean,Michael Hutsel, Stewart Skiles, Eric Kuster, Michael Brinkmann, Anthony Sanchez, November 2014
The outdoor range at the U. S. Army’s Electronic Proving Ground Antenna Test Facility features a large reflector in order to facilitate radar cross-section and antenna performance evaluation with large targets. Constructed during the late 1980s and early 1990s, this range features a 67-foot diameter reflector to satisfy quiet zone size specifications. The reflector is composed of 138 individual panels with nominal panel separation of 0.06 inches. This research investigates the impact of these gaps between reflector panels on the field received at the quiet zone. GTRI’s physical optics computational code was used to analyze the existing range design at the frequencies of interest, from C- through Ka-band, taking into account edge diffraction from the panels. In research presented at AMTA 2013, a range modification of the ground between the range source antenna and the reflector was performed to minimize ground reflections. This range modification has been incorporated with current research to provide quiet zone field analysis which includes reflector gaps as well as ground reflections.
"RF DNA" Fingerprinting for Non-Destructive Antenna Acceptance Testing
Mathew Lukacs,Peter Collins, Michael Temple, November 2014
Abstract- Quality control is critical for all industrial processes, but often times defect detection is labor intensive. A novel approach to industrial defect detection is to use a random noise radar (RNR), coupled with Radio Frequency "Distinctive Native Attributes (RF-DNA)" fingerprinting processing algorithms to non-destructively interrogate microwave devices and classify defective units from properly functioning units.  Example applications include assembly line inspection of automotive collision avoidance systems, wireless or cellular antenna defect detection during manufacture, and phased array element defect detection prior to RF system assembly. The RNR is uniquely suitable since it uses an Ultra Wideband noise waveform as an active interrogation method that will not cause destructive damage to microwave components. Additionally, it has been demonstrated that multiple RNRs can operate simultaneously in close proximity, allowing for significant parallelization of defect detection systems resulting in increased process throughput. Using this method, 100% sampling for quality control may be attainable in many cases. RF-DNA has previously demonstrated “serial number” discrimination of Orthogonal Frequency Division Multiplexed (OFDM), Direct Sequence Spread Spectrum (DSSS) network signals, GSM, WiMAX signals and others with classification accuracies above 80% using Multiple Discriminant Analysis and Generalized Relevance Learning Vector Quantification classification algorithms. Those cases all involved discrimination of passive emissions. This approach proposes to couple the classification successes of the RF-DNA fingerprinting with a non-destructive active interrogation waveform.
Modelling and Simulation of a Resistance Loaded Bow Tie Antenna
Matthew Galdeen,Peter Collins, November 2014
The modelling and simulation of a modified bow tie antenna optimized for radar cross section measurement is described.  The bow tie antenna shows improved transient response for radiating Ultra Wide Band pulses with decreased late time ringing.  In applications such as radar cross section measurement, late time ringing caused by reflections at the open ends can mask objects of interest in close proximity.  The antenna reduces reflections by resistive loading based on works by Lestari and Wu-King.  Full wave modelling and simulation is done using CST Microwave Studio.  S-Parameter and VSWR optimization by modification of the conductivity profile is demonstrated.  Experimental verification of the model has been carried out and confirms both the properties of the antenna and the simulation.
Polarimetric Weather Radar Antenna Calibration Using Solar Scans
Richard Ice,Adam Heck, Jeffrey Cunningham, Walter Zittel, Robert Lee, November 2014
The US NEXRAD weather surveillance Doppler radar (WSR-88D) was recently upgraded to polarimetric capability.  This upgrade permits identification of precipitation characteristics and type, thus providing the potential to significantly enhance the accuracy of radar estimated rainfall, or water equivalent in the case of frozen hydrometeors.  However, optimal benefits are only achieved if errors induced by the radar hardware are properly accounted for through calibration.  Hardware calibration is a critical element in delivering accurate meteorological information to the forecast and warning community.  The calibration process must precisely measure the gain of the antenna, the Polarimetric bias of the antenna, and the overall gain and bias of the receive path.  The absolute power measurement must be accurate to within 1 dB and the bias between the Polarimetric channels must be known to within 0.1 dB.  These requirements drive a need for precise measurement of antenna characteristics. Engineers and scientists with the NEXRAD program employ solar scanning techniques to ascertain the absolute gain and bias of the 8.53 m parabolic center fed reflector antenna enclosed within a radome.  They are also implementing use of daily serendipitous interference strobes from the sun to monitor system calibration.  The sun is also used to adjust antenna gain and pedestal pointing accuracy.  This paper reviews the methods in place and under development and identifies some of the challenges in achieving the necessary calibration accuracies.
Electromagnetic Scattering Analysis of Possible Targets for Orbital Debris Remediation
Russell Vela,James Park, Brian Kent, Anthony Griffith, Rebecca Johanning, November 2014
After decades of international launches and varying space expeditions the low Earth orbit (LEO) has become littered with man-made objects and debris. With over 22,000 objects larger than a softball, and hundreds of thousands in smaller size existing, remediation efforts must take place to ensure the continuation of both collision free space flight and orbits. While smaller objects are difficult to track, and would consume more resources, the larger bodied debris offer a means to collect greater volumes of orbital debris clutter with less operations. In an effort to assist in the architectural design of microwave remote sensors, for the detection, tracking and identification of the large tumbling bodies, apriori knowledge of their relevant electromagnetic scattering parameters is essential. This paper work focus on the scattering phenomenology from possible large bodied orbital debris, such as rocket bodies, whose geometries are publically available. The results will strengthen existing data sets, Radar architectures, required signal processing, and even guidance navigation and control (GNC) routines that would be supported by resultant sensor information. Data products developed from commercially available electromagnetic simulation software will be presented, and the induced phenomenological scattering differences from the geometric variations between the possible targets will also be discussed.
Revising the Relationships between Phase Error and Signal-to-Noise Ratio
Ryan Cutshall,Jason Jerauld, November 2014
Within RF measurement systems, engineers commonly wish to know how much phase ripple will be present in a signal based on a given signal-to-noise ratio (SNR). In a past AMTA paper (Measurement Considerations for Antenna Pattern Accuracy, AMTA 1997), John Swanstrom presented an equation which demonstrated how the bound on the phase error could be calculated from the peak SNR value. However, it can be shown that the Swanstrom bound is broken when the signal has a peak SNR value of less than approximately 15 dB. This paper introduces a new equation that bounds the maximum phase error of a signal based on the signal’s peak SNR value. The derivation of this new bound is presented, and comparisons are made between the old Swanstrom bound and the new bound. In addition, the inverse relationship (i.e., calculating the SNR value of a signal from phase-only measurements) is investigated. In the past, analytical equations for this relationship have been presented by authors such as Robert Dybdal (Coherent RF Error Statistics in IEEE Trans. on Microwave Theory and Techniques) and Jim P.Y. Lee (I/Q Demodulation of Radar Signals with Calibration and Filtering in a Defense Research Establishment Ottawa publication). The analytical equations for calculating the SNR value using phase-only measurements are reviewed and discussed, and a brand new numerical relationship based on a polynomial curve fitting technique is proposed.
Distinguishing Localized and Non-Localized Scattering for Improved Near-Field to Far-Field Transformations
Scott Rice,Lee Harle, November 2014
Historically, the inverse synthetic aperture radar (ISAR) reflectivity assumption has been used in the implementation of Image-Based Near Field-to-Far Field Transformations (IB-NFFFT) to estimate monostatic far field radar cross-sections (RCS) from monostatic near field radar measurements. The ISAR assumption states that all target scattering occurs at the location of the incident field excitations, i.e., the target is composed entirely of non-interacting localized scatters. Certain non-localized scattering phenomenon cannot be effectively handled by the IB-NFFFT approach with the ISAR assumption. Here we have used the adaptive Gaussian representation, which is a joint time-frequency decomposition technique, to coherently decompose near field measured data into two subsets of scattering features: one subset of localized scatterers and the other of non-localized scatterers. The localized scattering features are processed through the IB-NFFFT as typical, which includes compensating for the R4 fall-off present in the near field measured data. The non-localized scattering features, more appropriately scaled, are then coherently added back in to the post-NFFFT localized scattering phase history. Although this does not properly transform the non-localized scattering features into the far field, it does avoid the over-estimation error associated with improperly compensating distributed non-localized scattering features by a R4 power fall off based strictly on downrange position.
Radar Echoes from Dielectric Strings and Ropes
Pax Samuel Wei (retired), November 2015
At the Boeing 9-77 Range, we often encountered the need to support test objects of light to heavy weights with dielectric strings and fishing ropes of varying sizes from small to large.  Unlike a metallic material, which reflects the waves from its surface, the dielectric material is a volume scatterer [1].  Usually, the radar echoes from the strings or ropes at broadside to the wave-front are the highest, then they fall off quickly with angles away from normal.  In this paper we discuss several interesting cases learned, namely:   a). To deduce the dielectric constant of a rope by the ratio of co-pol to x-pol echoes.   b). To estimate the effective radius of a rope after being stretched under a heavy load.   c). Observation of interference between two or more scatterers in the same scene.   d). To process the angular dependent radar data of a tightly stretched rope as a         field-probe along that rope.   This paper is prepared in memory of and dedicated to a great teacher and friend on RCS [2]. ---------------------------------------------- ** Sam Wei  is at:  4123 - 205th Ave. SE, Sammamish, WA  98075-9600.     Email:,  Tel. (425) 392-0175   [1].  E. F. Knott, "Radar Cross Section Measurements," (Van Nostrand Reinhold, New York, 1993), Chapter 3, Target Support Structures, Section 3.2, String Supports, pp. 85-98.  [2].  In Memoriam: Eugene Knott, IEEE Antennas and Propagation Magazine, vol. 56,    No. 3, June 2014, pp. 132-133.
Dual-calibration Processing Based on Minimum Weighted Mean Squared Error (MWMSE) in RCS Measurement
Xiaojian Xu,Yongze Liu, November 2015
Dual-calibration was first proposed by Chizever et al. in 1996 [AMTA'1996] and had get wide applications in evaluation of the uncertainty in radar cross section (RCS) measurement and calibration. In 2013, LaHaie proposed a new technique based on jointly minimizing the mean squared error (MMSE) [AMTA'2013] among the calibrated RCS of multiple calibration artifacts, which estimates both the calibration function and the calibration uncertainty for each artifact. MMSE greatly improves the estimation accuracy for the radar calibration function as well as results in lower residual and RCS calibration errors. This paper presents a modified version of LaHaie's MMSE by minimizing the weighted mean squared error (MWMSE) for RCS calibration processing from  multiple calibrator measurements, which is related to the following functions and parameters: the calibration function; the theoretical and measured RCS; the number of calibration artifacts the number of frequency samples and the weight for ith calibration artifacts which may be defined in terms of the theoretical RCS of all the calibration artifacts. For example, if the weight is defined as the inverse of the total theoretical RCS of the ith calibration artifacts for all frequency samples, the error then represents the total relative calibration error instead of an absolute error as in MMSE. MWMSE then means that an optimal calibration function is found in terms of minimum total relative calibration error, which is expected for most applications. Numerical simulation results are presented to demonstrate the usefulness of the proposed technique.

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