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The need for thermal stability in a test chamber is a well-established requirement to maintain the accuracy and repeatability sought for high frequency planar near-field (PNF) scanner measurements. When whole chamber thermal control is impractical or unreliable, there are few established methods for maintaining necessary precision over a wide temperature range. Often the antenna under test (AUT) itself will require a closed-loop thermal control system for maintaining stable performance due to combined effects from transmission heat dissipation and the environment. In this paper, we propose a new approach for near-field system design that leverages this AUT stability, while relaxing the requirement of strict whole chamber thermal control. Fixed reference monuments strategically placed around the AUT aperture perimeter, when measured periodically with a sensing probe on the scanner, allow for the modeling and correction of the scanner positioning errors. This process takes advantage of the assumed stability of the reference monuments and attributes all apparent monument position changes to distortions in the scanner structure. When this monument measurement process is coupled with a scanner structure that can tolerate wide thermal variations, using expansion joints and kinematic connections, a robust structural error correction model can be generated using a bilinear mapping function. Application of such a structure correction technique can achieve probe positioning performance similar to scanners that require tightly controlled environments. Preliminary results as well as a discussion on potential design variations are presented.
The principles of near-field antenna measurements in Cartesian, cylindrical, and spherical coordinates are well established and documented in the literature and in standards used on antenna ranges throughout government, industry, and academia. However the measurement methods used and the mathematics that are applied to compute the gain and radiation of the pattern of the test antenna from the near-field data assume that the antenna is operating in free space. This leaves several questions open when dealing with antennas operating over a lossy ground plane, such as the ocean. In this paper, we shall discuss a possible avenue for addressing this problem : the use of Complex Image Theory (CIT). The CIT approach allows the lossy earth to be removed and an image of each equivalent source point in the space above it to be constructed in the now empty space below it, but where the depth of that image is in general a complex number. While it might appear confusing to define a complex depth, such a depth is merely a mathematical construct that accounts for a magnitude and phase shift that occurs due to the presence of the lossy ground. The depth is computed so that the boundary condition at the surface of the original lossy ground is maintained; in this way, an equivalent problem is formulated. We propose an approach based on CIT that can be applied to the problem of a spherical nearfield antenna measurement taken over seawater. A limiting case of measurements taken over a metal ground plane shall be presented, along with thoughts about some practical concerns involved in the performance of such measurements.
Prior literature in the subject area of far-field antenna measurements has demonstrated an extrapolation technique to isolate and correct the errors due to near-zone proximity effects as well as multi-path range reflections, thus allowing data to be collected at distances much less than the conventionally defined far-field criteria. This paper describes a modern, indoor, far-field antenna measurement range specifically designed to support this extrapolation technique. A multi-axis positioning system featuring a mobile horn tower capable of motion along the chamber Z-axis is emphasized. High-speed RF instrumentation and advanced software control support the full automation of the extrapolation method. This contemporary approach is demonstrated, and measurement examples are provided for an X-band slotted waveguide array. The resultant far-field gain calculations are also compared to similar data extracted using near-field scanning techniques.
The current method of measuring system G/T performance is with the use of celestial noise sources (sun and cold sky). This paper details a method using man-made noise sources to measure system performance within an anechoic chamber, followed by an outdoor measurement to obtain G/T performance in a real world operational environment. A simple method is presented and equations derived that relate system performance in unknown environments to performance with known noise sources.
The performance of a compact antenna test range is evaluated by field-probing measurements of the quiet zone. The comparison between the simulated and measured data, however, is misleading due to the finite measurement accuracy and the limited nature of the numerical model. In order to allow a comparison, the uncertainty terms of the field-probing measurements and the numerical model are identified based on the National Institute of Standards and Technology 18-term uncertainty analysis technique. The individual terms are evaluated with simulations or measurements using the equivalent-stray-signal model. Bearing the differences between the model and the actual measurements in mind, the electrical field can be estimated precisely within the overlapping region of both uncertainty budgets.
In this paper, a novel algorithm for designing 3D-printed shaped inhomogeneous dielectric lens antennas is provided. The synthesis approach is based on a novel combination of Geometrical Optics (GO) and the Particle Swarm Optimization (PSO) method. The GO method can trace rays through inhomogeneous media and calculate the amplitude, phase, and polarization of the electric field. The algorithm is used to design an inhomogeneous lens antenna to produce an electronically scanned revolving conical beam to replace a mechanically scanned parabolic reflector antenna for spaceborne weather radar satellite antenna applications. Two breadboard model on-axis fed lens designs are presented and measured results given to validate the approach. A representative optimum off-axis design is presented which produces the revolving conically scanned beam. Imposition of a Body-of-Revolution restriction allows the optimization to be performed at a single offset feed location. The complex inhomogeneous engineered materials that results from optimization are printed using new 3D printers.
This paper presents two methods for measuring dynamic antenna patterns of phased arrays in a compensated compact range. The first method uses the turntable of the compact range to counter steer the antenna beam. The dynamic pattern is created by measuring single points of the pattern over time. This method is successfully tested, and the measurement results show the effect of phase jumps during the steering process. The second method extends the range of application to fast steering phased arrays by decoupling the antenna scan angle and the azimuth angle of the turntable.
This contribution details a procedure to collect and process necessary data to describe the antenna patterns of PAAs using a planar near-field (NF) range. It is proposed that a complete characterization methodology involves not only capturing beam-steered antenna patterns, but also measuring embedded element patterns, exhaustive testing of the excitation hardware of the antenna under test (AUT), and performing a phased array calibration technique. Moreover, to demonstrate the feasibility of the proposed approach, the methodology is applied onto a 2x8 microstrip patch PAA, proving its utility and effectiveness. Finally, by means of the collected data, any array pattern could be predicted by post-processing, as proven by the great agreement found between a measured pattern and its computed predicted version.
New requirements in the field of autonomous driving and large bandwidth telecommunication are currently driving the research in millimeter-wave technologies, which resulted in many novel applications such as automotive radar sensing, vital signs monitoring and security scanners. Experimental data on scattering phenomena is however only scarcely available in this frequency domain. In this work, a new mono- and bistatic radar cross section (RCS) measurement facility is detailed, addressing in particular angular dependent reflection and transmission characterization of special RF material, e.g. radome or absorbing material and complex functional material (frequency selective surfaces, metamaterials), RCS measurements for the system design of novel radar devices and functions or for the benchmark of novel computational electromagnetics methods. This versatile measurement system is fully polarimetric and operates at W-band frequencies (75 to 110 GHz) in an anechoic chamber. Moreover, the mechanical assembly is capable of 360° target rotation and a large variation of the bistatic angle (25° to 335°). The system uses two identical horn lens antennas with an opening angle of 3° placed at a distance of 1 m from the target. The static transceiver is fed through an orthomode transducer (OMT) combining horizontal and vertical polarized waves from standard VNA frequency extenders. A compact and lightweight receiving unit rotating around the target was built from an equal OMT and a pair of frequency down-converters connected to low noise amplifiers increasing the dynamic range. The cross-polarization isolation of the OMTs is better than 23 dB and the signal to noise ratio in the anechoic chamber is 60 dB. In this paper, the facility including the mm-wave system is deeply studied along with exemplary measurements such as the permittivity determination of a thin polyester film through Brewster angle determination. A polarimetric calibration is adapted, relying on canonical targets complemented by a novel highly cross-polarizing wire mesh fabricated in screen printing with highly conductive inks. Using a double slit experiment, the accuracy of the mechanical positioning system was determined to be better than 0.1°. The presented RCS measurements are in good agreement with analytical and numerical simulation.
For features of very weak scattering while masked by background and clutter, care must be taken in the measurement design as well as data processing, in order to extract the true RCS values. A good example of flush-mounted fasteners for a low-observable (LO) aircraft, arranged in an array, was reported by Lutz, Mensa, and Vaccaro [1]. In the Boeing 9-77 range, retro-reflectors (called retros) were routinely taped on a test-body for monitoring its 3-D locations and angles during measurements. Though the retro’s RCS may be several orders below that of a test-body, a challenge was to discover their exact values. In the millimeter wave range (MMWR), we measured 2-D arrays of retros arranged in both square and hexagonal lattices taped onto a flat metal surface pitched 20o down. RCS measurements were made as a function of frequency and aspect angle. From the 2-D FFT images, the nominal RCS for a retro in VV polarization was found to be -75 dBsm, independent of the geometry and number of retros even down to one unit-cell [2]. But for the HH polarization, there is no backscattering from such a flat metal surface. References [1]. J. Lutz, D. Mensa, and K. Vaccaro, "RCS measurements of LO features on a test body," Proc. 21st AMTA, pp. 320-325 (1999). [2]. P. S. P. Wei and J. P. Rupp, " RCS measurements on arrays of weak scatterers enhanced by diffraction," Proc. 26th AMTA, pp. 263-268 (2004). ---------------------------------------------- ** Sam Wei is at: 4123 - 205th Ave. SE, Sammamish, WA 98075-9600. Email: paxwei3@gmail.com, Tel. (425) 392-0175
The portable antenna measurement system PAMS was developed for arbitrary and irregular near-field scanning. The system utilizes a crane for positioning of the near-field probe. Inherent positioning inaccuracies of the crane mechanics are handled with precise knowledge of the probe location and a new transformation algorithm. The probe position and orientation is tracked by a laser while the near-field is being sampled. Far-field patterns are obtained by applying modern multi-level fast multipole techniques. The measurement process includes full probe pattern correction of both polarizations and takes into account channel imbalances. Because the system is designed for measuring large antennas the RF setup utilizes fiber optic links for all signals from the ground instrumentation up to the gondola, at which the probe is mounted. This paper presents results of the Ka-band test campaign in the scope of an ESA/ESTEC project. First, the new versatile approach of characterizing antennas in the near-field without precise positioning mechanics is briefly summarized. The setup inside the anechoic chamber at Airbus Ottobrunn, Germany is shown. Test object was a linearly polarized parabolic antenna with 33dBi gain at 33GHz. The near-fields were scanned on a plane with irregular variations of over a wavelength in wave propagation. Allowing these phase variations in combination with a non-equidistant grid gives more degree of freedom in scanning with less demanding mechanics at the cost of more complex data processing. The setup and the way of on-the-fly scanning are explained with respect to the crane speed and the receiver measurement time. Far-fields contours are compared to compact range measurements for both polarizations to verify the test results. The methodology of gain determination is also described under the uncommon near-field constraint of coarse positioning accuracy. Finally, the error level assessment is outlined on the basis of the classic 18-term near-field budgets. The assessment differs in the way the impact of the field transformation on the far-field pattern is evaluated. Evaluation is done by testing the sensitivity of the transformation with a combination of measured and synthetic data.
At NIST, we have developed a precision, wide-band, mmWave modulated-signal source with traceability to primary standards. We are now extending the traceability path for this modulated-signal source into free space to be used for verifying over-the-air measurements in 5G, wireless receivers. However, to obtain a traceable modulated signal in free space, the full scattering matrix of the radiating antenna must be measured. We have extended the extrapolation methods used at NIST, based on the work of Newell, et al. [1]. The extrapolation measurement provides a very accurate, far-field, on-axis, scattering matrix between two antennas. When combined with scattering-matrix measurements made with permutations of pairs of three antennas, far-field scattering, and, thus, gain, is obtained for each antenna. This allows an accurate extrapolation of the antenna’s near-field pattern. We have incorporated the extrapolation fitting algorithms into a Monte Carlo uncertainty engine called the NIST Microwave Uncertainty Framework (MUF) [2]. The MUF provides a framework to cascade scattering matrices from various elements, while propagating uncertainties and maintaining any associated correlations. By incorporating the extrapolation measurements, and the three-antenna method into the MUF, we may provide traceability of all measurement associated with the gain, including the scattering parameters. In this process, we studied several aspects of the gain determination. In this work, we show simulations determining the efficacy of filtering to reduce the effect of multiple reflection on the extrapolation fits. We also show comparisons of using only amplitude (as is traditionally done) to using the full complex data to determine gain. Finally, we compare uncertainties associated with choices in the number of expansion terms, systematic alignment errors, uncertainties in vector network analyzer calibrations and measurements, and phase error introduced by cable movement. With these error mechanisms and their respective correlations, we illustrate the NIST MUF analysis of the antenna scattering-matrix with data at 118 GHz. [1] A. C. Newell, R. C. Baird, and P. Wacker “Accurate Measurement of Antenna Gain and Polarization at reduced distances by an extrapolation technique” IEEE Transactions on Antennas and Propagation. Vol. 21, No 4, July 1973 pp. 418-431. [2] D. F. Williams, NIST Microwave Uncertainty Framework, Beta Version. NIST, Boulder, CO, USA, Jun. 2014. [Online]. Available: http://www.nist.gov/pml/electromagnetics/related-software.cfm
Microstrip patch antennas are well known in the field of communications and other areas where antennas are used. They consist of a metallic conducting surface deposited onto a grounded dielectric substrate and are widely used in situations where a conformal antenna is desired. They are also popular antennas for array applications. But most patch antennas are typically resonant structures owing to the standing wave of current that forms on them. This resonant behavior limits the impedance bandwidth of the antenna to a few percent. In this paper we shall present an approach for improving the bandwidth of a resonant patch antenna which employs an engineered anisotropic superstrate. By proper design of this superstrate and its tensor, and proper alignment of it with the axis of the patch, an antenna with improved impedance bandwidth results. Some of the challenges associated with the measurement of the anisotropic superstrate will be discussed, ranging from 3D simulations to physical models tested in the laboratory. A final working model of the antenna will be discussed; this model consists of a stacked patch arrangement and was designed to operate at the GPS L1 and L2 frequencies. Data collected from 3D simulations using CST Microwave Studio along with laboratory and anechoic chamber measurements will be presented, showing how the bandwidth at both of these frequencies can be increased while maintaining circular polarization in both passbands. Tolerance to errors in alignment and fabrication will also be presented. Additionally, some lessons learned on anechoic chamber measurements of the antenna’s gain and axial ratio will be discussed.
In millimeter wave near-field measurements dual polarized probe system can be used with some of the advantages: the two electric field components are simultaneously measured within a single scan, amplitude and phase drift affects the two polarization components in the same way and there is no need of mechanical rotation of the probe. Today at DTU-ESA Facility we have dual-polarized probes in range 400MHz-40GHz and this study is part of extending the operational frequency range of the DTU-ESA Facility up to 60GHz. First order µ = ± 1 rotationally symmetric probes are desired because they employ an efficient data-processing and measurement scheme. In this work we design and test at DTU-ESA Facility a dual polarized first order probe system at 60GHz - a conical horn, including the elements: a pin diode SPDT (single pole double throw) switch up to 67GHz from Ducommun an OMT (ortho-mode transducer) from Sage Millimeter in 50-75GHz band with square waveguide antenna port (3.75mm) a square to circular transition (3.75mm to 3.58mm) from Sage Millimeter which is integrated between the OMT and conical horn 1.85mm connector cables up to 75GHz and two coaxial to waveguide adapters to connect the switch to the OMT from Flann Microwave To ensure accurate measurements at 60GHz, the hardware components were selected to provide a low cross polarization of the probe, the switch and the OMT having 40dB isolation between ports. The path loss at 60GHz is 83dB for a 6m distance and to compensate for such a loss, a 26dB gain is desired for the conical horn, which is simulated using WIPL-D software and in-house manufactured. The 60GHz dual-polarized probe is currently being assembled and will be tested in both planar and spherical near-field setups. In the full version of the paper calibration results will be shown but also results from using the probe as a probe for the measurement of a 60GHz AUT.
In this paper, we explore the possibility of using a photonics-based E-field sensor as a near-field probe. Relative to open-ended waveguide (OEWG) probes, a photonics probe could offer substantially larger bandwidths. In addition, since it outputs an optical signal, a photonics probe can offer signal transport through optical fiber with much lower loss than what can be achieved using RF cables. We begin with a discussion of the theory of the device followed by a summary of results of a photonics sensor that was tested in a spherical near-field (SNF) range. In these tests, data were collected with the photonics probe in the test antenna position to characterize various probe parameters including polarization discrimination, probe gain, effective dynamic range, and probe patterns. In the same set of tests, the photonics device was placed in the probe position in the range and used to measure patterns of two different antennas: a standard gain horn and a slotted waveguide array antenna. The resultant patterns are shown and compared to patterns collected with traditional RF probes. We conclude the paper with a discussion of some of the advantages and disadvantages of using a photonics probe in a practical system based on the lessons learned in the SNF testing.
Dual polarized probes with wide bandwidth operational capabilities are convenient for accurate and time efficient Planar Near Field (PNF) antenna testing. Nevertheless, traditional probe designs are often limited in terms of bandwidth and their electrically large size leads to high scattering in PNF measurements with short probe-AUT distances. An innovative octave band probe design is presented in this paper with minimum scattering characteristics. The scattering minimization is mainly obtained by an electrically small and axially symmetric aperture of 0.4? diameter at the lowest frequency. The aperture provide a near constant directivity in the full bandwidth and very low cross polar. The probe is fed by a balanced ortho-mode junction (OMJ) with external feeding circuitry to obtain high polarization purity. This paper discuss the design considerations, technical and implementation trade-offs and show experimental results on the manufactured hardware.
When antenna near-field (NF) measurement within small electrical distance is needed, such as miniaturization of the measurement device or measurement of a low-frequency DUT, the coaxial cables connected to the probes will significantly but inevitably disturb the fields. The measurement accuracy is therefore compromised. In this paper, a novel probe design is proposed by replacing coaxial cable with optical fiber to minimize the disturbance. In this design, the RF-over-Fiber (RoF) technology is applied in signal transmission with Vertical-Cavity Surface-Emitting Laser (VCSEL) and photodiode (PD) as the transmitter and receiver respectively. The VCSEL is powered via optical fiber with Power-over-Fiber (PoF) technology. A power laser emits optical power which is guided by optical fiber to illuminate a miniaturized photovoltaic (PV) element. The PV element serves as a voltage source for the VCSEL. A spherical, multi-probe, NF measurement design with 60cm-diameter is built for portable DUT operated between 0.6 to 2.6GHz. There are 64 probes installed along the two arches for both theta and phi polarizations, so mechanical rotation is needed only on phi axis. Thanks to the high RF transparency of the probes, there is no need to wrap absorbers around the probes to shield the cables. Another spherical NF measurement prototype is also under development. It is half-spherical (10m-diameter) for large DUT, such as vehicles, with low frequency antenna, namely, 70MHz to 600MHz. At this frequency range, to the best of our knowledge, there is no effective and accurate way to measure the radiation performance because the disturbance on the EM fields by the coaxial cables is obviously not negligible.
Gain is a principal property of antennas; it is essential in establishing the link budget for communication and sensing systems through its presence in Friis’ transmission formula and the radar range equation. The experimental determination of antenna gain is most often based on a gain-transfer technique involving a reference antenna for which the gain has been calibrated to high accuracy; this is typically a pyramidal horn antenna [1]. The required accuracy of antenna gain obviously depend on the application; in some cases it can very high, ±0.1 dB or less, and this implies an even higher accuracy, of the order of ±0.01dB, for the gain reference antenna. This work investigates the accuracy to which a gain reference antenna can be calibrated; the investigation is based on experimental spherical near-field antenna measurements [2] and computational integral equation / method of moments simulations [3]. While calibration of gain reference antennas has been studied in many previous works, even works from early 1950s [4]-[6], this work is novel in systematically supporting measurements with full-wave simulations. Such simulations facilitate the study of e.g. the effect of multiple reflections between antennas at short distances. We study two absolute calibration techniques for the gain of pyramidal horn antennas. The first technique determines gain as the product of directivity and radiation efficiency; this technique has been referred to as the pattern integration technique [7] (which is not an entirely adequate designation since gain cannot be determined from the radiation pattern). The second technique determines the gain from Friis’ transmission formula [8] for two identical antennas; this technique is generally referred to as the two-antenna technique [1]. These two calibration techniques involve very different steps and contain very different sources of error; for both techniques our investigation involves measurements as well as simulations. For the pattern integration technique we compare experimental and computational results for the directivity and demonstrate agreement within one-hundredth of a dB. The radiation efficiency is calculated by different techniques based on the surface impedance boundary condition for the metallic walls of the pyramidal horn. This technique is not influenced by proximity effects or by impedance mismatch between the measurement system and the gain reference antenna. For the two-antenna techniques we compare experimental and computational results for the gain and we compare the calculated distance-dependence with that of the extrapolation technique [9]. It is demonstrated how the use of the phase center distance in Friis’ transmission formula notably decreases the necessary separation between the antennas for a required accuracy, but that multiple reflections may then become a limiting factor. This technique is highly influenced by the impedance mismatch that must be accurately accounted for. We compare the gain values resulting from the pattern integration technique and the two-antenna technique, including their very different uncertainty estimates, for a C-band standard gain horn. The work is related to an on-going ESA project at the DTU-ESA Spherical Near-Field Antenna Test Facility for the on-ground calibration of the scatterometer antennas of the EUMETSAT MetOp Second Generation B-series satellites. IEEE Standard – Test Procedures for Antennas, Std 149-1979, IEEE & John Wiley & Sons, 1979. J.E. Hansen, “Spherical Near-Field Antenna Measurements”, Peter Perigrinus Ltd., London 1987. www.wipl-d.com W.C. Jakes, “Gain of Electromagnetic Horns”, Proceedings of the IRE, pp. 160-162, February 1951. E.H. Braun, “Gain of Electromagnetic Horns”, Proceedings of the IRE, pp. 109-115, January 1953. W.T. Slayton, “Design and Calibration of Microwave Antenna Gain Standards”, Naval Research Laboratory, Washington D.C., November 1954. A. Ludwig, J. Hardy, and R. Norman, “Gain Calibration of a Horn Antenna Using Pattern Integration”, Technical Report 32-1572, Jet Propulsion Laboratory, California Institute of Technology, October 1972. H.T. Friis, “A Note on a Simple Transmission Formula”, Proceedings of the I.R.E. and Waves and Electrons, pp. 254-256, May 1946. A.C. Newell, R.C. Baird, P.F. Wacker, “Accurate Measurement of Antenna Gain and Polarization at Reduced Distances by an Extrapolation Technique”, IEEE Transactions on Antenna and Propagation, vol. 21, no. 4, pp. 418-431, July 1973.
The expanding market for millimeter wave antennas is drivinga need for high performance near-field antenna measurement systems at these frequencies. Traditionally at millimeter waves, acquisition of two orthogonal polarizations have been achieved through mechanical rotation of a single polarized probe and an associated frequency conversion module. This generally results in the collection of two complete spherical data sets, one for each polarization,with both acquisitions significantly separated in time. To enable improvements in both measurement speed and accuracy, MVG have developed a new high performance dual polarized feed in V-band (50GHz-75GHz). This probe has been integrated in a millimeter wave Spherical Near-Field (SNF) system via two parallel receiver channels that are simultaneously sampled. This architecture more than doubles the acquisition speed and additionally ensures that the two polarization components are sampled at precisely the same point in space and time. This is particularly important when performing accurate polarization analysis (e.g. conversion of dual linear polarization to spherical/elliptical polarizations). The two measurement channels are calibrated via radiated boresight measurements over a range of polarization angles, generating a four term “ortho-mode” correction matrix vs. frequency. The SNF probe is based on an axially corrugated aperture providing a medium gain pattern (14dBi). The probe provides symmetric cuts and low cross-polarization levels in the diagonal planes. The directivity/beam-width of the aperture has been tailored to the measurement system, ensuring proper AUT illumination and sufficient gain to compensate for free space path loss. Dual polarization capability is achieved with an integrated turnstile OMT feeding directly into the probe circular waveguide and a conical matching stub at the bottom. Thanks to the balanced feed used for each polarization, the port-to-port coupling is sufficiently low to allow for simultaneous acquisition of the two linear field components. Input ports are based on standard WR-15 waveguide to simplify the integration with the front-end (dual channel receiver). The paper will present the detailed description and measured performances of the new dual polarized SNF probe. Additionally, measurement time and achieved accuracy will be compared between the single polarization probe architecture and the dual polarized probe installed in the same spherical near-field antenna measurement system.
Rotating the coordinate system of an antenna pattern can be problematic due to the need to interpolate complex data in spherical coordinates. Common approaches to 2D interpolation often introduce errors because of polarization discontinuities at the spherical coordinate system poles. To overcome these difficulties, it is possible to transform an antenna pattern from field-space into spherical mode-space, perform the desired coordinate system rotation in mode-space, and then transform the modes in the rotated coordinate system back into field-space. This method, while more computationally intensive, is exact and alleviates all of the interpolation-related issues associated with rotations in field-space. Although rotations in mode-space have been implemented in commercially available software (e.g., the ROSCOE algorithm provided by TICRA), these algorithms may not be well understood by the general antenna measurement community. Therefore, the first goal of this paper is to present an easy-to-understand algorithm for performing rotations in mode-space. Next, the paper will address the challenge of computing the rotation coefficients, which are required by the mode-space coordinate system rotation algorithm. Although J. E. Hansen presented a method for recursively computing the rotation coefficients, this method is numerically unstable for large values of N (where N is the upper limit of the polar index). Therefore, this paper will present a numerically stable method for the recursive computation of the rotation coefficients. Finally, this paper will show the relationships between Euler angles and both Az-over-El angles and El-over-Az angles. These relationships are quite useful because it is often desired to rotate an antenna pattern based on Elevation and Azimuth angles, whereas the inputs for the mode-space rotation algorithm are Euler angles. Knowing these relationships, the Euler angles may be computed from the Azimuth and Elevation angles, which can then be used as the inputs to the mode-space rotation algorithm.