System and method for determining corrected total radiated power (TRP) or corrected total isotropic sensitivity (TIS) of an offset antenna under test

Abstract

A method for determining a corrected TRP or TIS for an AUT in a near-field test chamber, the AUT having a phase center offset from the center of rotation of the test chamber, includes: making EIRP or EIS measurements of the AUT at first sampling grid points on a first closed surface geometry centered on the center of rotation; mapping second sampling grid points to the first closed surface geometry to obtain mapped sampling grid points on the first closed surface geometry, the second sampling grid points being on the second closed surface geometry centered on the phase center of the AUT; determining an estimated EIRP or EIS at the mapped sampling grid points using the EIRP or EIS measurements; scaling the estimated EIRP or EIS at the mapped sampling grid points to obtain a scaled EIRP or EIS; and calculating a corrected TRP or TIS based on the scaled EIRP or EIS.

Description

[Background Technology]

[0001] In testing wireless communication devices, a device under test (DUT) may have multiple antennas under test (AUTs), such as multiple antenna arrays, located at different locations on the DUT. For example, the DUT may have a first AUT transmitting an uplink (UL) transmission of a radio frequency (RF) signal and a second AUT receiving a downlink (DL) transmission of the RF signal. Alternatively, the DUT may have multiple UL / DL antennas for multiple-input and multiple-output (MIMO) or diversity. The DUT may be tested in a test chamber using white-box or black-box testing, which performs measurements in the radiative near-field or radiative far-field. In white-box testing, the array phase center of each antenna array being tested is aligned with the center of a quiet zone in the test chamber, and the offset of the array phase center of each antenna array is known. On the other hand, black-box testing does not require prior knowledge of which antenna arrays are active or the detailed location of the active antenna arrays in the DUT relative to the test chamber's quiet zone center. For example, the geometric center of the DUT may be aligned with the quiet zone center, with the DUT itself offset from the quiet zone center by an unknown amount. In the case of direct near-field and far-field test chambers, black-box testing can have a significant impact on measurements based on UL power or DL ​​power due to the unknown offset from the quiet zone center to the active antenna array. This offset can result in a large path loss difference and, therefore, a large difference in power-based metrics.Taking various measurements of the DUT is more difficult when the AUT is offset from the center of the quiet zone, especially when the offset is an unknown distance and therefore requires black box testing.

[0002] Measurement types that are adversely affected by AUT offset include total radiated power (TRP), total isotropic sensitivity (TIS), equivalent isotropic radiated power (EIRP), and effective isotropic sensitivity (EIS). In particular, although EIRP and EIS conventionally describe radiated power and sensitivity measurements in the far field, for simplicity of explanation, EIRP and EIS are used herein to describe radiated power and sensitivity measurements in the near field as well.

[0003] TRP is a common metric used to evaluate the total power radiated by a DUT, while TIS measures the average sensitivity of the transmitter and receiver system that includes the DUT. The TRP of a DUT can be calculated according to equation (1), where θ is the elevation angle and φ is the azimuthal angle.

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[0004] The integral function in equation (1) is typically solved using numerical integration techniques. For example, a derivation from the closed surface TRP integral to the traditional discrete summation formula used for over the air (OTA) transmission is obtained as shown in equation (3).

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[0005] TIS is derived from equation (2) as shown in equation (4).

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[0006] However, equations (3) and (4) and other conventional numerical integration techniques do not consider the degradation in accuracy of the TRP or TIS calculation caused by the offset of the AUT from the quiet zone center in the test chamber. For AUT measurements in the near field, the uncertainty in the TRP measurement increases especially because the AUT offset is no longer a small fraction of the measurement distance. Summary of the Invention [Problem to be solved by the invention]

[0007] Because the offset distance of the AUT is usually relatively small compared to the far-field measurement distance, the offset of the AUT is generally less of a problem in far-field measurements than in near-field measurements. However, there are several drawbacks to performing far-field measurements. For example, in the case of measurement solutions based on DFF measurements, 5G base stations and user equipment utilize large antenna arrays to support massive MIMO functionality. This requires a very large measurement distance for far-field OTA measurements, and therefore requires a very large anechoic chamber in which to perform the OTA measurements. Such a large anechoic chamber is expensive to set up and may have practical space requirements challenges. Furthermore, the longer the far-field measurement distance, the larger the resulting propagation loss, which is another challenge for accurate OTA measurements, especially for tests with high downlink power and low uplink power. Therefore, near-field measurements are more desirable. [Means for solving the problem]

[0008] The illustrative embodiments are best understood from the following detailed description when read in conjunction with the accompanying drawings. It is emphasized that the various features are not necessarily drawn to scale. In fact, dimensions may be arbitrarily increased or decreased for clarity of discussion. Wherever applicable and practical, like reference numerals refer to like elements. [Brief explanation of the drawings]

[0009] [Figure 1] 1 is a simplified block diagram of a device under test (DUT) having an antenna under test (AUT) offset from the quiet zone center of a test chamber, according to a representative embodiment. [Figure 2] FIG. 1 is a simplified block diagram illustrating a test system for performing EIRP or EIS measurements on a DUT including at least one AUT offset from a quiet zone center to determine TRP or TIS, according to a representative embodiment. [Figure 3] 1 is a schematic diagram of an AUT having TRP or TIS grid points around the center of the test chamber's quiet zone center and around the AUT's array phase center, according to a representative embodiment. [Figure 4] 1 is a flow diagram illustrating a method for determining corrected TRP and / or TIS (i.e., TRP or TIS or both) of an AUT in a near-field test chamber according to a representative embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0010] In the following detailed description, for purposes of explanation and not limitation, exemplary embodiments disclosing specific details are described to provide a thorough understanding of an embodiment according to the present teachings. Descriptions of known systems, devices, materials, methods of operation, and methods of manufacture may be omitted to avoid obscuring the description of the exemplary embodiments. Nevertheless, systems, devices, materials, and methods within the understanding of those skilled in the art are within the scope of the present teachings and may be used in accordance with the exemplary embodiments. It is to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting. Defined terms have meanings commonly understood and accepted in the technical field of the present teachings, in addition to the scientific and technical meaning of the defined term.

[0011] Although terms such as first, second, and third may be used herein to describe various elements or components, it should be understood that these elements or components should not be limited by these terms. These terms are used only to distinguish one element or component from another. Thus, a first element or component discussed below could be referred to as a second element or component without departing from the teachings of the present disclosure.

[0012] The terms used herein are for the purpose of describing particular embodiments only and are not intended to be limiting. As used in this specification and the appended claims, the singular terms "a," "an," and "the" are intended to include both the singular and the plural, unless the context clearly dictates otherwise. Furthermore, the term "comprises" and / or similar terms (i.e., terms and / or similar terms) when used herein specify the presence of referenced features, elements, and / or components (i.e., features, elements, and / or components), but do not exclude the presence or addition of one or more other features, elements, components, and / or groups thereof. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0013] Unless otherwise specified, when an element or component is said to be "connected to," "coupled to," or "adjacent to" another element or component, it is understood that the element or component can be directly connected or coupled to the other element or component, or that there may be intervening elements or components. That is, these and similar terms include instances where one or more intervening elements or components may be used to connect the two elements or components. However, when an element or component is said to be "directly connected" to another element or component, this only includes instances where the two elements or components are connected to each other without any intervening or intervening elements or components.

[0014] The present disclosure is therefore intended to achieve one or more of the advantages, as specifically mentioned below, through one or more of its various aspects, embodiments, and / or specific features (aspects, embodiments, and / or specific features) or subcomponents. For purposes of explanation and not limitation, exemplary embodiments disclosing specific details are described to provide a thorough understanding of an embodiment in accordance with the present teachings. However, other embodiments that depart from the specific details disclosed herein but are not inconsistent with the present disclosure may still be within the scope of the appended claims. Furthermore, descriptions of known devices and methods may be omitted so as not to obscure the description of the exemplary embodiments. Such methods and devices are within the scope of the present disclosure.

[0015] Generally, according to various embodiments, the total radiated power (TRP), equivalent isotropically radiated power (EIRP), and / or effective isotropic sensitivity (EIS) (i.e., TRP, EIRP, and / or EIS) of a device under test (DUT) can be determined in a near-field test chamber, where the DUT has at least one antenna under test (AUT) offset from the center of the quiet zone of the near-field test chamber. Other measurements, such as transmitted signal quality (EVM), are not excluded.

[0016] FIG. 1 is a simplified block diagram of a DUT having an AUT offset from the quiet zone center of a test chamber, according to a representative embodiment.

[0017] Referring to FIG. 1 , the DUT 110 is positioned within the quiet zone 120 of the near-field test chamber 105. The geometric center 115 of the DUT 110 is aligned with the quiet zone center 125 of the quiet zone 120. The DUT 110 includes a representative AUT 130 located at an outer edge of the DUT 110 away from the geometric center 115. The AUT 130 has beamforming capabilities. In the illustrated embodiment, the AUT 130 includes an antenna array 131 having an antenna pattern 140. For purposes of illustration and not limitation, the antenna array 131 includes eight antenna elements arranged in a 2×4 array with an array phase center 135 at the physical and / or electronic center (i.e., the physical center, the electronic center, or both) of the 2×4 array. The array phase center 135 is offset from the quiet zone center 125 by an offset OS to an offset location at an angle and distance from the quiet zone center 125 to the array phase center 135 of the antenna array 131. The amount of offset may be known or unknown depending on the embodiment described herein. Thus, measurements of the AUT 130, such as TRP, EIRP, and / or EIS, for example, are made in accordance with black-box testing. Of course, the antenna array 131 may include more or fewer antenna elements arranged in various patterns without departing from the scope of the present teachings. Also, the AUT 130 may be an uplink antenna and / or a downlink antenna (i.e., an uplink antenna or a downlink antenna).

[0018] Each antenna element of the antenna array 131 can have multiple codebooks, resulting in an antenna pattern 140 with various beams, represented by representative first beam 141, second beam 142, third beam 143, and fourth beam 144. For example, in compliance testing, only the beam with the best performance needs to be evaluated, and this beam is locked onto before integrating over the antenna pattern of the AUT 130 to prevent the DUT 110 from selecting a different beam during measurement. The best beam is typically the beam with the best EIRP or EIS performance in the declared test direction in the far field. Each of the first beam 141 through fourth beam 144 has a corresponding beam peak direction from the array phase center 135. In the illustrated example, the beam peak direction 148 of the first beam 141 is indicated by an arrow extending substantially perpendicular to the antenna array 131.

[0019] 2 is a simplified block diagram illustrating a test system for performing EIRP or EIS measurements to determine TRP or TIS of a DUT including at least one AUT offset from the quiet zone center, according to a representative embodiment. Again, as used herein, EIRP and EIS refer to measurements of radiated power and sensitivity, respectively, in the near field as well as the far field.

[0020] Referring to FIG. 2 , system 200 includes a test chamber 205 and a processing unit 250 in communication with test chamber 205. Test chamber 205 may be, for example, an anechoic chamber. DUT 110 is located within a quiet zone (not shown) of test chamber 205, including AUT 130 offset from the center of the quiet zone, as discussed above. In the illustrated example, test chamber 205 includes at least one near-field probe antenna, represented by near-field probe antenna 260. Test chamber 205 may also include a far-field probe antenna (not shown), e.g., a conventional probe antenna positioned in the far field or a compact antenna test range reflector-based methodology, to introduce a far-field environment at DUT 110. Probe antenna 260 may be movable to different locations within test chamber 205 both laterally and radially relative to DUT 110 and may have a variable range length. In one embodiment, the test chamber 205 can be a near-field test chamber or a far-field test chamber having at least one near-field probe antenna located in the near field without departing from the scope of the present teachings.

[0021] The processing unit 250 includes a processor device 255, a memory 256, and an interface 257, along with a display 258. The processor device 255, together with the memory 256, can be configured to implement a method for determining at least one of the TRP, EIRP, and EIS of the DUT 110 in the test chamber 205 and to perform and / or control (i.e., execute and / or control) all or a portion of the process steps illustrated in FIGS. 4 and 8 and discussed below. In various embodiments, the processor device 255 can include a general-purpose computer using any combination of hardware, software, firmware, hardwired logic circuitry, or a central processing unit (CPU), one or more processors, microprocessors or microcontrollers, state machines, programmable logic devices, field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), or combinations thereof. The term "processor" specifically encompasses electronic components capable of executing programs or machine-executable instructions. When referring to a processor, this should be interpreted to include two or more processors or processing cores, such as a multi-core processor and / or a parallel processor (i.e., a multi-core processor and / or a parallel processor). A processor can also refer to a collection of processors within a single computer system or a collection of processors distributed among multiple computer systems, such as a cloud-based or other multi-site application. A program has software instructions that are executed by one or more processors, which may be within the same computing device or distributed across multiple computing devices. Processor device 255 may have its own memory (not shown) and is in communication with memory 256.

[0022] Memory 256 stores instructions / computer-readable code (e.g., software, software modules) that enable the performance of various functions described herein. For example, memory 256 can store software instructions / computer-readable code executable by processor device 255 (e.g., computer processor) to perform some or all aspects of the methods described herein. Memory 256 can be embodied, for example, by any number, type, and combination of random access memory (RAM) and read-only memory (ROM) and can store various types of information, such as software algorithms, AI models, including ANNs and other neural network-based models, and computer programs, all of which are executable by processor device 255. The various types of ROM and RAM may include any number, kind, and combination of computer-readable storage media, such as disk drives, flash memory, electrically programmable read-only memory (EPROM), electrically erasable and programmable read only memory (EEPROM), registers, hard disks, removable disks, tapes, compact disk read only memory (CD-ROM), digital versatile disks (DVD), floppy disks, Blu-ray disks, universal serial bus (USB) drives, or any other form of storage media known in the art.Memory 256 may be secure and / or encrypted (i.e., secure and / or encrypted) or non-secure and / or unencrypted (i.e., unsecure and / or unencrypted). Memory 256 represents one or more memories and databases, including processing memory, as well as multiple memories and databases, including distributed and networked memories and databases.

[0023] Memory 256 is a tangible storage medium that stores data and executable software instructions and is non-transitory for the time the software instructions are stored therein. As used herein, the term "non-transitory" should not be interpreted as a permanent characteristic of a state, but as a characteristic of a state that continues for a period of time. The term "non-transitory" does not particularly concern transient characteristics, such as characteristics of a carrier wave or signal or other form of thing that exists only temporarily at a given location at a given time. Memory 256 can store software instructions and / or computer-readable code (i.e., software instructions, computer-readable code, or both) that enable the performance of various functions. The terms "memory" and "database" are examples of computer-readable storage media and should be interpreted as multiple memories or databases, as the case may be. The memories or databases may, for example, be multiple memories or databases local to a computer and / or distributed among multiple computer systems or computing devices (i.e., multiple memories or databases local to a computer, or distributed among multiple computer systems or computing devices, or all of the above).

[0024] Interface 257 may include a user interface and / or a network interface (i.e., a user interface or a network interface, or both) that provides information and data output by processor device 255 and / or memory 256 (i.e., processor device 255, memory 256, or both) to a user and / or receives information and data input by a user. That is, interface 257 allows a user to input data to control or manipulate aspects of the process for determining the EIRP and / or EIS (i.e., EIRP and / or EIS) of the DUT, and also allows processor device 255 to indicate the effects of the user's control or manipulation (e.g., on display 258). Interface 257 may include one or more of a port, a disk drive, a wireless antenna, or other type of receiver circuitry. The interface 257 may further connect one or more user interfaces, such as a mouse, keyboard, mouse, trackball, joystick, microphone, video camera, touchpad, touchscreen, recognizers for voice or gestures captured by the microphone or video camera, or any other peripheral or control device that allows user feedback from and interaction with the processing unit 250.

[0025] Display 258 may be, for example, a monitor such as a computer monitor, television, liquid crystal display (LCD), organic light emitting diode (OLED), flat panel display, solid-state display, or cathode ray tube (CRT) display, or an electronic whiteboard. Display 258 may also provide a graphical user interface (GUI) for displaying information to and receiving information from a user. Interface 257 may include a user interface and / or network interface for providing information and data output by processor device 255 and / or memory 256 to a user and / or receiving information and data input by a user. That is, interface 257 allows a user to input data to control or manipulate aspects of the processes described herein, and also allows processor device 255 to indicate the effects of the user's controls or manipulations. The interface 257 may further connect one or more user interfaces, such as a mouse, keyboard, mouse, trackball, joystick, haptic device, microphone, video camera, touchpad, touch screen, recognizer for voice or gestures captured by the microphone or video camera, etc.

[0026] The accuracy of TRP and TIS measurements is adversely affected when the AUT's antenna array phase center is offset from (not aligned with) the test chamber's measurement system rotation center (quiet zone center), and the offset size is not negligible compared to the test distance. This is particularly true for TRP measurements in the near field, which can be more sensitive to antenna array offsets. For convenience, the following embodiments discuss determining TRP. However, it will be appreciated that these embodiments are equally applicable to determining TIS.

[0027] FIG. 3 is a simplified block diagram of an AUT having grid points of TRP or TIS around the center of the test chamber quiet zone center and around the array phase center of the AUT, according to a representative embodiment.

[0028] 3, the AUT 130 is positioned within the quiet zone 220 of the test chamber 205 discussed above. The quiet zone 220 includes a quiet zone center P0 at its center, which may also be referred to as the center of rotation of the test chamber 205. The AUT 130 may be positioned, for example, at an array phase center P0 that is offset from the quiet zone center P0 by a known offset OS. ANT That is, the offset OS may be pre-declared by, for example, a manufacturer or a user, or may be determined using any known technique. The AUT 130 may be an uplink antenna and / or a downlink antenna.

[0029] To determine the TRP or TIS of the AUT 130, first sampling grid points are placed on a first closed surface geometry 310 having a geometric center at the quiet zone center P0. The first sampling grid points can be placed on the first closed surface geometry 310 using a constant step size or a constant density. A constant step size grid type has azimuth and elevation angles uniformly distributed across the surface of the closed surface geometry, while a constant density grid type has measurement points uniformly distributed on the surface of the closed surface geometry. In the illustrated embodiment, the first closed surface geometry 310 is a first sphere, and the first sampling grid points are represented by eight uniformly distributed sampling grid points 311, 312, 313, 314, 315, 316, 317, and 318 around the periphery of the first sphere.

[0030] Similarly, the second sampling grid point is the array phase center P ANTThe second sampling grid points are disposed on a second closed surface geometry 340 having a geometric center at . There are no restrictions on the arrangement of the second sampling grid points disposed on the second closed surface geometry 340, and therefore the arrangement of the second sampling grid points and the grid density of the second sampling grid points on the second closed surface geometry 340 are arbitrary. Thus, the second sampling grid points do not need to follow the same grid type, grid spacing, and / or grid density as the first sampling grid points disposed on the first closed surface geometry 310 (i.e., the same grid type, grid spacing, and / or grid density as the first sampling grid points). In the illustrated embodiment, the second closed surface geometry 340 is a second sphere, and the second sampling grid points are represented by eight equally spaced second sampling grid points 341, 342, 343, 344, 345, 346, 347, and 348 around the second sphere. Of course, the first closed surface geometry 310 and the second closed surface geometry 340 can be shapes other than spherical without departing from the scope of the present teachings. Also, the first closed surface geometry 310 and the second closed surface geometry 340 can be the same shape and size as one another, or different shapes and sizes. Similarly, the first sampling grid points and the second sampling grid points can each be fewer or more than eight without departing from the scope of the present teachings. It is further understood that, although shown in only two dimensions, the first sampling grid points and the second sampling grid points can be arranged in three dimensions on the first closed surface geometry and the second closed surface geometry, respectively.

[0031] The EIRP or EIS of the AUT 130 is measured at each of the first sampling grid points 311-318 in the near-field test chamber 205. The EIRP or EIS of the AUT 130 can be measured using one or more probe antennas, such as the probe antenna 260, by, for example, moving the probe antenna 260 to the location of the first sampling grid points or by moving the DUT 110 so that the position of the probe antenna 260 relative to the AUT 130 is at each of the first sampling grid points. When determining the TRP of the AUT 130, the EIRP is measured at each of the first sampling grid points 311-318, and when determining the TIS of the AUT 130, the EIS is measured at each of the first sampling grid points 311-318.

[0032] Each of the second sampling grid points 341-348 corresponds to the array phase center P ANT The second closed surface geometric shape 340 is mapped to the first closed surface geometric shape 310 using distance and angle information of the offset OS from the second closed surface geometric shape 340 to each of the second sampling grid points 341-348. As mentioned above, the arrangement of the second sampling grid points 341-348 on the second closed surface geometric shape 340 and the grid density of the second sampling grid points 341-348 are arbitrary. For illustrative purposes, FIG. 3 shows the arrangement of the second sampling grid points 341-348 on the second closed surface geometric shape 340 as a grid density of the array phase center P ANT 3 shows second sampling grid point 341 mapped from second closed surface geometry 340 to first closed surface geometry 310 by drawing line L1 from second closed surface geometry 340 through second sampling grid point 341 to first closed surface geometry 310. The resulting point on first closed surface geometry 310 is shown by mapped sampling grid point 341′. Similarly, second sampling grid point 346 is mapped to array phase center P ANTThe second closed surface geometry 340 is mapped to the first closed surface geometry 310 by drawing a line L3 from the second closed surface geometry 340 through the second sampling grid point 346 to the first closed surface geometry 310. The resulting point on the first closed surface geometry 310 is indicated by the mapped sampling grid point 346′.

[0033] Mapping the second sampling grid points 341 onto, for example, the first closed surface geometry 310, in three dimensions, provides an array phase center P ANT 3, the elevation angle θ is determined from the array phase center P of the AUT 130 to the second sampling grid point 341, and from the quiet zone center P to the mapped sampling grid point 341′. ANT The elevation angle θ′ is indicated by a line L1 extending from the quiet zone center P0 through the second sampling grid point 341 to the first closed surface geometry 310. The mapped sampling grid point 341′ is located on the first closed surface geometry 310 where the line L1 intersects with the first closed surface geometry 310 at the determined elevation angle θ. The elevation angle θ′ is indicated by a line L2 extending from the quiet zone center P0 and the mapped sampling grid point 341′. From a mathematical calculation of the elevation angle θ and azimuth angle φ of the second sampling grid point 341 and the offset information, a new elevation angle θ′ and azimuth angle φ′ of the mapped sampling grid point 341′ can be calculated.

[0034] Array phase center P ANT If the line L1 passing through the first closed surface geometric shape 310 from the first second sampling grid point 341 is extended in the opposite direction, it may intersect with the first closed surface geometric shape 310 at two points. In this case, the mapped sampling grid point is located on the first closed surface geometric shape 310 at the point of these two that is closer to the initial second sampling grid point 341 (in the illustrated example, this point is the mapped sampling grid point 341′ as shown).

[0035] Once the mapped sampling grid points 341′ and 346′ are positioned on the first closed surface geometry 310, an estimated EIRP or EIS of the AUT 130 at the mapped sampling grid points 341′ and 346′ is determined by interpolating between the measured EIRP or EIS from among the first sampling grid points 311-318. For example, the estimated EIRP or EIS for the mapped sampling grid point 341′ may be a simple interpolation between the EIRP or EIS of the immediately adjacent first sampling grid points 311 and 312. Similarly, the estimated EIRP or EIS for the mapped sampling grid point 346′ may be a simple interpolation between the EIRP or EIS of the immediately adjacent first sampling grid points 315 and 316. Of course, other interpolation techniques may be incorporated, as well as other techniques for estimating the EIRP or EIS of the mapped sampling grid points, without departing from the scope of the present teachings. This process is repeated for each of the remaining second sampling grid points 342-345 and 347 and 348, which are individually mapped onto the first closed surface geometry 310, and the corresponding estimated EIRP or EIS is found by interpolating between the appropriate measured EIRP or EIS of the first sampling grid points 311-318.

[0036] The estimated EIRP or estimated EIS at the mapped sampling grid points corresponding to the second sampling grid points 341-348 are scaled to obtain corresponding scaled EIRP or scaled EIS at each of the second sampling grid points 341-348 on the second closed surface geometric shape 340. The scaling provides a weight for the EIRP or EIS at the second sampling grid points 341-348, respectively. The scaled EIRP or scaled EIS is indicated in the illustrated example by the different sizes of the second sampling grid points 341 and 346 corresponding to the respective values ​​of the scaled EIRP or scaled EIS. That is, the value of the scaled EIRP or scaled EIS at the second sampling grid point 346 is larger than the value of the scaled EIRP or scaled EIS at the second sampling grid point 341, as indicated by the larger dot at the second sampling grid point 346. Generally, the scaling is performed by scaling the second sampling grid points P ANT to each of the mapped sampling grid points.

[0037] In one embodiment, scaling the estimated EIRP or estimated EIS at the mapped sampling grid points includes: (i) scaling the array phase center P ANTand (ii) determining a ratio between each of the first distances between the quiet zone center and each of the mapped sampling grid points and each of the second distances between the quiet zone center and each of the mapped sampling grid points. Each of these ratios is multiplied by the estimated EIRP or estimated EIS at the mapped sampling grid point, respectively. So, for example, scaling the estimated EIRP or estimated EIS at the mapped sampling grid point 341′ can be done by determining the ratio of lines L1 and L2 and multiplying the estimated EIRP or estimated EIS at the mapped sampling grid point 341′ by this ratio.

[0038] The corrected TRP or corrected TIS of the AUT 130 is then calculated based on the scaled EIRP or scaled EIS, respectively, at the second sampling grid points 341-348 on the second closed surface geometry 340. The corrected TRP and corrected TIS are accurate even though the AUT 130 is offset from the quiet zone center P0 because the offset OS is taken into account.

[0039] 4 is a simplified flow diagram illustrating a method for determining a corrected TRP or TIS of an AUT in a near-field test chamber according to a representative embodiment. As discussed above, the AUT includes an antenna array having an array phase center offset from the center of the quiet zone of the near-field test chamber by a known offset. For example, the geometric center of the DUT can be aligned with the center of the quiet zone, while one or more AUTs are positioned at outer portions of the DUT offset from the center of the quiet zone. The method can be performed, for example, by a processing unit 250 in which the method steps are provided as instructions stored in memory 256 executable by a processor device 255.

[0040] 4, in block S411, EIRP or EIS measurements of the AUT are taken at a plurality of first sampling grid points (e.g., first sampling grid points 311-318), respectively. These first sampling grid points are arranged on a first closed surface geometry (e.g., first closed surface geometry 310) having a corresponding first center located at the quiet zone center (center of rotation) of a quiet zone within the test chamber.

[0041] In block S412, a plurality of second sampling grid points disposed on a second closed surface geometry (e.g., second closed surface geometry 340) are respectively mapped to the first closed surface geometry to obtain corresponding mapped sampling grid points on the first closed surface geometry. The second closed surface geometry has a second center located at the array topological center of the AUT. In one embodiment, the first closed surface geometry and the second closed surface geometry include a first sphere and a second sphere, respectively.

[0042] In one embodiment, mapping the second sampling grid points from the second closed surface geometry to the first closed surface geometry includes determining an angle between a direction of offset from the quiet zone center of the test chamber and a line extending from the array phase center of the AUT through each of the second sampling grid points onto the first sphere, and placing each mapped sampling grid point on the first sphere where the line intersects with the first sphere at the determined angle. When the line from the array phase center of the AUT through the second sampling grid point intersects with the first sphere at two points, the mapped sampling grid point is located on the first sphere at the point that is closer to the corresponding second sampling grid point.

[0043] In block S413, the EIRP or EIS of the AUT at the mapped sampling grid points is determined using the EIRP or EIS measurements made in block S411 at one or more first sampling grid points on the first closed surface geometry. For example, the estimated EIRP or EIS of each mapped sampling grid point may be estimated by interpolating between the EIRP or EIS of two immediately adjacent first sampling grid points on either side of the mapped sampling grid point. Of course, any other suitable type of interpolation may be incorporated without departing from the scope of the present teachings.

[0044] The estimated EIRP or estimated EIS at the mapped sampling grid points is scaled in block S414 to obtain scaled EIRP or scaled EIS at second sampling grid points, respectively, on the second closed surface geometric shape. As described above, scaling compensates for path loss errors. In one embodiment, scaling the estimated EIRP or estimated EIS at the mapped sampling grid points includes determining a ratio between (i) each of the first distances between the array phase center and each of the mapped sampling grid points and (ii) each of the second distances between the quiet zone center and each of the mapped sampling grid points. Each of these ratios is multiplied by the estimated EIRP or estimated EIS at the mapped sampling grid points, respectively. Scaling the estimated EIRP or estimated EIS at the mapped sampling grid points may further include compensating for off-axis differences in antenna gain of a probe antenna used to make EIRP or EIS measurements of the AUT at each of the first sampling grid points. For different directions between the probe antenna and the first sampling grid point, the antenna gain is different.

[0045] In block S415, a corrected TRP or a corrected TIS of the AUT is calculated based on the scaled EIRP or the scaled EIS, respectively, at second sampling grid points on the second closed surface geometry. The corrected TRP can be calculated using the scaled EIRP in equations (1) and (3) above. The corrected TIS can be calculated using the scaled EIS in equations (2) and (4) above. The corrected TRP and the corrected TIS have improved accuracy over the TRP and TIS, respectively, measured directly from the first sampling grid points.

[0046] While the invention has been illustrated and described in detail in the drawings and the foregoing description, such illustrations and descriptions are to be considered illustrative or exemplary and not restrictive, and the invention is not limited to the disclosed embodiments. Those skilled in the art will understand and can make other variations to the disclosed embodiments when practicing the claimed invention, from a study of the drawings, the disclosure and the appended claims. In the claims, the word "comprising" does not exclude other elements or steps, and the indefinite article "a" or "an" does not exclude a plurality. The mere recitation of certain measures in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

[0047] Aspects of the present invention may be embodied as an apparatus, a method, or a computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, microcode, etc.), or an embodiment combining software and hardware aspects, all of which may be collectively referred to herein as a "circuit," "module," or "system." Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer-readable medium(s) having computer-executable code embodied therein.

[0048] While exemplary embodiments are disclosed herein, those skilled in the art will recognize that many variations are possible in accordance with the present teachings and are within the scope of the appended claims. Accordingly, the invention is not to be limited except as by the scope of the appended claims. The claims as originally filed are as follows: Claim 1: 1. A method for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having a phase center offset from a center of rotation of the test chamber; performing equivalent isotropic radiated power (EIRP) or effective isotropic sensitivity (EIS) measurements of the AUT at a plurality of first sampling grid points, the plurality of first sampling grid points being arranged according to a constant step size or a constant density on a first closed surface geometric shape centered on the center of rotation of the test chamber; mapping a plurality of second sampling grid points onto the first closed surface geometric shape to obtain a plurality of mapped sampling grid points respectively on the first closed surface geometric shape, the plurality of second sampling grid points being disposed on the second closed surface geometric shape centered on the phase center of the AUT; determining an estimated EIRP or an estimated EIS of the AUT at the plurality of mapped sampling grid points using the EIRP or the EIS measurement at one or more first sampling grid points of the plurality of first sampling grid points; scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points to obtain a scaled EIRP or the estimated EIS or a scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively, wherein the scaling compensates for path loss error; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively; A method comprising: Claim 2: 2. The method of claim 1, wherein determining the estimated EIRP or the estimated EIS of the AUT at the plurality of mapped sampling grid points comprises interpolating between two of the EIRP or EIS measurements. Claim 3: The method of claim 1 , wherein the first closed surface geometric shape and the second closed surface geometric shape comprise a first sphere and a second sphere, respectively. Claim 4: Mapping the plurality of second sampling grid points onto the first geometric shape includes: determining an angle between a direction of the offset from the center of rotation of the test chamber and a line from the phase center of the AUT to the first sphere that passes through each of the plurality of second sampling grid points; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first sphere such that the line intersects the first sphere at the determined angle for each of the plurality of second sampling grid points; 4. The method of claim 3, comprising: Claim 5: 5. The method of claim 4, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first sphere at two points, the mapped sampling grid point is located on the first sphere at the point of the two points that is closer to the second sampling grid point. Claim 6: Scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points may include: determining a ratio between (i) a respective first distance between the phase center and each of the plurality of mapped sampling grid points and (ii) a respective second distance between the center of rotation and each of the plurality of mapped sampling grid points; multiplying each of the ratios by the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points, respectively, to obtain the scaled EIRP or the scaled EIS; 2. The method of claim 1, comprising: Claim 7: Scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points may include: Compensating for off-axis differences in antenna gain of probe antennas used to perform the EIRP or EIS measurements of the AUT at the plurality of first sampling grid points, respectively. Further comprising: The method of claim 6 , wherein the antenna gain is different for different directions between the probe antenna and the first plurality of sampling grid points. Claim 8: 1. A system for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having a phase center offset from a center of rotation of the test chamber; at least one probe antenna configured to make equivalent isotropically radiated power (EIRP) or effective isotropic sensitivity (EIS) measurements of the AUT at a plurality of first sampling grid points, the plurality of first sampling grid points being arranged on a first closed surface geometric shape having a first center about the center of rotation of the test chamber; a processor device; A memory for storing instructions Equipped with The instructions, when executed by the processor device, mapping a plurality of second sampling grid points onto the first closed surface geometric shape to obtain a plurality of mapped sampling grid points respectively on the first closed surface geometric shape, the plurality of second sampling grid points being disposed on the second closed surface geometric shape centered on the phase center of the AUT; determining an estimated EIRP or an estimated EIS of the AUT at the plurality of mapped sampling grid points using the EIRP or the EIS measurement at one or more first sampling grid points of the plurality of first sampling grid points; scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points to obtain a scaled EIRP or the estimated EIS or a scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively, wherein the scaling compensates for path loss error; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively; A system that causes the processor device to perform the above. Claim 9: 9. The system of claim 8, wherein the first sampling grid points are placed on the first closed surface geometry using a constant step size or a constant density. Claim 10: 9. The system of claim 8, wherein the instructions cause the processor device to determine the estimated EIRP or the estimated EIS of the AUT at the plurality of mapped sampling grid points by interpolating between two of the EIRP or EIS measurements. Claim 11: 9. The system of claim 8, wherein the first closed surface geometric shape and the second closed surface geometric shape comprise a first sphere and a second sphere, respectively. Claim 12: The instruction: determining an angle between a direction of the offset from the center of rotation of the test chamber and a line from the phase center of the AUT to the first closed surface geometric shape that passes through each of the plurality of second sampling grid points; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first closed surface geometric shape such that the line intersects the first closed surface geometric shape at the determined angle for each of the plurality of second sampling grid points; 9. The system of claim 8, wherein the processor device maps the plurality of second sampling grid points onto the first geometric shape by: Claim 13: 13. The system of claim 12, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first closed surface geometry at two points, the mapped sampling grid point is located on the first closed surface geometry at the point closer to the second sampling grid point of the two points. Claim 14: The instruction: determining a ratio between (i) a respective first distance between the phase center and each of the plurality of mapped sampling grid points and (ii) a respective second distance between the center of rotation and each of the plurality of mapped sampling grid points; multiplying each of the ratios by the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points, respectively, to obtain the scaled EIRP or the scaled EIS; 9. The system of claim 8, further comprising: causing the processor device to scale the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points by: Claim 15: 15. The system of claim 14, wherein the instructions further cause the processor device to scale the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points by compensating for off-axis differences in antenna gain of a probe antenna used to make the EIRP or EIS measurements of the AUT, respectively, at the plurality of first sampling grid points, wherein the antenna gain is different for different directions between the probe antenna and the plurality of first sampling grid points. Claim 16: 1. A non-transitory computer-readable medium storing instructions for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having an array phase center offset by an offset distance from a quiet zone center of a quiet zone in the test chamber, the instructions, when executed by a processor device, providing first sampling grid points arranged according to a fixed step size or a fixed density on a first sphere centered on the quiet zone center, wherein an equivalent isotropic radiated power (EIRP) or an effective isotropic sensitivity (EIS) of the AUT is measured at each of the first sampling grid points in the near field of the test chamber; providing second sampling grid points arbitrarily positioned on a second sphere centered at the array phase center; mapping each second sampling grid point onto the first sphere using the offset distance and angular information from the array phase center to each of the second sampling grid points; for each of the mapped second sampling grid points on the first sphere, determining an interpolated EIRP or an interpolated EIS at each of the mapped second sampling grid points by interpolating between the measured EIRP or the measured EIS at selected first sampling grid points; scaling the interpolated EIRP or the interpolated EIS at each of the mapped second sampling grid points on the first sphere to obtain a scaled EIRP or a scaled EIS at each of the second sampling grid points on the second sphere, respectively; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the second sampling grid points on the second sphere, respectively; A non-transitory computer-readable medium that causes the processor device to execute a process including: Claim 17: 17. The non-transitory computer-readable medium of claim 16, wherein scaling the interpolated EIRP or the interpolated EIS at each of the mapped second sampling grid points comprises determining a ratio of (i) a first distance between the array phase center and the mapped second sampling grid point on the first sphere and (ii) a second distance between the quiet zone center and the mapped second sampling grid point on the first sphere. Claim 18: 17. The non-transitory computer-readable medium of claim 16, wherein, for each of the mapped second sampling grid points, interpolating between the measured EIRP or the measured EIS at the selected first sampling grid points comprises performing a linear fit between the first sampling grid points immediately adjacent to each of the mapped second sampling grid points on the first sphere. Claim 19: Mapping each second sampling grid point onto the first sphere includes: determining an angle between a direction of the offset from the quiet zone center and a line from the array phase center of the AUT through each of the plurality of second sampling grid points to the first sphere; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first sphere such that the line intersects the first sphere at the determined angle for each of the plurality of second sampling grid points; 17. The non-transitory computer-readable medium of claim 16, comprising: Claim 20: 20. The non-transitory computer-readable medium of claim 19, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first sphere at two points, the mapped sampling grid point is located on the first sphere at the point of the two that is closer to the second sampling grid point.

Claims

1. 1. A method for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having a phase center offset from a center of rotation of the test chamber; performing an equivalent isotropic radiated power (EIRP) measurement or an effective isotropic sensitivity (EIS) measurement of the AUT at each of a plurality of first sampling grid points, the plurality of first sampling grid points being arranged according to a fixed step size or a fixed density on a first closed surface geometry centered about the center of rotation of the test chamber; mapping a plurality of second sampling grid points onto the first closed surface geometry to obtain a plurality of mapped sampling grid points respectively on the first closed surface geometry, the plurality of second sampling grid points being disposed on a second closed surface geometry centered on the phase center of the AUT; determining an estimated EIRP or an estimated EIS of the AUT at the plurality of mapped sampling grid points using the EIRP or EIS measurements at one or more first sampling grid points of the plurality of first sampling grid points; scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points to obtain a scaled EIRP or the estimated EIS or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively, wherein the scaling compensates for path loss error; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometry, respectively; A method comprising:

2. 2. The method of claim 1, wherein determining the estimated EIRP or the estimated EIS of the AUT at the plurality of mapped sampling grid points comprises interpolating between two of the EIRP measurements or two of the EIS measurements.

3. The method of claim 1 , wherein the first closed surface geometric shape comprises a first sphere and the second closed surface geometric shape comprises a second sphere.

4. Mapping the plurality of second sampling grid points onto the first geometric shape includes: determining an angle between a direction of the offset from the center of rotation of the test chamber and a line from the phase center of the AUT to the first sphere that passes through each of the plurality of second sampling grid points; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first sphere such that the line intersects the first sphere at the determined angle for each of the plurality of second sampling grid points; The method of claim 3, comprising:

5. 5. The method of claim 4, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first sphere at two points, the mapped sampling grid point is located on the first sphere at the point of the two points that is closer to the second sampling grid point.

6. Scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points includes: determining a ratio between (i) a respective first distance between the phase center and each of the plurality of mapped sampling grid points and (ii) a respective second distance between the center of rotation and each of the plurality of mapped sampling grid points; multiplying each of the ratios by the estimated EIRP or the estimated EIS, respectively, at the plurality of mapped sampling grid points to obtain the scaled EIRP or the scaled EIS, respectively; The method of claim 1 , comprising:

7. Scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points includes: and compensating for off-axis differences in antenna gain of probe antennas used to perform the EIRP or EIS measurements of the AUT at the plurality of first sampling grid points. Further comprising: The method of claim 6 , wherein the antenna gain is different for different directions between the probe antenna and the first plurality of sampling grid points.

8. 1. A system for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having a phase center offset from a center of rotation of the test chamber; at least one probe antenna configured to make equivalent isotropically radiated power (EIRP) or effective isotropic sensitivity (EIS) measurements of the AUT at a plurality of first sampling grid points, the plurality of first sampling grid points being arranged on a first closed surface geometric shape having a first center about the center of rotation of the test chamber; a processor device; A memory for storing instructions Equipped with The instructions, when executed by the processor device, mapping a plurality of second sampling grid points onto the first closed surface geometric shape to obtain a plurality of mapped sampling grid points respectively on the first closed surface geometric shape, the plurality of second sampling grid points being disposed on the second closed surface geometric shape centered on the phase center of the AUT; determining an estimated EIRP or an estimated EIS of the AUT at the plurality of mapped sampling grid points using the EIRP or EIS measurements at one or more first sampling grid points of the plurality of first sampling grid points; scaling the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points to obtain a scaled EIRP or the estimated EIS or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometric shape, respectively, wherein the scaling compensates for path loss error; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the plurality of second sampling grid points on the second closed surface geometry, respectively; A system that causes the processor device to perform the above.

9. The system of claim 8 , wherein the first sampling grid points are placed on the first closed surface geometry using a constant step size or a constant density.

10. 9. The system of claim 8, wherein the instructions cause the processor device to determine the estimated EIRP or the estimated EIS of the AUT at the plurality of mapped sampling grid points by interpolating between two of the EIRP measurements or two of the EIS measurements.

11. The system of claim 8 , wherein the first closed surface geometric shape comprises a first sphere and the second closed surface geometric shape comprises a second sphere.

12. The instruction: determining an angle between a direction of the offset from the center of rotation of the test chamber and a line from the phase center of the AUT to the first closed surface geometric shape that passes through each of the plurality of second sampling grid points; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first closed surface geometric shape such that the line intersects the first closed surface geometric shape at the determined angle for each of the plurality of second sampling grid points; 9. The system of claim 8, wherein the processor device maps the plurality of second sampling grid points onto the first geometric shape by:

13. 13. The system of claim 12, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first closed surface geometry at two points, the mapped sampling grid point is located on the first closed surface geometry at the closer of the two points to the second sampling grid point.

14. The instruction: determining a ratio between (i) a respective first distance between the phase center and each of the plurality of mapped sampling grid points and (ii) a respective second distance between the center of rotation and each of the plurality of mapped sampling grid points; multiplying each of the ratios by the estimated EIRP or the estimated EIS, respectively, at the plurality of mapped sampling grid points to obtain the scaled EIRP or the scaled EIS, respectively; 9. The system of claim 8, further comprising: causing the processor device to scale the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points by:

15. 15. The system of claim 14, wherein the instructions further cause the processor device to scale the estimated EIRP or the estimated EIS at the plurality of mapped sampling grid points by compensating for off-axis differences in antenna gain of a probe antenna used to make the EIRP or EIS measurements of the AUT, respectively, at the plurality of first sampling grid points, wherein the antenna gain is different when directions between the probe antenna and the plurality of first sampling grid points are different, respectively.

16. 1. A non-transitory computer-readable medium storing instructions for determining a corrected total radiated power (TRP) or a corrected total isotropic sensitivity (TIS) of an antenna under test (AUT) in a near-field test chamber, the AUT having an array phase center offset by an offset distance from a quiet zone center of a quiet zone in the test chamber, the instructions, when executed by a processor device, providing first sampling grid points arranged according to a fixed step size or a fixed density on a first sphere centered on the quiet zone center, wherein an equivalent isotropic radiated power (EIRP) or an effective isotropic sensitivity (EIS) of the AUT is measured at each of the first sampling grid points in the near field of the test chamber; providing second sampling grid points arbitrarily positioned on a second sphere centered on the array phase center; mapping each second sampling grid point onto the first sphere using the offset distance and angular information from the array phase center to each of the second sampling grid points; for each of the mapped second sampling grid points on the first sphere, determining an interpolated EIRP or an interpolated EIS at each of the mapped second sampling grid points by interpolating between the measured EIRP or the measured EIS at selected first sampling grid points, respectively; scaling the interpolated EIRP or the interpolated EIS at each of the mapped second sampling grid points on the first sphere to obtain a scaled EIRP or a scaled EIS at each of the second sampling grid points on the second sphere, respectively; calculating the corrected TRP or the corrected TIS of the AUT based on the scaled EIRP or the scaled EIS at the second sampling grid points on the second sphere, respectively; A non-transitory computer-readable medium that causes the processor device to execute a process including:

17. 17. The non-transitory computer-readable medium of claim 16, wherein scaling the interpolated EIRP or the interpolated EIS at each of the mapped second sampling grid points comprises determining a ratio of (i) a first distance between the array phase center and the mapped second sampling grid point on the first sphere and (ii) a second distance between the quiet zone center and the mapped second sampling grid point on the first sphere.

18. Mapping each second sampling grid point onto the first sphere includes: determining an angle between a direction of the offset from the quiet zone center and a line from the array phase center of the AUT through each of the plurality of second sampling grid points to the first sphere; locating each mapped sampling grid point of the plurality of mapped sampling grid points on the first sphere such that the line intersects the first sphere at the determined angle for each of the plurality of second sampling grid points; 20. The non-transitory computer-readable medium of claim 16, comprising:

19. 20. The non-transitory computer-readable medium of claim 18, wherein when the line from the phase center of the AUT through a second sampling grid point intersects the first sphere at two points, the mapped sampling grid point is located on the first sphere at the point of the two that is closer to the second sampling grid point.

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