Root Sum of the Squares

Description

Root Sum of the Squares (RRS) is a mathematical and statistical technique specified within the 3GPP physical layer testing framework, primarily documented in TS 36.143 for LTE. It is a critical component in Over-the-Air (OTA) testing methodologies, specifically for evaluating the Total Radiated Power (TRP) of User Equipment (UE). The method is employed to combine the power measurements taken from multiple spatial points around the device under test in an anechoic chamber. The fundamental principle involves measuring the power radiated by the UE's antenna(s) across a spherical grid of points, capturing the three-dimensional radiation pattern. These individual point measurements, typically expressed in linear power units (e.g., milliwatts), are first squared, then summed across all measurement points. The final TRP value is derived by taking the square root of this sum, effectively providing the root mean square (RMS) of the radiated power over the entire sphere. This process accounts for the vectorial nature of electromagnetic fields and provides a single, comprehensive metric for the device's total output power, independent of its directivity.

The application of RRS is crucial for compliance with regulatory limits on maximum transmit power and for validating baseband and RF performance. In testing, a UE is placed on a positioning system that rotates it through azimuth and elevation angles while a measurement antenna connected to a receiver (like a spectrum analyzer or vector signal analyzer) records the received power at each point. The RRS calculation integrates these discrete measurements to approximate the continuous integral of power over the full 4π steradian solid angle. This method is more accurate than simply taking a peak or average of a few measurements, as it properly weights each measurement point according to the solid angle it represents in the spherical coordinate system. The resulting TRP figure is a key performance indicator used to verify that the UE's power amplifier and antenna system are functioning correctly and efficiently, impacting both the device's radio range and its battery consumption.

Beyond basic TRP, the RRS principle underpins other OTA figures of merit like Total Isotropic Sensitivity (TIS). Its standardization in 3GPP ensures consistency and reproducibility of test results across different laboratories and equipment manufacturers. The methodology is designed to be robust against measurement noise and the discretization errors inherent in sampling a continuous radiation pattern. By providing a standardized calculation, RRS enables fair and accurate comparisons between different UE models and ensures that devices meet the stringent performance requirements set by network operators and regulatory bodies worldwide, forming a foundational element of device certification and type approval processes.

Purpose & Motivation

The purpose of standardizing the Root Sum of the Squares (RRS) method within 3GPP was to establish a uniform, accurate, and repeatable procedure for evaluating the total radiated power of mobile devices in an OTA environment. Prior to such standardization, different test houses and manufacturers might have used varying methodologies (e.g., simple averaging, peak detection, or different numerical integration techniques) to compute TRP from spherical measurements. This lack of consistency led to potential discrepancies in test results, making it difficult to objectively compare device performance or ensure compliance with regulatory emission limits. The RRS method was introduced to solve this problem by providing a definitive mathematical formula that all parties must follow, thereby guaranteeing that a TRP value reported for a UE is calculated in exactly the same way regardless of where the test is performed.

Historically, as mobile devices became more complex with multiple antennas (e.g., for MIMO and diversity reception) and operated across an expanding range of frequency bands, traditional conductive testing methods (connecting a cable directly to the antenna port) became insufficient. Conductive testing ignores the performance of the antenna itself and the effects of the device's physical casing and user's hand (the "handset effect"). OTA testing emerged as the necessary solution to evaluate the device as a whole system in its operational state. The RRS calculation is the cornerstone of this OTA power measurement. It addresses the fundamental challenge of converting a finite set of discrete point measurements into a single, accurate estimate of the total power radiated in all directions. Its creation was motivated by the need for a physics-correct and statistically sound integration technique that is both implementable in automated test systems and unambiguous in its interpretation.

By mandating RRS in specifications like TS 36.143, 3GPP ensured that the industry moved towards a common testing language. This not only streamlined the device certification process but also gave network operators confidence in the reported performance metrics of the UEs they approve for their networks. Accurate TRP, derived via RRS, directly correlates to a device's effective coverage and link budget, impacting overall network planning and quality of service. Therefore, the RRS method exists to eliminate measurement ambiguity, foster interoperability in testing, and ultimately ensure that end-user devices perform reliably in real-world wireless environments.

Key Features

• Standardized mathematical formula for calculating Total Radiated Power (TRP) from spherical measurements
• Integrates discrete power measurements over a full 4π steradian solid angle
• Provides a root mean square (RMS) value of radiated power, accounting for spatial distribution
• Foundation for Over-the-Air (OTA) compliance and performance testing
• Ensures measurement consistency and reproducibility across different test laboratories
• Applicable to testing of multi-antenna systems and MIMO configurations

Evolution Across Releases

Defining Specifications