Phased Array Calibration based on Rotating-Element Harmonic Electric-Field Vector with Time Modulation

TL;DR

This work tackles amplitude–phase imbalance in phased arrays by introducing the Rotating-element Harmonic Electric-field Vector (RHEV), a time-modulation–based calibration method that uses relative delays between two 1-bit phase shifters to synthesize high-resolution phase shifts at the +1st harmonic. Amplitude-ambiguity is resolved by sequentially modulating individual elements so the +1st-harmonic power scales linearly with the modulated element's amplitude, enabling inter-element calibration independent of array size. The authors provide a detailed mathematical formulation, extensive numerical simulations, and both in-channel and OTA experiments confirming sub-degree phase calibration accuracy and sub-dB amplitude calibration, with an equivalent 8-bit phase control realized via time-delay discretization (N_{be} up to 8). The approach demonstrates robustness to phase-shifter errors and preserves calibration performance as the array grows, offering practical benefits for real-world phased-array systems and 6G/OTA deployments.

Abstract

Calibration is crucial for ensuring the performance of phased array since amplitude-phase imbalance between elements results in significant performance degradation. While amplitude-only calibration methods offer advantages when phase measurements are impractical, conventional approaches face two key challenges: they typically require high-resolution phase shifters and remain susceptible to phase errors inherent in these components. To overcome these limitations, we propose a Rotating element Harmonic Electric-field Vector (RHEV) strategy, which enables precise calibration through time modulation principles. The proposed technique functions as follows. Two 1-bit phase shifters are periodically phase-switched at the same frequency, each generating corresponding harmonics. By adjusting the relative delay between their modulation timings, the phase difference between the $+1$st harmonics produced by the two elements can be precisely controlled, utilizing the time-shift property of the Fourier transform. Furthermore, the +1st harmonic generated by sequential modulation of individual elements exhibits a linear relationship with the amplitude of the modulated element, enabling amplitude ambiguity resolution. The proposed RHEV-based calibration method generates phase shifts through relative timing delays rather than physical phase shifter adjustments, rendering it less susceptible to phase shift errors. Additionally, since the calibration process exclusively utilizes the $+1$st harmonic, which is produced solely by the modulated unit, the method demonstrates consistent performance regardless of array size. Extensive numerical simulations, practical in-channel and over-the-air (OTA) calibration experiments demonstrate the effectiveness and distinct advantages of the proposed method.

Figures (22)

• Figure 1: Flowchart diagram of the proposed calibration method.
• Figure 2: (a) The periodic phase modulation functions of the reference element and the current element, (b) the power of the $+1$st order harmonics varies with the relative time delay.
• Figure 3: The schematic of the synthesized vector ${{G}_{n}}$, when the amplitude ratio of the channel under test to the reference channel $\left| \Delta {{A}_{n}} \right|$ is greater than 1 in cases (a-c) or less than 1 in cases (d-f), and the phase difference in the $+1$ harmonic introduced by the time delay and the phase difference between the two channels satisfy (a,d) $2\pi \eta =0$, (b,e) $2\pi \eta =\Delta {{\varphi }_{n}}$, (c,f) $2\pi \eta =\pi +\Delta {{\varphi }_{n}}$ respectively.
• Figure 4: Variation of the power for the $+1$st order harmonics with the relative time delay between the modulation timing of the $2$nd element and the reference element. And the power spectrum of the array-synthesized signal when the $+1$th harmonic power reaches its maximum and minimum values (inset figures).
• Figure 5: Spectrum of the signal received by the AUT when only one element is periodically phase modulated: the reference element (black solid line), and the second element (red dashed line). The preset value of the amplitude ratio between the second element to that of the reference element is -1.39 dB.