Beamforming Gain with Nonideal Phase Shifters

TL;DR

This work develops a universal framework to quantify beamforming gain when phase shifters are nonideal, applicable to transmitter, receiver, and RIS configurations. It proves a fundamental bound: $g_{\mathcal{W}} \ge \frac{\mathrm{perimeter}(\mathrm{Conv}\,\mathcal{W})}{2\pi} \; g_{\text{ideal}}$, linking the gain loss to the geometry of the feasible coefficient set via convex hull perimeter; the shortfall in dB is $20 \log_{10}\left( \frac{\mathrm{perimeter}(\mathrm{Conv}\,\mathcal{W})}{2\pi} \right)$. The authors provide two proofs—the main derivation using the support function and a geometric Minkowski-sum argument—and offer a tighter finite-$N$ refinement for polygonal $\mathrm{Conv}\,\mathcal{W}$. In i.i.d. fading, the shortfall concentrates to this bound as $N$ grows, establishing a fundamental limit that informs practical phase-shifter design and RIS-enabled systems.

Abstract

This research sets forth a universal framework to characterize the beamforming gain achievable with arbitrarily nonideal phase shifters. Precisely, the maximum possible shortfall relative to the gain attainable with ideal phase shifters is established. Such shortfall is shown to be fundamentally determined by the perimeter of the convex hull of the set of feasible beamforming coefficients on the complex plane. This result holds regardless of whether the beamforming is at the transmitter, at the receiver, or at a reconfigurable intelligent surface. In i.i.d. fading channels, the shortfall hardens to the maximum possible shortfall as the number of antennas grows.

Figures (4)

• Figure 1: Response of $3$-bit phase shifter. The dots are the elements of $\mathcal{W}$ while the shaded region is their convex hull. The dashed circle is $\mathcal{W}_{\text{ideal}}$. As the perimeter of the heptagonal convex hull equals 5.01, there exists some ${\boldsymbol{w}}$ attaining $5.01/2\pi \approx 80\%$ of $g_{\text{ideal}}$, which corresponds to a $2$-dB shortfall in beamforming gain. Part of this shortfall is the power loss caused by the magnitude of the elements of $\mathcal{W}$ being less than unity.
• Figure 2: Width of a convex set in the direction specified by $\theta$.
• Figure 3: Visual comparison of the constants in \ref{['crude_constant']} and \ref{['best_constant']}, which respectively correspond to the perimeter of the circle in light gray and to the perimeter of the regular $M$-gon in charcoal gray, both divided by $2\pi$. As a reference, the unit circle is shown as a dashed line.
• Figure 4: Construction of a convex polygon whose perimeter is at least that of the original convex polygon.

Theorems & Definitions (25)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4