A Direct Calibration Algorithm for ADC Interleaving

TL;DR

The paper tackles interleaving-induced mismatches in ADC arrays, specifically phase delay $\tau_p$, gain $\gamma_p$, and offset $\alpha_p$, which degrade high-bandwidth signal reconstruction. It introduces a direct, Fourier-based calibration method that uses a sinusoidal reference $f(t) = A\cos(\Omega t)$ to extract mismatch parameters from the frequency-domain samples, yielding closed-form relationships between the observed Fourier components and the unknowns. Key results include $\tilde{\mathcal{R}}_m = \sum_{p=0}^{P-1} e^{-2\pi i m p / P} \Gamma_p$ with $\Gamma_p = \gamma_p e^{i \Omega \tau_p}$ and $\tilde{\mathcal{A}}_m = \sum_{p=0}^{P-1} e^{-2\pi i m p / P} \alpha_p$, enabling recovery via inverse DFT. The proposed algorithm operates with complexity $\mathcal{O}(N \min(\log N, P))$, is easily implementable in hardware or software, and is scalable to large numbers of interleaved ADCs, making it well-suited for real-time calibration in VLBI and related high-throughput sensing domains.

Abstract

We introduce a novel direct calibration algorithm to address phase delay, gain, and offset mismatches in Analog-to-Digital Converter (ADC) time interleaving systems. These mismatches, common in high-speed data acquisition, degrade system performance and signal integrity, particularly in applications such as radio astronomy and very long baseline interferometry (VLBI). Our proposed algorithm uses a sinusoidal reference signal and Fourier analysis to isolate and correct each type of mismatch, providing a computationally efficient solution. Extensive numerical simulations validate the algorithm's effectiveness and demonstrate its ability to significantly enhance signal reconstruction accuracy compared to existing methods. This work provides a robust and scalable solution for maintaining signal fidelity in interleaved ADC systems and has broad applications in fields that require high-speed data acquisition.

Figures (3)

• Figure 1: Illustration of the impact of mismatches in a 4-ADC interleaving system. The black solid curve in the shows the input signal. Open circles mark the ideal sampling points for a mismatch-free system, while filled circles show the actual sampling locations when mismatches are present. The top panel shows the ideal case so the black curve is $\cos(t)$. We introduce a phase delay mismatch, which samples the signal in a non-uniform fashion. The second panel highlights the effect of a phase delay mismatch in $\mathrm{ADC}_1$, which results in a distortion in the reconstructed signal. The third panel adds a gain mismatch in $\mathrm{ADC}_2$, further modifying the amplitude of the sampled signal. The fourth panel introduces an offset mismatch in $\mathrm{ADC}_3$, demonstrating how combined mismatches lead to significant distortion. Accurate calibration of these mismatches is essential for high-fidelity signal reconstruction in interleaved ADC systems.
• Figure 2: (Left) Frequency spectrum of the signal before applying the proposed direct calibration algorithm, illustrating the effects of phase delay, gain, and offset mismatches in a 4-ADC interleaving system. Dashed and dotted vertical lines in different colors mark the frequencies at $\Omega + m \Omega_\Delta$ (see equation \ref{['eq:classI']}) and $m' \Omega_\Delta$ (see equation \ref{['eq:classII']}), respectively. The top panel shows the full spectrum across a broad range of angular frequencies, while the bottom panel provides a zoomed-in view of a specific band, clearly revealing spurious peaks primarily caused by phase delay mismatches. (Right) Same as the left, but with additive noise. The noise level is comparable to that of the reference signal. Although the spurious peaks remain visible, their amplitudes are partially masked by the noise.
• Figure 3: Convergence properties of the proposed direct calibration algorithm as a function of the number of samples. (Left column) The number of periods is fixed at $N_T = 256$, while the number of samples per period $N_\delta$ is varied. (Right column) The number of samples per period is fixed at $N_\delta = 256$, while the number of periods $N_T$ is varied. Colored circles indicate the RMS errors in recovering each mismatch parameter: phase delay $\tau_p$, gain $\gamma_p$, and offset $\alpha_p$. The dotted and dash-dotted lines show reference slopes for ideal scaling behavior, allowing comparison to theoretical convergence rates. These plots demonstrate that different mismatch parameters exhibit distinct but predicted convergence characteristics. See section \ref{['sec:noise']} for details.