We consider the joint estimation of multipath channels obtained with a set of receiving antennas and uniformly probed in the frequency domain. This scenario fits most of the modern outdoor communication protocols for mobile access (ETSI Std. 125 913) or digital broadcasting (ETSI Std. 300 744) among others. Such channels verify a sparse common support (SCS) property which was used in the work of Barbotin (2012) to propose a finite rate of innovation (FRI)-based sampling and estimation algorithm. In this paper, we improve the robustness and computational complexity aspects of this algorithm. The method is based on projection in Krylov subspaces to improve complexity and a new criterion called the partial effective rank (PER) to estimate the level of sparsity to gain robustness. If $P$ antennas measure a $K$-multipath channel with $N$ uniformly sampled measurements per channel, the algorithm possesses an ${\cal O}(KPN\log N)$ complexity and an ${\cal O}(KPN)$ memory footprint instead of ${\cal O}(PN^{3})$ and ${\cal O}(PN^{2})$ for the direct implementation, making it suitable for $K\ll N$. The sparsity is estimated online based on the PER, and the algorithm therefore has a sense of introspection being able to relinquish sparsity if it is lacking. The estimation performances are tested on field measurements with synthetic additive white Gaussian noise, and the proposed algorithm outperforms nonsparse reconstruction in the medium to low- signal-to-noise ratio range ($\leq$ 0 dB), increasing the rate of successful symbol decoding by 1/10 in average, and 1/3 in the best case. The experiments also show that the algorithm does not perform worse than a nonsparse estimation algorithm in nonsparse operating conditions, since it may fall-back to it if the PER criterion does not detect a sufficient level of sparsity. The algorithm is also tested against a method assuming a “discrete” sparsity model as in compressed sensing. The conducted test indicates a trade-off between speed and accuracy.