Phase Retrieval for Sparse Signals

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurement. We first investigate the minimal number of measurements for the success of the recovery of sparse signals without the phase information. We completely settle the minimality question for the real case and give a lower bound for the complex case. We then study the recovery performance of the $\ell_1$ minimization. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of $\ell_1$ minimization for $k$-sparse phase retrievable.Comment: 14 page

Evaluation of Factors Affecting Earth Pressures on Buried Box Culverts

Factors affecting the earth pressures acting on buried box culverts under deep embankments were evaluated by field instrumentation and numerical analyses. Two instrumented cast-in-place concrete culverts were designed and constructed differently. Long term earth pressures under constant embankment height were observed after the completion of construction. Both observed earth pressures and those predicted by numerical analyses were compared with the current AASHTO earth pressure recommendations, as well as the AASHTO design pressures in effect at the time a failed box culvert was designed in the mid 1970s. Field measurements suggested that the previous AAHSTO design pressure (1977, 12th edition) significantly underestimated both vertical and horizontal earth pressures, whereas the current AASHTO (1996, 16th edition) provides more appropriate simplified earth pressures. Both measured and predicted earth pressures indicated that the level of compactive effort had a significant influence on the earth pressure distribution, especially the horizontal earth pressure acting on the culvert wall. A parametric finite element study suggested that the stiffness of the gravel backfill beside the culvert, which is dependent on the degree of compaction, had the greatest influence on the magnitude and distribution of earth pressure. An additional parametric study suggested that the modulus of the soil below the culvert also has a significant effect on the horizontal earth pressures. Dynamic horizontal earth pressures induced by different construction equipment operating close to the structure were also recorded. The measurements suggested that the dynamic strain in the structure in response to the maximum transient loading (about 70 kPa) was small and had negligible effect on the culvert. High residual compaction earth pressures measured after compaction were found to decrease rapidly with time to reach a steady value under constant embankment height. Analytical evaluation of the culvert orientation with respect to the embankment alignment suggested that the largest horizontal earth pressure acting on the culvert wall occurs when the culvert alignment is perpendicular to the alignment of embankment. The investigation of factors affecting the earth pressures on cast-in-place box culverts suggested that the design pressures are not only dependent upon the height of the embankment, but the relative stiffness of the surrounding materials is also important