When continuous wave (CW) interference is introduced into the Costas loop, the loop may lock onto the interference signal if the amplitude of the interference reaches a certain threshold, or if the carrier frequency of the interference signal is very close to that of the desired signal. This phenomenon in the Costas loop has not been thoroughly investigated to date. Although there have been studies on this phenomenon in Phase-Locked Loop (PLL), their results cannot be directly applied to the variant of PLL, Costas loop. This article investigates the necessary condition for the classic Costas loop to maintain tracking of the desired signal in the presence of CW interference. The necessary condition refers to the required relationship between the frequency difference and amplitude ratio of the interference and desired signals. Since the first harmonic model used for PLL cannot be directly applied to the Costas loop, this article proposes a second harmonic model. The proposed model is substituted into the steady-state equation of the Costas loop and solved using the harmonic balance method. The necessary condition for the Costas loop to maintain tracking of the desired signal is derived. Simulations demonstrate that the theoretical condition for tracking the desired signal is consistent with the simulation results.

Shape fitting, including straight line and ellipse fitting, plays an important role in the (cylinder-) shape control system of silicon single crystal growth, because the straight lines and ellipse in the crystal image contain the important horizontal circle center and diameter information. This information can be used as control variables so that the grown crystal approximates to a perfect cylinder, and thus can be used as high-quality source materials. In this paper, we develop new straight line and ellipse fitting algorithms. The key points are as follows. We formulate the two-dimensional (2-D) binary image into a single-snapshot array signal of a virtual sensor array, and casts the angle estimation problem of straight lines into the direction finding one of virtual incoming sources. Based on the virtual array manifold and potential incoming angles, the relevant over-complete dictionary is constructed, and thus a sparse regression problem is formed. To solve such a regression problem, we introduce the weight vector sparsity term into the conventional linear least-squares support vector regression framework to estimate the angles of these straight lines. Based on the estimated angles and potential offsets, another over-complete dictionary is constructed, and thus the image can be looked upon as the sparse representation of these dictionary atoms. Since the constructed dictionary is of the same size as the image, we use the compressed sensing theory to reduce the relevant dimensionality and then apply the aforementioned sparse regression method to obtain the relevant offsets of these straight lines. We derive a new second-order polynomial of ellipse equation to obtain the ellipse parameters to avoid the trival solution from the conventional polynomial model. Some simulation and experimental examples are given to illustrate the effectiveness of the proposed algorithms.