Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric $K$-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known {\em a priori} to lie in certain frequency bands. Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin

Exploring the Influence of Wind Turbine\u27s Blades on Its Output and Efficiency

Wind turbines are machines that convert wind energy into electricity. The efficiency of this conversion is measured by comparing the incoming wind\u27s speed and the output power. This paper focuses on how the properties of blades affect the output and power of wind turbines. The attributes of turbine blades that affect output and efficiency, such as blade size and angle of entry, are considerable. Although results generally match with theory models, we find a size limit with blade length