Conditioning of rectangular Vandermonde matrices with nodes in the unit disk

Let W-N = W-N(z(1),z(2),...,z(n)) be a rectangular Vandermonde matrix of or der n x N, N greater than or equal to n, with distinct nodes z(j) in the un it disk and z(j)(k-1) as its (j, k) entry. Matrices of this type often aris e in frequency estimation and system identification problems. In this paper , the conditioning of W-N is analyzed and bounds for the spectral condition number kappa(2)(W-N) are derived. The bounds depend on n, N, and the separ ation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided t he nodes are close to the unit circle but not extremely close to each other and provided the number of columns of W-N is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examp les.