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Abstract-A semiblind iterative algorithm to construct the best linear unbiased estimate (BLUE) of the channel impulse response (CIR) vector h for communication systems that utilize a periodically transmitted training sequence within a continuous stream of information symbols is devised. The BLUE CIR estimate for the general linear model y = Ah + w, where w is the correlated noise, is given by the Gauss-Markoff theorem. The covariance matrix of the correlated noise, which is denoted by C(h), is a function of the channel that is to be identified. Consequently, an iteration is used to give successive approximations h (k) , k = 0, 1, 2, . . . to h BLUE , where h (0) is an initial approximation given by the correlation processing, which exists at the receiver for the purpose of frame synchronization. A function F (h) for which h BLUE is a fixed point is defined. Conditions under which h BLUE is the unique fixed point and for which the iteration proposed in the algorithm converges to the unique fixed point h BLUE are given. The proofs of these results follow broadly along the lines of Banach fixed-point theorems.

We present a low-complexity method for approximating the semi-blind best linear unbiased estimate (BLUE) of a channel impulse response (CIR) vector for a communication system, which utilizes a periodically transmitted training sequence. The BLUE, for h, for the general linear model, y ¼ Ah þ w þ n, where w is correlated noise (dependent on the CIR, h) and the vector n is an Additive White Gaussian Noise (AWGN) process, which is uncorrelated with w is given by h ¼ (A T C(h) À1 A) À1 A T C(h) À1 y. In the present work, we propose a Taylor series approximation for the function F(h) ¼ (A T C(h) À1 A) À1 A T C(h) À1 y. We describe the full Taylor formula for this function and describe algorithms using, first-, second-, and third-order approximations, respectively. The algorithms give better performance than correlation channel estimates and previous approximations used, at only a slight increase in complexity. Our algorithm is derived and works within the framework imposed by the ATSC 8-VSB DTV transmission system, but will generalize to any communication system utilizing a training sequence embedded within data.

T h i s paper presents a method for optimum placement of nodes in grid generation for process simulation together with an algorithm for updating the grid after each addition of a new node to ensure that the Delaunay property is satisfied. Placement of nodes is decided on by considering the optimum error in evaluating the integral sv C(z)dz. The best error estimate is obtained when the node coincides with the centroid (the center of mass) of its own Voronoi region and moreover when the Voronoi region is symmetric. After addition of a node the grid is updated to maintain the Delaunay property using the Delaunay-Voronoi algorithm. I t ro duct ionIn process simulation, we are concerned with solving the diffusion equation _ -a ' -V.(DVC), t 2 0, z E Q c Rd, d = 2,3 8t where D is the diffusion coefficient. In solving the problem numerically, we are concerned with the error in computing the integral sv C(z)& where V is the region of influence of the node Z (the Voronoi region (see [4]) of the node Z.) Previously, we have considered the problem of node removal in a grid where mesh is too fine for the problem, (see [I], [2], and [3].) The removal algorithm identified nodes to be removed by their low discretizaton error. We now turn our attention to addition of nodes. When a grid is adapted, new nodes are added in areas where, due to larger variation in C(Z), it is deemed that greater resolution is needed. To decide on the optimum placement of these new nodes, we consider which position w i l l give the best error in evaluating Jv C ( z ) d x .The best error for this calculation is O(h3k, hk3) when C(z) is approximated by its Taylor series evaluated at the node f , the generator of the Voronoi region V, and moreover, when the node is the centroid (center of mass) of its own Voronoi region. Such a Voronoi tesselation is called a centroidal Voronoi tesselation. (see [5]). After addition of a new node, the grid quality must be maintained. A desirable property of the grid is that it has the Delaunay property. The grid is updated, after addition of each new node, to ensure that it has the Delaunay property. The algorithm to achieve this is called the Delaunay-Voronoi algorithm.The actual position of a new node is at the centroid of surrounding nodes to ensure that the node is at the centroid of its own Voronoi region. This position is decided on by wnsidering the smallest order error in evaluating the integral Jv C(z)dz using a Taylor series as stated a h . Hence the error in evaluating Jv C(z)dz is used as an error indicator in deciding on the positioning of nodes.In adding nodes during grid adaption, a check is used whereby a new node is added if and only if the geometric grid quality is i m p r d . If the geometric grid quality is not improved, then the new node is not added. Error Indicator for Evaluation of s , C(z)dzThe best error for this calculation of JvC(z)dz.is O(h3k, hk3), when C(z) is approximated by its Taylor series evaluated at the node Z , the generator of the Voronoi region, V, and moreover, when the node is t...