Least-Squares (LS) Interpolation

The generalized linear model uses a set basis functions to estimate the channel. The channel transfer function $H_k$ can be modelled as a linear weighted sum of basis functions evaluated at the k-th instance as: $$H_k = \sum_{i=0}^{N-1} {\theta _i \cdot \phi _i (t_k) }$$ where
$\phi _i (t_k)$ is the i -th basis function evaluated at that k -th instance of time $t_k$ ,
$\theta _i$ is the weighing factor of the basis function $\phi _i (t_k)$ ,
N is the number of basis functions used in the linear model.

For the basis function either Orthogonal polynomials or Legendre polynomials can be selected. In this demo the Legendre polynomials up to order 3 are selected.

From the raw channel observations at pilot locations a Least squares estimate of the weighing factors $\hat{\theta _i}$ can be calculated. The least-square estimate $\hat{\theta _i}$ can then be used to estimate the channel $\hat{H_k}$, at regular time instances $t_k$ as: $$\hat{H_k} = \sum_{i=0}^{N-1} {\hat{\theta _i} \cdot \phi _i (t_k) }$$