Covariance matrix of an adaptive filter input
The Covariance Matrix is commonly defined as 

$$\mathbf Q = E\left[ (\mathbf x -\mathbf\mu_{x})(\mathbf x -\mathbf\mu_{x})^*\right]$$
with $\mu$ denoting the mean value, i.e. $\mu_{x}=E\left[\mathbf x\right]$, and $\mathbf x$ being column vectors. The fact that you define the covariance matrix as

$$\mathbf{R}_i = E\left[\textbf{u}_i^*\textbf{u}_i \right]$$

indicates that your $\mathbf u_i$ have zero mean and are row vectors; that might be a result of the signal model you use, but it's pretty non-standard.

Furthermore, the product of a column vector with its hermitian is inherently hermitian (symmetric if real).

Now, one way to estimate an expectation value from multiple observations. Wikipedia has an article on estimating covariance matrices, so here's just the gist: the most intuitive estimator is the sample covariance matrix; adapted to your formula, that'd be

$$\hat{\mathbf R} = \frac1{N-1}\sum\limits_{i=1}^N \mathbf u_i^*\mathbf u_i$$