Solving for the Minimum Mean Squared Error Weights

If ${\bf R} = \frac{1}{L} \sum_{n= l_1}^{l_1+L-1} {\bf x}[n] {\bf x}^T[n]$, ${\bf p} = \frac{1}{L} \sum_{n= l_1}^{l_1+L-1} d[n]{\bf x}[n]$ and $\sigma_d^2 = \frac{1}{L} \sum_{n= l_1}^{l_1+L-1} d^2[n]$ as defined in the video, then the vector of optimum weights ${\bf w}$ is expressed as

The mean-squared error is a quadratic function of the filter weights.

The matrix ${\bf R}$, vector ${\bf p}$ and scalar $\sigma_d^2$ defined in question 1 correspond to sample estimates of the correlation matrix ${\bf R} = E\{{\bf x}[n]{\bf x}^T[n]\}$, cross-correlation vector ${\bf p} = E\{{\bf x}[n] d[n]\}$, and second moment $\sigma_d^2 = E\{d^2[n]\}$.

The solution to the system of $L>N$ equations in $N$ unknowns ${\bf X}^T{\bf w} \approx {\bf d}$ where ${\bf X}^T$ is $L$ by $N$, ${\bf w}$ is $N$ by $1$, and ${\bf d}$ is $L$ by $1$ may be expressed as…

The MMSE filtering problem is similar, but not equivalent to a least squares problem involving $L>N$ equations in $N$ unknowns.