## 3 Wl-Rls-Dcd
The steps of the WL-RLS-DCD are described in Table 1, where we denoted by \(\mathbf{R}^{(1)}(n)\) the first column of the matrix \(\mathbf{R}(n)\) and by \(\epsilon\) a small positive constant employed to initialize the correlation matrix in a non-singular form. The updates in Steps 1 and 2 can be performed by relying on the time-shift property of the input vector \(\widetilde{\mathbf{x}}(n)\), which is also translated to \(\mathbf{R}(n)\). The upper-left \((2L-2)\times(2L-2)\) sub-matrix of \(\mathbf{R}(n)\) can be copied to the lower-right \((2L-2)\times(2L-2)\) sub-matrix, and only the first two columns and first two rows must be computed. If we take into consideration that the matrix is Hermitian, the first two rows can be obtained from the first two columns. Moreover, for every pair of consecutive columns, starting with an odd position (e.g., column 1, column 3, etc. ), the sets of values comprising one of the columns can also be found in the other column in a conjugate form, at a distance of \(\pm 1\) row. Consequently, the only arithmetic operations necessary for updating \(\mathbf{R}(n)\) can be employed for a single column (i.e., a set of \(2L\) values). The described methodology shows how the arithmetic complexity corresponding to Steps 1 and 2 from Table 1, which is apparently \(\mathcal{O}(4L^{2})\), can be kept to value proportional to \(2L\) (i.e., the filter's length). Further possible optimizations, such as the option of storing only a part of the matrix, are up to the developers performing the hardware implementations and deciding on the associated compromises. The latter depend considerably on the value of \(L\) (i.e., the associated memory requirements versus the necessary extra logic elements required to exploit the time-shift properties). Steps 3 and 4 of the WL-RLS-DCD represent the standard filtering operation and the trivial computation of the error signal. In Step 5, the _residual component_\(\mathbf{p}_{0}(n)\) is updated for the current time index by applying the forgetting factor to the so-called _residual vector_\(\mathbf{r}(n-1)\). The residual component is then employed in Step 6 as part of an auxiliary system of equations, which can be solved by exploiting the properties of the correlation matrix \(\mathbf{R}(n)\) with the DCD iterations. Step 6 produces two sets of results. Firstly, the _solution vector_, denoted by \(\Delta\widetilde{\mathbf{h}}(n)\), is used in Step 7 by adding it to the filter coefficients \(\widetilde{\mathbf{h}}(n-1)\) determined at the previous time index, thus generating the updated estimate \(\widetilde{\mathbf{h}}(n)\) of the unknown complex valued system. Secondly, the residual vector \(\mathbf{r}(n)\) is computed to reflect the changes performed in the filter update process through \(\Delta\widetilde{\mathbf{h}}(n)\). As can be noticed in Steps 5 and 6, the vector \(\mathbf{r}(n)\) will be relevant at the next time index as part of the next DCD operation. Table 2 shows a description of the DCD iterations with a _leading element_. It can be noted that \(\Delta\widetilde{\mathbf{h}}(n)\) is initialized with \(2L\times 1\) zeros values, and at the end of the entire DCD procedure, it represents the solution vector to a system which also comprises the matrix \(\mathbf{R}(n)\) and the \(2L\times 1\) residual component \(\mathbf{p}_{0}(n)\). The DCD algorithm works under the presumption that \(\mathbf{R}(n)\) is comprised of complex values on all positions except the main diagonal, where all values are real positive numbers. Moreover, on the main diagonal, all values are statistically considerably higher than all other real or imaginary values in the rest of the matrix (in terms of absolute values). By exploiting this property, Step 1 of the algorithm chooses the position with the greatest absolute value in \(\mathbf{r}_{aux}\) (all real and imaginary parts) and performs a comparison with its counterpart (i.e., same position) on the main diagonal of \(\mathbf{R}(n)\) in order to determine if an update should be made in the solution vector.