## 3 Wl-Rls-Dcd
This approach is considered a _greedy_ behaviour because it searches for the most probable position where an update could be triggered for \(\Delta\widetilde{\mathbf{h}}\). The parameters of the DCD iterations are the maximum expected amplitude \(H\), the step size \(\alpha\), the number of bits \(M_{b}\) used for the numerical representation of the real and imaginary values comprising \(\Delta\widetilde{\mathbf{h}}\), and _the maximum number of allowed updates_\(N_{\mu}\). The value of \(H\) is usually chosen as a power of two (e.g., \(H=1\)), and it is used to initialize the value of the step size. Considering how the latter is updated in Step 2 of Table 2, this choice ensures that the multiplication operations performed with \(\alpha\) are equivalent to bit shifts. In Step 1 of the DCD iterations, a search is performed amongst all real and imaginary parts comprising the vector \(\mathbf{r}_{aux}\) in order to determine the maximum absolute value and its corresponding position. The target value \(r_{temp}\) is employed in Step 4 to decide if an update is triggered, by comparing it to the (real and positive) value \(R_{p,p}\) situated on the corresponding position of the main diagonal of \(\mathbf{R}(n)\), which is scaled using the step size. If an update is decided, then the same position in the solution vector (i.e., \(\Delta\widetilde{h}_{p}\)) is updated using \(\alpha\) and the sign of \(r_{temp}\). The operation is equivalent to setting a bit in the binary representation of one of the real and imaginary values comprising the solution vector \(\Delta\widetilde{\mathbf{h}}\). Moreover, the coefficient update is always followed by another change in the vector \(\mathbf{r}_{aux}\) on the same position \(p\) (Step 6). It is important to note that \(\mathbf{r}_{aux}\) is initialized at the beginning of the DCD with the values comprising the residual component \(\mathbf{p}_{0}(n)\), and after the changes performed by the algorithm, it will represent the residual vector \(\mathbf{r}(n)\). If the condition in Step 4 does not trigger an update, then a jump is made to Step 2, and the value of the step size is halved, making \(\alpha\) point to the next lesser significant bit possible. Such an operation can be performed until the algorithm reaches the least significant bit it can process (i.e., until \(m=M_{b}\)). Consequently, the algorithm can perform a total number of updates upper limited by \(N_{u}\) (see the \(for\) loop), or it can exit when it would need to perform updates in the solution vector equivalent to binary values smaller than the least significant bit used to represent the real and imaginary values comprising \(\Delta\widetilde{\mathbf{h}}(n)\) (Step 3). Due to the nature of the auxiliary system of equations, it is expected the WL-RLS-DCD will generate satisfactory performance with a small value of \(N_{u}\) (less than 10, or comparable to it, for larger systems, with thousands of coefficients). In addition, it is natural to assume at this point that the value \(M_{b}\) also denotes the number of bits used to represent the real and imaginary parts of the coefficients comprising the adaptive filter \(\widetilde{\mathbf{h}}(n)\). The overall complexity of the employed DCD algorithm (at any time index \(n\)) takes into account that bit shifts replace all potential multiplications, and only additions are employed. Moreover, a maximum number of \(N_{u}\) update operations can be performed on the solution vector. The final number of modified \(\Delta\widetilde{\mathbf{h}}(n)\) coefficients (which are all initialized with 0) is expected to be much smaller than the filter's length (i.e., \(N_{u}\ll 2L\)). In [5; 21], it was demonstrated that the DCD iterations can generate performance similar to other classical RLS methods for values of \(N_{u}\) smaller than 10. Consequently, the upper limit for the amount of additions is noted in Table 2.  
Information regarding the arithmetic complexity of the WL-RLS-DCD is also displayed in Table 1, in terms of real valued multiplications and additions.