## 3 Results
### Algorithms Comparison
As we do not apply either PU or gain in step-size, the second and fourth alternatives are identical, respectively, to the first and third algorithms. In Figure 9 we confirm that the learning curves of the four adaptive strategies are similar in terms of convergence rate. In the next step, we apply PU by setting a decimating factor \(N=2\). If we do not apply gain in step-size to compensate for the inherent reduction in convergence rate, that is, we make \(G_{\mu}=1\), the second and fourth learning curves should reduce the convergence rate by a factor of 2, as we can confirm in Figure 10.  
Finally, we apply PU by setting \(N=2\) to reduce computational costs, but we also make \(G_{\mu}=2\) to compensate the reduction in convergence rate. As expected (and we can confirm in Figure 11), the second strategy named (b) diverges because the gain in step-size can not be applied at full strength \(G_{\mu}=N\) due to the notch that appears at the second harmonic of \(x_{1}(n)\) (see Figure 7). On the other hand, the fourth strategy (d) works properly because the theoretical limit of the gain in step-size is well above the value used. Now, let us carry out the second version of the experiment dealing with \(x_{2}(n)\) and \(d_{2}(n)\). Here, the input signals contains harmonics at frequencies marked in green in Figures 7 and 8. Figure 7: Gain in step-size for a single tone when using sequential PU with LMS algorithm. The length of the filter is set to 64 taps. The decimation factor takes values 2 and 4. Circles mark frequencies where reference and desired signals have significant harmonics; red circles mark components of \(x_{1}(n)\) and green circles mark components of \(x_{2}(n)\). Figure 8: Gain in step-size for a single tone when using sequential PU with periodic LMS algorithm (matrix-based). The filter length is set to 64 taps. The decimation factor takes values 2 and 4. Apart from the very low frequency range, the step-size gain can be approximated by \(N^{2}\). Circles mark frequencies where reference and desired signals have significant harmonics; red circles mark components of \(x_{1}(n)\) and green circles mark components of \(x_{2}(n)\). Figure 9: Learning curves comparison. Version 1 of the experiment: desired signal \(d_{1}(n)\) and reference signal \(x_{1}(n)\). (**a**) LMS, (**b**) LMS with Seq PU; \(N=1\) and \(G_{\mu}=1\), (**c**) P-LMS, (**d**) P-LMS with Seq. PU; \(N=1\) and \(G_{\mu}=1\). Average of 50 runs. The performance measure used is the instantaneous squared error in logarithmic scale (dB). \[d_{1}(n)=2\cos\left(2\pi\frac{1}{8}n\right)+3\cos\left(2\pi\frac{1}{4}n\right)+4 \cos\left(2\pi\frac{3}{8}n\right)+awgn(n) \tag{41}\]0 Learning curves comparison. Version 1 of the experiment: desired signal \(d_{1}(n)\) and reference signal \(x_{1}(n)\). (**a**) LMS, (**b**) LMS with Seq PU; \(N=2\) and \(G_{\mu}=2\), (**c**) P-LMS, (**d**) P-LMS with Seq. PU; \(N=2\) and \(G_{\mu}=2\). Average of 50 runs. The performance measure used is the instantaneous squared error in logarithmic scale (dB). \[d_{1}(n)=2\cos\left(2\pi\frac{1}{8}n\right)+3\cos\left(2\pi\frac{1}{4}n\right)+4 \cos\left(2\pi\frac{3}{8}n\right)+awgn(n) \tag{41}\]1 Learning curves comparison. Version 1 of the experiment: desired signal \(d_{1}(n)\) and reference signal \(x_{1}(n)\). (**a**) LMS, (**b**) LMS with Seq PU; \(N=2\) and \(G_{\mu}=1\), (**c**) P-LMS, (**d**) P-LMS with Seq. PU; \(N=2\) and \(G_{\mu}=1\). Average of 50 runs. The performance measure used is the instantaneous squared error in logarithmic scale (dB). \[d_{1}(n)=2\cos\left(2\pi\frac{1}{8}n\right)+3\cos\left(2\pi\frac{1}{4}n\right)+4 \cos\left(2\pi\frac{3}{8}n\right)+awgn(n) \tag{41}\]2 Learning curves comparison. Version 2 of the experiment: desired signal \(d_{2}(n)\) and reference signal \(x_{2}(n)\). (**a**) LMS, (**b**) LMS with Seq PU; \(N=1\) and \(G_{\mu}=1\), (**c**) P-LMS, (**d**) P-LMS with Seq. PU; \(N=1\) and \(G_{\mu}=1\). Average of 50 runs. The performance measure used is the instantaneous squared error in logarithmic scale (dB).