## 3 Results
### Algorithms Comparison  
This section is devoted to a comparison of the learning curves of four different strategies dealing with active noise the control of periodic disturbances:  
* Algorithm 1: standard LMS algorithm
* Algorithm 2: LMS algorithm with sequential PU, applying, optionally, the gain in step-size. * Algorithm 3: Periodic LMS algorithm, based on a matrix formulation. * Algorithm 4: Periodic LMS algorithm, based on a matrix formulation. Sequential PU (and gain in step-size) are applied again as in alternative 2. In Figure 6 is shown the block diagram of the noise control problem addressed with the four adaptive strategies. The sampling frequency is set to 8000 samples/s. We execute 1920 iterations of the adaptive algorithms, and the weights of the filters are reset to zero  
Figure 5: Gain in step-size for a single tone when using the matrix-based version (from [PERSON] et al.) of the periodic LMS algorithm with seq. partial updates. The length of the filter is set to 160 taps. The decimation factor takes values 1, 2, 4 and 8.  
after 800 iterations so as to evaluate a second convergence process. The step-size chosen for the experiment is set near its maximum value, which ensures convergence. In the experiment, we deal with two different versions of the desired signal \(d(n)\). In both cases, referred to in sub-indexes 1 and 2, the desired signal consists of three different harmonics,  
\[d_{1}(n)=2\cos\left(2\pi\frac{1}{8}n\right)+3\cos\left(2\pi\frac{1}{4}n\right)+4 \cos\left(2\pi\frac{3}{8}n\right)+awgn(n) \tag{41}\]  
\[d_{2}(n)=2\cos\left(2\pi\frac{3}{16}n\right)+3\cos\left(2\pi\frac{5}{16}n \right)+4\cos\left(2\pi\frac{7}{16}n\right)+awgn(n) \tag{42}\]  
where \(awgn(n)\) is an additive white Gaussian noise that makes \(d(n)\) have a signal to noise ratio of 45 dB. The reference signal \(x(n)\) is also considered in this example in two different versions,  
\[x_{1}(n)=\cos\left(2\pi\frac{1}{8}n\right)+\cos\left(2\pi\frac{1}{4}n\right)+ \cos\left(2\pi\frac{3}{8}n\right) \tag{43}\]  
\[x_{2}(n)=\cos\left(2\pi\frac{3}{16}n\right)+\cos\left(2\pi\frac{5}{16}n\right) +\cos\left(2\pi\frac{7}{16}n\right) \tag{44}\]  
being in both cases a set of three tones of unitary amplitude. The two versions of the reference and the desired signals are designed in order to explain the behavior of the adaptive algorithm with regard to the gain in step-size in the second and fourth alternative. When the second algorithm (LMS with Seq. PU) is applied, one may infer from Figure 7 that the gain in step-size can be taken at its full strength \(G_{\mu}=N\) for the second version of the reference signal, \(x_{2}(n)\) (marked with green circles), but not for the first version, \(x_{1}(n)\) (marked with red circles) because some/all of its harmonics are located in notches of the gain. More precisely, if the decimating factor \(N\) is set to 2, the second harmonic of \(x_{1}(n)\) appears in a frequency affected by a notch (around 0.25 normalized Hz). If the decimating factor \(N\) is set to 4, the three harmonics of \(x_{1}(n)\) appear in frequencies where notches in gain in step-size are present. Figure 6: Block diagram of the noise attenuation experiment used to compare different adaptive algorithms. Nevertheless, if the fourth algorithm (periodic P-LMS with Seq. PU) is applied, we see from Figure 8 that the gain in step-size does not present notches as long as frequencies are above the very low frequency range. Then, in this fourth adaptive strategy, both versions of the reference signal \(x_{1}(n)\) and \(x_{2}(n)\) are expected to provide similar results. In the comparison of the four algorithms presented below (Figures 9-14), the performance measure used is the instantaneous squared error in logarithmic scale (dB). All simulation results are obtained by averaging 50 independent runs. First, we work with \(x_{1}(n)\) and \(d_{1}(n)\), that is, the input signals whose frequencies are marked in red in Figures 7 and 8. In the first example, we compare the fourth adaptive algorithms previously listed, making \(N=G_{\mu}=1\).