# An Improved Set-Membership Proportionate Adaptive Algorithm for a Block-Sparse System
## 4 Simulation and Result Analysis  
In this section, four experiments are constructed to verify the performances of the devised BS-SMPNLMS and BS-SMPNLMS algorithms. All of the experiments are created with white Gaussian noise (WGN), colored noise, and a speech signal as input signals, respectively. Herein, we assume that the signal-to-noise ratio (SNR) is 30 dB. The colored noise is generated from the WGN with a pole at 0.8. The sampling frequency of the speech signal is 8 KHz. In the first and second experiments, the influences of the block size on the BS-SMPNLMS and BS-SMIPNLMS are investigated. In the third experiment, the performances of the BS-SMPNLMS and the BS-SMIPNLMS are compared with other related algorithms. In the fourth experiment, the upper bound of estimation error is investigated in detail, which is the key parameter of the BS-SMPNLMS and BS-SMIPNLMS algorithms. Experiment 1. In this experiment, the estimation behaviors of the BS-SMPNLMS with different block sizes are studied. The response length of the block-sparse channel is assumed to be 1024. The single-clustering response as shown in Figure 1b is located at [257; 288] which has 32 taps, while the two-clustering response as shown in Figure 1c is located at [257; 272] and [769; 800] which has 16 and 32 taps [26]. The performances of the BS-SMPNLMS for WGN, colored noise, and the speech signal with different block sizes of 4, 16, 32, and 64 are shown in Figures 2-4, respectively. Each figure is divided into two parts. The first half of the 80,000 sample shows the performance of single-clustering, while the second half of the 80,000 sample shows the performance of two-clustering. The normalized misalignment is used to evaluate the estimating performance, and each simulation is repeated 10 times to get an average value. The simulation parameters are set as follows: when the input is the speech signal, the step sizes of the NLMS, PNLMS, and BS-PNLMS are \(\mu_{\text{NLMS}}=\mu_{\text{PNLMS}}=\mu_{\text{BSPNLMS}}=0.2\). When the input is colored noise, all of the step sizes of the algorithms above are 0.1. When the input is WGN, all the step sizes of the algorithms are 0.05. The small positive constants of the NLMS, PNLMS, BS-PNLMS, and BS-SMPNLMS are \(\varepsilon=\varepsilon_{\text{BS}}=0.01\). The parameters of the PNLMS, BS-PNLMS, and BS-SMPNLMS are \(\rho=5/N\), \(\delta=0.01\). The bound of the BS-SMPNLMS is \(\gamma=0.01\). Figure 2: Influence of block size on the BS-SMPNLMS for WGN. NLMS: normalized least mean square; PNLMS: proportionate NLMS; BS-PNLMS: block-sparse PNLMS; BS-SMPLMS: block-sparse set-membership PLMS; WGN: white Gaussian noise. From the simulation results we can see that independent of single-clustering or two-clustering, the BS-SMPNLMS with \(L=4\) behaves best, with the smallest block size among the four cases. On the contrary, the BS-SMPNLMS with \(L=64\) behaves the worst. The PNLMS and NLMS can be considered as the BS-PNLMS with \(L=1\) and \(L=N\). Therefore, to obtain good estimation performance, the block size should be selected appropriately. Overall, the proposed BS-SMPNLMS algorithm with single-clustering performs better than that with two-clustering. The normalized misalignment of the colored noise input signal is smaller than that of the speech signal but larger than that of WGN. The convergence speed of the colored noise is faster than that of speech signal but slower than that of WGN. Experiment 2. The estimation behaviors of the BS-SMPNLMS with different block sizes are studied in this experiment. The simulation environment is almost the same as Experiment 1. The adjusted parameter of the IPNLMS, BS-IPNLMS, and BS-SMPNLMS algorithms is \(c=0\). The  
Figure 4: Influence of block size on the BS-SMPNLMS for the speech signal. Figure 3: Influence of block size on the BS-SMPNLMS for colored noise. Experiment 3. In this experiment, the performances of the BS-SMPNLMS and BS-SMIPNLMS algorithms are studied by comparison with the NLMS, PNLMS, IPNLMS, BS-PNLMS, and BS-IPNLMS algorithms.