# Transient Analysis of a Selective Partial-Update LMS Algorithm
## 5 Results
### Extensions of the Proposed Framework Simulations  
In accordance with the extension approach for SPU-LMS-M-max, incorporating a variable tap length algorithm, it is noteworthy that this method inherently utilizes a greater number of steady-state coefficients than strictly essential. Consequently, such an algorithm demands inefficient utilization of computational resources during steady-state operation. In response to this challenge, the amalgamation of SPU and VTL is proposed, thereby facilitating diminished computational costs while preserving an equivalent asymptotic performance. Despite the inevitable consequence of a controllable reduction in the convergence rate due to this integration, this trade-off can be effectively managed by judiciously selecting the fraction of coefficients to update. Thus, the proposed SPU methodology was applied to an Acoustic Echo Cancellation (AEC) context, showcasing sustained asymptotic performance alongside diminished computational costs. This accomplishment is achieved at the cost of a controllable loss in the convergence rate, a parameter that remains adjustable by the designer. Figure 9a shows comparisons between empirical MSE for those algorithms VTL and SPU-VTL, where the 100 coefficients of Model 2 of [27] are employed as the ideal transfer function. All empirical results were obtained by performing \(2\times 10^{4}\) independent Monte Carlo trials, and the additive noise derives from a white Gaussian process. The total number of blocks is \(M=8\) and only \(85\%\) of the blocks were updated. The input signal is generated by passing a unity-variance white Gaussian noise through the filter \(H(z)=0.35-z^{-1}+0.35z^{-2}\). The selected parameters were \(\beta=10^{-2}\), \(\sigma_{v}^{2}=10^{-6}\), \(\gamma=20\), \(\alpha=4.10^{-4}\), \(\delta=3\), and \(\Delta=15\). While the transient regime underscores the superior convergence rate of Variable Tap Length (VTL), it is imperative to highlight that the utilization of Selective Partial Update with Variable Tap Length (SPU-VTL) concurrently diminishes computational complexity while maintaining an equivalent convergence rate during steady-state operation. Therefore, in the context of a Mean Squared Error (MSE) comparison, the judicious selection between VTL and SPU-VTL is pivotal for designers aiming to achieve specific targets. Figure 9b shows comparisons between empirical Tap Length over iterations used on VTL and SPU-VTL taking advantage of the same parameters as those employed in Figure 9a. In the configurations depicted in Figure 9a,b, the reduction in the number of multiplications resulting from the insertion of the SPU technique averaged \(7.5\%\). Further, a detailed examination of Figure 9b reveals that in terms of energy efficiency, the application of tap length measurements by MSE calculation over up to \(10^{4}\) iterations proved to be more effective during the transient regime. Moreover, in the steady-state regime, both algorithms demonstrated comparable performance. Figure 8: Comparison between empirical SPU-LMS-M-max (red) and SPU-LMS-M-min (blue) for the very intense impulsive noise scenario. (**a**) MSD (dB); (**b**) MSE (dB). Figure 10a,b present comparisons, measured in terms of Mean Squared Error (MSE), between traditional LMS and SPU algorithms applied to a practical scenario of Acoustic Echo Cancellation (AEC). In both instances, the parameters utilized included 20 coefficients representing the ideal transfer function, with SPU employing only 18 coefficients. The empirical results stem from \(2.10^{4}\) independent Monte Carlo trials, incorporating white Gaussian noise as the additive element. The input signal is generated by passing unitary-variance white Gaussian noise through the filter \(H(z)=1-0.2z^{-1}+z^{-2}\). The coefficients of the ideal plant are described by, defined as \(h_{\mathrm{k}}=\left[\cos\left(0.2\times\pi\times(k+1)\right)\right]^{-0.2 \times(k+1)}\). The parameters applied are \(\beta=10^{-2}\), \(\sigma_{\
u}^{2}=10^{-6}\) for Figure 10a and \(\sigma_{\
u}^{2}=10^{-2}\) for Figure 10b.