# Transient Analysis of a Selective Partial-Update LMS Algorithm
###### Abstract  
In applications where large-order filters are needed, the computational load of adaptive filtering algorithms can become prohibitively expensive. In this paper, a comprehensive analysis of a selective partial-update least mean squares, named SPU-LMS-M-min, is developed. By employing the partial-update strategy for a non-normalized adaptive scheme, the designer can choose an appropriate number of update blocks considering a trade-off between convergence rate and computational complexity, which can result in a more than 40% reduction in the number of multiplications in some configurations compared to the traditional LMS algorithm. Based on the principle of minimum distortion, a selection criterion is proposed that is based on the input signal's blocks with the lowest energy, whereas typical Selective Partial Update (SPU) algorithms use a selection criterion based on blocks with highest energy. Stochastic models are developed for the mean weights and mean and mean squared behaviour of the proposed algorithm, which are further extended to accommodate scenarios involving time-varying dynamics and suboptimal filter lengths. Simulation results show that the theoretical predictions are in good agreement with the experimental outcomes. Furthermore, it is demonstrated that the proposed selection criterion can be easily extended to active noise cancellation algorithms as well as algorithms utilizing variable filter length. This allows for the reduction of computational costs for these algorithms without compromising their asymptotic performance. adaptive algorithms; least mean squares; selective partial-update algorithms +
Footnote †: journal: Applied Optics  
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Footnote 0: _Applied_  
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Footnote 0: _Applied_  
that employs the observable data pair \(\{\mathbf{x}(k),d(k)\}\), where \(d(k)\in\mathbb{R}\) is the reference signal and \(\mathbf{x}(k)\in\mathbb{R}^{N}\) denotes the input vector, defined as:  
\[\mathbf{x}(k)\triangleq\big{[}x(k)\quad x(k-1)\quad\ldots\quad x(k-N+1)\big{]}^{T}. \tag{1}\]  
where \(N\) denotes the filter length. The LMS belongs to the class of supervised adaptive algorithms whose learning dynamics is driven by a feedback mechanism based on the error signal  
\[e(k)\triangleq d(k)-\mathbf{w}^{T}(k)\mathbf{x}(k), \tag{2}\]  
where \(y(k)\triangleq\mathbf{w}^{T}(k)\mathbf{x}(k)\) is the filter output at the \(k\)-th iteration and \(\mathbf{w}(k)\in\mathbb{R}^{N}\) contains the \(N\) adaptive coefficients \(\{w_{0}(k),w_{1}(k),\ldots,w_{N-1}(k)\}\). Moreover, the LMS update equation is traditionally derived from the stochastic gradient of the mean squared error, thereby yielding  
\[\mathbf{w}(k+1)=\mathbf{w}(k)+\beta\mathbf{x}(k)e(k), \tag{3}\]  
where \(\beta\in\mathbb{R}_{+}\) is the step size, whose choice implies a trade-off between asymptotic performance and convergence rate [4]. In a system identification task, the block diagram of the adaptive filter functioning is depicted in Figure 1, where the desired signal \(d(k)\) sometimes is acquired by ingenious ways and signal \(v(k)\) models the impact of inaccuracies in the measurement process, such as quantization noise. Although the LMS is one of the simplest adaptive filtering algorithms, its update equation requires computational complexity proportional to the length of the adaptive coefficient vector \(\mathbf{w}(k)\in\mathbb{R}^{N}\) (see Equations (2) and (3)). Unfortunately, as already discussed, long transfer functions that occur in important real applications might demand a prohibitively large computational effort. Several schemes have been presented in the open literature to address this issue, such as sign-error, sign-data, and selective partial-update (SPU) strategies [5; 6; 7; 8; 9]. The latter schemes are the focus of this paper. The SPU-NLMS algorithm, one of the most popular SPU-based adaptive algorithms, is typically presented as a solution of a deterministic optimization problem [10]. The resulting update equation implies a reduced computational burden by decreasing the amount of tap modifications at every iteration.