The Least Mean Square (LMS) algorithm and its variants are currently the most frequently used adaptation algorithms; therefore, it is desirable to understand them thoroughly from both theoretical and practical points of view. One of the main aspects studied in the literature is the influence of the step size on stability or convergence of LMS-based algorithms. Different publications provide different stability upper bounds, but a lower bound is always set to zero. However, they are mostly based on statistical analysis. In this paper we show, by means of control theoretic analysis confirmed by simulations, that for the leaky LMS algorithm, a small negative step size is allowed. Moreover, the control theoretic approach alows to minimize the number of assumptions necessary to prove the new condition. Thus, although a positive step size is fully justified for practical applications since it reduces the mean-square error, knowledge about an allowed small negative step size is important from a cognitive point of view.

The 2019 Coronavirus (COVID-19) virus has caused damage on people's respiratory systems over the world. Computed Tomography (CT) is a faster complement for RT-PCR during peak virus spread times. Nowadays, Deep Learning (DL) with CT provides more robust and reliable methods for classifying patterns in medical pictures. In this paper, we proposed a simple low training proposed customized Convolutional Neural Networks (CNN) customized model based on CNN architecture that layers which are optionals may be included such as the layer of batch normalization to reduce time taken for training and a layer with a dropout to deal with overfitting. We employed a huge dataset of chest CT slices images from diverse sources COVIDx-CT, which consists of a 16,146-image dataset with 810 patients of various nationalities. The proposed customized model's classification results compared to the VGG-16, Alex Net, and ResNet50 Deep Learning models. The proposed CNN model shows robustness by achieving an overall accuracy of 93% compared to 88%, 89%, and 95% for the VGG-16, Alex Net, and ResNet50 DL models for the classification of 3 classes. When this relates to binary classification, the classification accuracy of the proposed model and the VGG-16 models were identical (almost 100% accurate), with 0.17% of misclassification in the class of Non-Covid-19, the Alex Net model achieved almost 100% classification accuracy with 0.33% misclassification in the class of Non-Covid-19. Finally, ResNet50 achieved 95% classification accuracy with 5% misclassification in the Non-Covid-19 class.