Performance Analysis of Fractional Learning Algorithms

The least mean square (LMS) algorithms are of paramount importance in the field of signal processing since their emergence [61, 62, 60]. In particular, they are used profusely in adaptive filtering and signal analysis [27, 64, 24, 5]. The key aspects that make LMS algorithms attractive are their low complexity, stability, and an unbiased mean convergence to the so-called Wiener solution in stationary environments [48]. Unfortunately, its rate of convergence depends on the eigenvalue spread of the correlation matrix of the input signal in non-stationary environments [62, 27]. Accordingly, many variant algorithms were proposed to achieve better performance by curtailing the influence of the spectral properties of the input signal correlation matrix; see, for instance, the LMS-Newton algorithm [25], transform-domain algorithm [37], and affine projection algorithm [49]. On the other hand, a desire for computationally simpler algorithms has also led to the development of many variants such as quantized-error algorithms [6, 19, 30] and normalized LMS algorithms [65, 48, 36]. A decent list of these variant algorithms along with the details of their key features is provided in [24, Ch. 4]. We also refer to fairly recent survey articles [63, 26] on the history of adaptive filtering and the development of the LMS algorithms.