Performance Analysis of Deficient Length Quaternion Least Mean Square Adaptive Filters

Quaternion adaptive filters have been widely used for processing 3D and 4D phenomena. Deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown. However, their statistical behaviors are yet to be fully understood. As theoretical results on the class of 'full length' quaternion least mean square (QLMS) algorithms do not necessarily apply to their deficient length versions, this article fills this void and analyses the mean and mean square convergence of the deficient length QLMS algorithms, both for the strictly linear and widely linear cases. Transient and steady-state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors. A novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed-form solutions for correlated quaternion Gaussian inputs, thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data. The analysis also provides a general framework whereby the strictly linear and semi-widely linear QLMS algorithms can be seen as 'deficient-length' versions of the widely linear QLMS. Numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms.