Adaptive filters play an important role in the fields related to digital signal processing and
communication such as system identification noise cancellation channel equalization and
beamforming. In practical applications the computational complexity of an adaptive filter is
an important consideration. The Least Mean Square (LMS) algorithm is widely used because of its
low computational complexity ($O(N)$) and simplicity in implementation. The least squares
algorithms such as Recursive Least Squares (RLS) Conjugate Gradient (CG) and Euclidean
Direction Search (EDS) can converge faster and have lower steady-state mean square error (MSE)
than LMS. However their high computational complexity ($O(N^2)$) makes them unsuitable for
many real-time applications. A well-known approach to controlling computational complexity is
applying partial update (PU) method to adaptive filters. A partial update method can reduce the
adaptive algorithm complexity by updating part of the weight vector instead of the entire
vector or by updating part of the time. In the literature there are only a few analyses of
these partial update adaptive filter algorithms. Most analyses are based on partial update LMS
and its variants. Only a few papers have addressed partial update RLS and Affine Projection
(AP). Therefore analyses for PU least-squares adaptive filter algorithms are necessary and
meaningful.This monograph mostly focuses on the analyses of the partial update least-squares
adaptive filter algorithms. Basic partial update methods are applied to adaptive filter
algorithms including Least Squares CMA (LSCMA) EDS and CG. The PU methods are also applied to
CMA1-2 and NCMA to compare with the performance of the LSCMA. Mathematical derivation and
performance analysis are provided including convergence condition steady-state mean and
mean-square performance for a time-invariant system. The steady-state mean and mean-square
performance are also presented for a time-varying system. Computational complexity is
calculated for each adaptive filter algorithm. Numerical examples are shown to compare the
computational complexity of the PU adaptive filters with the full-update filters. Computer
simulation examples including system identification and channel equalization are used to
demonstrate the mathematical analysis and show the performance of PU adaptive filter
algorithms. They also show the convergence performance of PU adaptive filters. The performance
is compared between the original adaptive filter algorithms and different partial-update
methods. The performance is also compared among similar PU least-squares adaptive filter
algorithms such as PU RLS PU CG and PU EDS. In addition to the generic applications of
system identification and channel equalization two special applications of using partial
update adaptive filters are also presented. One application uses PU adaptive filters to detect
Global System for Mobile Communication (GSM) signals in a local GSM system using the Open Base
Transceiver Station (OpenBTS) and Asterisk Private Branch Exchange (PBX). The other application
uses PU adaptive filters to do image compression in a system combining hyperspectral image
compression and classification.
