The affine projection (AP) algorithm, achieving a good tradeoff between the convergence rate and computational cost, is widely used in echo cancellation, active noise control, and acoustic feedback cancellation. Because the computational burden of the AP algorithm increases with the projection order, various fast AP algorithms have been proposed to reduce the complexity in the last two decades.

However, those low-complexity methods are scattered throughout the literature and have not been well enough analyzed and compared. Hence, engineers may still not know which fast version is optimal for their specific applications.

To fill this gap, researchers from the Institute of Acoustics (IOA) of the Chinese Academy of Sciences carried out a thorough review of the fast affine projection algorithms, evaluated the complexity and performance of each fast version and indicated their advantages and limitations. The study was published in Digital Signal Processing.

Researchers provided an in-depth treatment of the fast techniques in the implementation of the AP algorithm, including the fast weight vector update, fast filtering, and solutions to the linear system of equations. The advantages and disadvantages of each fast implementation version were clarified based on an extensive performance evaluation.

Both engineers and experts in this field can benefit from conclusions listed as follows.

The fast approximated filtering schemes and the fast exact filtering schemes can provide similar convergence performance if their regularization parameters are tuned carefully.

The matrix inversion methods have a significant impact on the overall convergence performance. There is a tradeoff between the complexity and the accuracy of the solutions.

One iteration of Gauss–Seidel (GS) or conjugate gradient (CG) is not sufficient to obtain a stable solution in the simplified fast affine projection (SFAP) algorithm for a highly correlated signal, although this conclusion has been claimed in many previous publications.

It is proved that three versions of the pseudo AP (PAP) in the literature are indeed mathematically equivalent, although they are assumed to be different algorithms.

The complexity also depends on the implementation platform and the realization skills. It does not make sense to talk about the complexity without mentioning the specific platform.

Reference:

YANG Feiran, YANG Jun. A Comparative Survey of Fast Affine Projection Algorithms. Digital Signal Process (Volume 83, December 2018, Pages 297-322). DOI: 10.1016/j.dsp.2018.09.004.

Contact:

WANG Rongquan

Institute of Acoustics, Chinese Academy of Sciences, 100190 Beijing, China

E-mail: media@mail.ioa.ac.cn