![]() ![]() ![]() The affine projection algorithm (APA), suggested by Ozeki and Umeda, is one method to overcome this problem. However, they show a slow convergence speed with highly colored input signals. ![]() The least mean square (LMS) and normalized LMS (NLMS) are widely used owing to their simplicity and ease of implementation. Īdaptive filters (AF) have been extensively applied in many areas such as system identification, active noise control, and echo cancellation during the past decades. Owing to the convenience of GA-based models, GA has been studied in many applications, such as classification, direction of arrival estimation, and image processing. Besides, geometric calculus (GC) can perform calculus with hypercomplex numbers clearly and compactly. The product operation of GA, namely geometric product, allows a set of vectors to be mapped to scalars and hypersurfaces. In geometric algebra (GA)-based algorithms, the hypercomplex signals are transformed into multivectors, such as complex entries, quaternion entries, and higher dimensional entries, and handled holistically. However, they are constructed into vectors and processed as multi-channel signals in most existing literature. These signals derive from observations of different dimensions. The electromagnetic vector-sensor consists of 6 spatially arranged antennas, which measure the electric and magnetic field signals in the three directions of the incident wave. For example, 3-D wind speed, dynamic pressure, aircraft rotation axis (roll, pitch, yaw), and angle of attack are used to predict the attitude of the aircraft. With the development of sensor technology, there are more and more data sources for recording the same process. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The stability of the algorithm is analyzed based on the mean-square deviation. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Any help is appreciated.Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. I believe all my questions stem from not understanding the answer to 1). He finally adds that it is a general fact that the projection of a projective variety from a linear subspace disjoint from it is finite.ġ) How is the projection from a linear subspace defined? I am familiar with the notion of projection from a point.Ģ) Is this projection a regular morphism?ģ) Why does a regular morphism in this context has finite fibers?Ĥ) Why is the projection of a projective variety from a linear subspace disjoint from it is finite in general? He then proceeds to project $J$, claiming that as $J\cap L=\emptyset$, the projection is regular map. Harris mentions a projective space $\mathbb\longrightarrow L$, what he refers to as "the projection from $L$", which is a term I both never heard of, nor ever encountered in my reading. I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument. ![]()
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