A geometrical derivation of the excess mean square error for Bussgang algorithms in a noiseless environment
Introduction
The performances of adaptive filters are characterized in terms of convergence rate, that is, the speed of the transient response and in steady-state mean square error level. It is well-known that fair comparison of two algorithms is a very difficult task taking into account these two antagonist behaviors. Numerous works in the literature concern the performance of adaptive filters. In [5], the author proposed a new measure of performance to allow fair comparison between several algorithms. Another parameter could also be used and gives another kind of information, which could be seen as the noise of the algorithm. This parameter called excess mean square error (EMSE) is the one we are studying in this paper.
Bussgang algorithms could easily be described geometrically; and a simple application of the Pythagoras theorem gives a very general relation similar to the one called “fundamental energy conservation” derived in [4]. This relation allows us to find a general closed form of the EMSE including the presence of the cost function, and could then be used to compute the EMSE of a large family of algorithms such as the constant modulus algorithm (CMA) as done thereafter.
Section 2, presents the problem formulation and sets the notations. The next section describes the geometrical interpretation and applies the Pythagoras theorem to the right triangles. Then the derivation of the EMSE in both real and complex cases is led by Section 4. And, in Section 5, before concluding, an example is given by computing, thanks to our method the EMSE of the CMA, whose expression could already be found in the literature.
Section snippets
Problem formulation and preliminaries
Before proceeding any further, let us define the notation used throughout the paper: vectors are denoted by bold letters; and denote, respectively, the conjugate and real part of z, and is the scalar product of and .
Thanks to the notation given in Fig. 1, Bussgang algorithms which follow the classical stochastic gradient algorithm are described by the general update formulawith the vector of the input filter samples, the function φ is called the
Pythagoras and LMS
First of all, we should define several points of the Fig. 2. zopt is the output of the optimal filter and zk is the output of the filter at time k and zk+1 the output of the filter after the update (1). Mathematically speaking, we have , and .
As classically done, we define two errors: ea the a priori error, and ep the a posteriori error. These errors represent the difference between zopt and zk or zk+1. Noting , we have and
EMSE computation
During the steady state phase, we remark thatfrom which we deriveTherefore, with the classical independence assumption between the data and the error, we obtain . The main idea of the following is to develop an approximation of |φ(z)|2 and from (7) near the optimum zopt. As zk−zopt is the a priori error ea, we will find a simple relation between ea and the value of the function φ at the optimum. Thanks to this relation, the EMSE will
Constant modulus algorithm application
The CMA, one of the most widely studied algorithm, has been developed by Godard [3] as a Sato algorithm extension for constant module modulations like PSK; but surprisingly it works for other kinds of modulations. Its cost function can be written asThus, in the CMA case, the functions needed for the EMSE computation areIn a noiseless SIMO context, the CMA have a zero-forcing solution. Therefore, we
Conclusion
This paper, develops a very simple geometrical approach for the steady-state analysis of Bussgang algorithms. More precisely, we obtain a general form of the excess mean square error which includes the cost function and therefore could be applied to many algorithms. The application to constant modulus algorithm done in this paper, proved the validity of this approach. We may add that this approach could be extended from a straightforward manner to other adaptive schemes.
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