**5. Adaptive step-size normalised orthogonal gradient adaptive algorithms**

Based on filtered gradient adaptive algorithm, adaptive algorithms employing orthogonal gradient filtering can provide with the development of simple and robust filter across a wide range of input environments. This section is therefore concerned with the development of simple and robust adaptive frequency-domain equalisation by defining normalised orthogonal gradient adaptive algorithm.

In this section, we describe the orthogonal gradient adaptive (OGA) algorithm that is a class of the filtered gradient adaptive (FGA) algorithm using an orthogonal constraint. This employs the mixed-tone criterion described above in Section 4 in order to improve the convergence speed presented in Section 5.1, respectively.

The idea for low complexity adaptive step-size algorithms with the mixed-tone cost function is described in Section 5.2. For a large prediction error, the algorithm will increase the step-size to track the change of system whereas a small error will result in the decreased step-size [15], [16].
