Orthogonal LMS algorithms for fast line echo canceller training

An LMS-like fast algorithm, called the orthogonal LMS (OLMS) algorithm, is proposed. The OLMS algorithm is capable of estimating an FIR system. The estimation error will drop below the noise floor with the number of iterations in about two times the number of taps in the FIR system. This algorithm was designed originally to train the echo cancellers for the voiceband modems. The main feature of OLMS is that it uses an orthogonal training sequence and by exploiting the orthogonality, it accomplishes the exact same computations of the RLS algorithm. A simplified version of OLMS, called the SOLMS algorithm, is shown to be a normalized LMS algorithm with an orthogonal sequence. SOLMS has exactly the computation complexity of the LMS algorithm while converging at a speed compatible to the RLS algorithm. The steady-state error of the SOLMS is almost the same as that of the RLS algorithm. This difference in error actually goes to zero as the channel length goes to infinity. On the other hand, OLMS, which requires 50% more memory than SOLMS while being only slightly more complex than SOLMS, is shown to be exactly equivalent to the RLS algorithm. Using a deterministic linear algebraic formulation of the system identification problem of FIR systems, new insights about LMS-like algorithms and their relationship with the RLS algorithms are obtained. According to the linear algebraic framework, the LMS-like algorithms are shown to be based on the under-determined system approach, while the RLS algorithm is based on the over-determined system approach and these two approaches become the same when the training sequence is orthogonal or when the system is actually exactly determined. Numerical implementation results are provided to demonstrate the theoretical results presented in this paper.