Adaptive filtering without a desired signal

In least-squares estimation problems, a desired signald(n)is estimated using a linear combination of L successive data samples, [x(n), x(n - 1), . . . , x(n-L+1)]. The weight set Woptwhich minimizes the mean-square error betweend(n)and the estimate is given by the product of the inverse data covariance matrix and the cross-correlation between the data vector and the desired signal, i.e. the P-vector. For those cases in which time samples of both the desired and data vector signals are available, a variety of adaptive methods have been proposed which will guarantee that an iterative weight vectorW_{a}(n)converges (in some sense) to the optimal solution. Two which have been extensively studied are the recursive least-squares (RLS) method and the LMS gradient approximation approach. There are several problems of interest in the communication and radar environment in which the optimal least-squares weight set is of interest and in which time samples of the desired signal are not available. Examples can be found in array processing in which only the direction of arrival of the desired signal is known and in single channel filtering where the spectrum of the desired response is known a priori. One approach to these problems which has been suggested is the P-vector algorithm which is an LMS-like approximate gradient method. Although it is easy to derive the mean and variance of the weights which result with this algorithm, there has never been an identification of the corresponding underlying error surface which the procedure searches. The purpose of this paper is to suggest an alternative approach to providing adaptive solutions to problems in which samples ofd(n)are unavailable. The method is based on the use of linearly-constrained minimum mean-square error methods. The constraint used is simply that the inner product of the filter weights with the known P-vector must be unity. The criterion employed is then minimization of total output power, subject to this constraint. Once this problem has been formulated, it can be readily implemented in either scalar or multi-channel form using the Generalized Sidelobe Canceller method. Both LMS-like and RLS algorithms may be employed to update the coefficients.