Convolutive reduced rank Wiener filtering

If two wide-sense stationary time series are correlated then one can be used to predict the other. The reduced rank Wiener filter is the rank-constrained linear operator which maps the current value of one time series to an estimate of the current value of the other time series in an optimal way. A closed-form solution exists for the reduced rank Wiener filter. This paper studies the problem of determining the reduced rank FIR filter which optimally predicts one time series given the other. This optimal FIR filter is called the convolutive reduced rank Wiener filter, and it is proved that determining it is equivalent to solving a weighted low rank approximation problem. In certain cases a closed-form solution exists, and in general, the iterative optimisation algorithm derived here can be used to converge to a locally optimal convolutive reduced rank Wiener filter