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This paper deals with the problem of noise reduction in stereo sound systems where the objective is not only to reduce noise, but also to preserve the spatial information of both the desired speech and noise sources so that the listener can still localize the speech and noise sources by listening to the enhanced binaural outputs. To achieve this objective, we use the widely linear (WL) framework developed previously and convert the problem of binaural noise reduction into one of monaural filtering with complex signals. We then present a way to decompose both the complex speech and noise signal vectors into two orthogonal components: one correlated and the other uncorrelated with the corresponding current signal sample. With this decomposition, the problem of noise reduction with preservation of the spatial information of speech and noise sources is formulated as an optimization problem with two constraints: one on the desired speech and the other on the preservation of the noise signal. We then derive a WL linearly constrained minimum variance (LCMV) filter, which can take advantage of the statistics and noncircularity of the complex speech signal to achieve noise reduction. In contrast to the WL Wiener and minimum variance distortionless response (MVDR) filters developed previously that can only preserve the characteristics and spatial information of the desired sound source, this new WL LCMV filter has the potential to reduce noise while preserving the characteristics and spatial information of both the desired and noise sources at the same time. Experimental results are provided to justify the claimed merits of the proposed WL LCMV filter.