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On a wireless link with multiple antennas at both transmitter and receiver, the interference at the receiving antenna array can exhibit a strong spatial coloring in the presence of a small number of dominant interferers. In such a situation, a receiver whose detection algorithms are designed for spatially white interference could exhibit degraded performance. We propose an iterative algorithm to compute a spatial whitening filter for a given covariance matrix. Unlike well-known techniques for computing such a whitening filter (e.g., inverting the lower-triangular Cholesky factor of the covariance), the algorithm we propose requires only matrix additions and multiplications and no nontrivial division or square root operations, making it well suited to VLSI implementation. By analyzing the dynamics of the proposed algorithm, we provide simple conditions under which it is guaranteed to converge to a desired solution. We also describe a simple technique to estimate the spatial covariance of the interference at the receiver, using pilot signals from the transmitter. We demonstrate by simulation that the proposed iterative whitening algorithm achieves virtually the same BER-versus-SNR performance as an exact whitening filter, with a small number of iterations.