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In this paper, a novel subband-selective generalized sidelobe canceller (GSC) with cosine-modulation for partially adaptive broadband beamforming is proposed. The columns of the blocking matrix are derived from a prototype vector by cosine-modulation, and the broadside constraint is incorporated by imposing zeros on the prototype vector appropriately. These columns constitute a series of bandpass filters, which select signals with specific direction of arrivals and frequencies. This results in high pass-type bandlimited spectra of the blocking matrix outputs, which is further exploited by subband decomposition and discarding the low-pass subbands appropriately prior to running independent unconstrained adaptive filters in each non-redundant subband. By these steps, the computational complexity of our GSC implementation is greatly reduced compared to fully adaptive GSC schemes, while performance is comparable or even enhanced due to subband decorrelation in both spatial and temporal domains.