In this paper, a new lower bound on the mean-square-error of unbiased estimators of deterministic parameters is developed. The proposed bound is derived from a class of bounds presented in our recent work using the kernel of the Fourier transform, multiplied by a "weighting" function. The "weighting" function is defined on the parameter space and its significance in the parameter space and frequency domain is discussed throughout the paper. We show that the proposed bound is computationally manageable and can be easily implemented using the fast Fourier transform. The proposed bound is applied for the problem of direction-of-arrival estimation. It is shown by simulations that in comparison to other existing bounds in the literature, the proposed bound provides better prediction of the signal-to-noise ratio threshold region, exhibited by the maximum-likelihood estimator.