We address the problem of cancelling a stationary noise component from its static mixtures with a nonstationary signal of interest. Two different approaches, both based on second-order statistics, are considered. The first is the blind source separation (BSS) approach which aims at estimating the mixing parameters via approximate joint diagonalization of estimated correlation matrices. Proper exploitation of the nonstationary nature of the desired signal, in contrast to the stationarity of the noise, allows parameterization of the joint diagonalization problem in terms of a nonlinear weighted least squares (WLS) problem. The second approach is a denoising approach, which translates into direct estimation of just one of the mixing coefficients via solution of a linear WLS problem, followed by the use of this coefficient to create a noise-only signal to be properly eliminated from the mixture. Under certain assumptions, the BSS approach is asymptotically optimal, yet computationally more intense, since it involves an iterative nonlinear WLS solution, whereas the second approach only requires a closed-form linear WLS solution. We analyze and compare the performance of the two approaches and provide some simulation results which confirm our analysis. Comparison to other methods is also provided.