We consider the estimation of the Fourier transform of multidimensional deterministic signals from a finite number of random samples. First, we consider a scenario where the sampling instants are taken from a continuous-time observation window. Under this class of Fourier transform estimation we analyse three estimation schemes, i.e. the total random estimation, stratified estimation and antithetical stratified estimation. We compare the derived estimators in terms of the mean-square error they introduce to the estimated Fourier transform. Also, we compare the rates of convergence of the estimates with respect to the number of random samples. Second, we examine two Fourier transform estimation schemes where the sampling points are selected from a predefined dense and uniformly distributed grid of time instants. The schemes are named as the total random on grid estimation and stratified on grid estimation. Accuracy of these estimates is shown and compared with each other.