The Hilbert transformHsatisfies the Bedrosian identityH(fg)=$=$fHgwhenever the supports of the Fourier transforms off,g∈$\in$L2$L^{2}$(ℝ$\mathbb{R}$) are respectively contained inA=$=$[-a,b] andB=$=$ℝ$\mathbb{R}$∖$\setminus$(-b,a), where0≤$\leq$a,b≤$\leq$+∞$\infty$. Attracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded Fourier multiplier operator onL2$L^{2}$(ℝd$\mathbb{R}^{d}$) and for general support setsAandB. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for the partial Hilbert transforms are all found. In particular, for the Hilbert transform to satisfy the Bedrosian identity, the support sets must be given as above.