Absolute value operation in frequency domain

Question

Let $f\in L^2(\mathbb{R}^d)$ be a real, positive function and $h\in L^2(\mathbb{R}^d)$ a complex function with compact support in frequency domain and $0\notin \text{supp }\hat{h}$. I am looking for an expression of $\mathcal{F}(|f\star h|)$ in terms of $\mathcal{F}(f)$ and $\mathcal{F}(h)$ without explicitly calculating the inverse Fourier transform.