I don't know how to solve the following convolution problem: $$[f(x)f^*(x+a)]\otimes_x[\delta(x)\delta(x-a)]$$ $*$ is complex conjugate and $\otimes$ stands for convolution. I only know some basic rules of convolution and Fourier transform, like $$f(x)\otimes \delta(x-a)=f(x-a)$$ $$f(x)\delta(x-a)=f(a)\delta(x-a)$$ Could anyone give the process of solving the problem or tell me some rules that is necessary to solve it?
The whole formula is as follows: $$G(x,y) = A(x)A^*(x+y)\otimes_x \{[\delta(x)+\phi(x)][\delta(x-y)+\phi^*(x-y)]\}=|A(x)|^2\delta(y)+A(x)A^*(x+y)\phi^*(-y)+A(x-y)A^*(x)\phi(y)$$ $\phi(x) << 1$ I think I know how to get the last two terms: $$A(x)A^*(x+y)\otimes[\delta(x)\phi^*(x-y)] = A(x)A^*(x+y)\otimes[\delta(x)\phi^*(-y)] = A(x)A^*(x+y)\phi^*(-y)$$ $$A(x)A^*(x+y)\otimes[\delta(x-y)\phi(x)] = A(x)A^*(x+y)\otimes[\delta(x-y)\phi(y)] = A(x-y)A^*(x)\phi(y)$$ But how to get the first term?