How to calculate wignner transformation or distribution for 2D function f(x,y) that means for four variables W(x,px,y,p​y)=int2(-i​nf...inf,-​inf..inf)f​(x+s/2.y+l​/2)f*(x-s/​2,y-l/2)ex​p(-i*(px*s​+py*l))?

Hi

I'm a new user. I want to perform wignner transfromation for 2D fuction f(x,y) that means first I have to calculatie correlation fuction which is f(x1,y1)f*(x2,y2) or f(x+s/2,y+l/2)f*(x-s/2,y-l/2). W(x,px,y,py)=int2(-inf..inf,-inf..inf)f(x+s/2,y+l/2)f*(x-s/2,y-l/2)exp(-i(pxs+pyl)). For sake of understanding I started with 1D function with lead to 2D wigner transformation for example f(x)=sin(n*pi*x) then w(x,p)=int(-inf...inf)sin(n*pi*(x+s/2))sin(n*pi*(x-s/2))*exp(-ips). For that first I calculateH(x1,x2)=f(x1)*f(x2) then I change valriable by assuming x1=x+s/2 and x2=x-s/2 then to get the wigner transfromation W(x,p)=FFT(H(x+s/2,x-s/2)), but I dont get the correct answer. Also I'm a bit confused how to do zero padding because after I do then i dont get the result in the range i specified even I use FFTShift after FFT. If some one could help me with the 4D wignner transformation, or anyone have better approach that would be greatn.

May thank for your help.