I have a problem in DFT. It was one of my past-year exam papers questions.
Question:
Let $F(u,v)$ be the 2-D Fourier transform of a 2-D continuous function $f(x,y)$. Derive in terms of $F(:,:)$ the 2-D Fourier transform of each of the following functions
1) $f(x,-2y)$
2) $f(x+2y,y)$
I know how to do 1-D Fourier transforms but not 2-D. I'm not sure how to start on it and need some guidance.
For the second part, this was my approach. Please let me know if it's right or correct me if it's wrong.
Let $τ= x + 2y$ hence $x = τ-2y$ and $dx = dτ$ $$ \begin{align} \mathfrak{F}\{f(x+2y,y)\}&=∬ f(τ,y)e^{−j2π(u(τ-2y) +vy )} dx\ dv\\ \mathfrak{F}\{f(x+2y,y)\}&= ∬ f(τ,y)e^{−j2π(uτ+(-2u+v)y )} dx\ dτ \\ \mathfrak{F}\{f(x+2y,y)\}&= F(u,-2u+v) \end{align} $$