Problem
My goal is to show the following relationship holds: $$ \mathcal{S} \triangleq \frac{\int\int_{-\infty}^{\infty}P(x,y)e^{j\phi(x,y)}dxdy} {\int\int_{-\infty}^{\infty}P(x,y) dx dy} =\frac{\int\int_{-\infty}^{\infty}\mathcal{H}(f_x, f_y) df_x df_y} {\int\int_{-\infty}^{\infty}\mathcal{H}_0(f_x, f_y) df_x df_y} \tag{1}\label{eq:goal} $$ where $$ \mathcal{H}(f_x, f_y)= \mathcal{F}\{h(u,v)\} \tag{2}\label{eq:H}\\ \mathcal{H}_0(f_x,f_y)= \mathcal{F}\{h_0(u,v)\} $$ Note that $$ h(u,v) = \frac{1}{\lambda^2z^2}\left|\mathcal{F}\{P(x,y)e^{j\phi(x,y)}\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2 \\ h_0(u,v) = \frac{1}{\lambda^2z^2}\left|\mathcal{F}\{P(x,y)\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2 \tag{3}\label{eq:h} $$
My attempt
First, I simply substitute \eqref{eq:h} into \eqref{eq:H} to get $$ \mathcal{H}(f_x, f_y)= \frac{1}{\lambda^2z^2}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)e^{j\phi(x,y)}\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\}\\ \mathcal{H}_0(f_x, f_y)= \frac{1}{\lambda^2z^2}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\} \tag{4}\label{eq:H_simplify} $$ I'd like to somehow reshape the RHS of \eqref{eq:goal} into LHS of \eqref{eq:goal}. So, using \eqref{eq:H_simplify}, it follows that $$ \frac{\int\int_{-\infty}^{\infty}\mathcal{H}(f_x, f_y) df_x df_y} {\int\int_{-\infty}^{\infty}\mathcal{H}_0(f_x, f_y) df_x df_y} =\frac{\int\int_{-\infty}^{\infty}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)e^{j\phi(x,y)}\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\}df_x df_y} {\int\int_{-\infty}^{\infty}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\}df_x df_y}\tag{5}\label{end} $$ The remaining task is to somehow show the following relationship: $$ \frac{\int\int_{-\infty}^{\infty}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)e^{j\phi(x,y)}\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\}df_x df_y} {\int\int_{-\infty}^{\infty}\mathcal{F}\left\{\left|\mathcal{F}\{P(x,y)\}_{f_x=\frac{u}{\lambda z}, f_y=\frac{v}{\lambda z}} \right|^2\right\}df_x df_y} =\frac{\int\int_{-\infty}^{\infty}P(x,y)e^{j\phi(x,y)}dxdy} {\int\int_{-\infty}^{\infty}P(x,y) dx dy}\tag{6}\label{wannashow} $$ but I have no idea how to proceed. I'm sure I need to use the Parseval's theorem in some way but the first thing is to simplify \eqref{end} in some way. Does anyone have an idea?
