I have a signal in the 2D spatial frequency domain
$$S(K_X,K_Y) = e^{j(R_B K_Y-K_X x_t)}$$
My goal is to bring this signal from the spatial frequency domain into the spatial coordinate domain by performing a 2D inverse Fourier transform.
$$S(X_a,Y_s)= \int \int S(K_X,K_Y) e^{j(K_Y Y_s + K_X X_a)} \space dK_Y \space dK_X$$
When I perform the integration I end up with
$$S(X_a,Y_s) = \frac{-1}{(R_B+Y_s)(X_a-x_t)} e^{j(R_B + Y_s)K_Y} e^{j(X_a-x_t)K_X}$$
The spatial frequency terms still exist in the final equation. I am not sure how to correct this mistake and obtain the proper analytical solution. What is the proper method to convert the signal into the spatial coordinate domain?
I have also tried to make use of principle of stationary phase and have been unsuccessful in getting the proper solution
This can be obtained by using the following Fourier Pair relationship,
$$e^{j\omega_0t} \Leftrightarrow 2\pi\delta(\omega-\omega_0)$$
After applying the above pair, we obtain the final solution,
$$S(X_a,Y_s)=4\pi^2\delta(X_a-x_t)\delta(Y_s-R_b)$$