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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

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1 Answer 1

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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,

$$S(X_a,Y_s)=4\pi^2\delta(X_a-x_t)\delta(Y_s-R_b)$$

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