Fourier transform of shifted and scaled functions

[hanafnaf]

fourier transform please help me

if the Fourier transform of f(x,y) is f(u,v) what is the Fourier transform of the following:

f(x+6,y)
f(x,-y)
f(2x+6,y)

please solve it and help me

 

[elibj123]

Just try substituting these into the definition

$$F[f(x,y)](u,v)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(x,y)e^{-i(ux+vy)}dxdy$$

f.e. let's do the first one:

f(x+6,y):

$$F[f(x+6,y)]=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(x+6,y)e^{-i(ux+vy)}dxdy$$

I'm going to make the next change of variables:
$$x->\tilde{x}-6$$

The boundaries of the integral remains unchanged, and so does the differential, but we get:

$$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(\tilde{x},y)e^{-i(u\tilde{x}-6u+vy)}d\tilde{x}dy$$

Notice that $$e^{i6u}$$ is constant wrt to the integration, so we pull it out and what we have left of the integral is simply the Fourier transform of f(x,y) [because x* is just a dummy variable]:

$$e^{i6u}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(\tilde{x},y)e^{-i(u\tilde{x}+vy)}d\tilde{x}dy=e^{i6u}f(u,v)$$

This technique of variable change is the standard technique to observe how shift & scale of the time-domain (the original function) affects the frequency-domain (its Fourier transform).

 

[hanafnaf]

i can't understand, please explain more
thanks for your attention

 

[TheLagrangian]

You obviously don't understand simple substitution in integrals, maybe you should brush up on that first.