# Fourier transform me

1. Dec 12, 2009

### hanafnaf

fourier transform plz 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)

plz solve it and help me

2. Dec 12, 2009

### elibj123

Re: fourier transform plz help me

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. lets 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 gonna 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).

3. Dec 12, 2009

### hanafnaf

Re: fourier transform plz help me

i can't understand, plz explain more