# How to Prove the Fourier Transform of Vertical Line Mathematically?

I've tried to do Fourier transform in Matlab of vertical line. Basically, I make matrix A having size 100x100 and give the value 1s in certain number of coloumn vector, whereas the others is 0s. Here is the script. (You can see the plot result in the image bellow).

A = zeros(100,100);
A(:,10)=1;
A_FFT = abs(fft2(A));
figure, imshow(A,[]);
title('vertical line (A)');
figure, imshow(A_FFT,[]);
title('FFT of Vertical Line (A)');

From the result, I get horizontal line matrix A_FFT(u,v) which is A_FFT(1,:)=100 and the other matrix components are 0s. From the theory, yes, I can understand why the result is like that. But, I need to prove it mathematically. I have tried to write and solve the DFT equation of matrix A, but still don't get the right result. Can anyone help me how to prove this mathematically ?

Thank you very much....

Best regards,
Ardian.

-Click this link to see the image-
left : original vertical line, matrix A; right : DFT of A, by taking fft2

• Are you aware that if $f(x,y)=g(x)h(y)$ then its Fourier transform is $\hat f(\xi,\eta)=\hat g(\xi)\hat h(\eta)$? – Rahul Oct 27 '16 at 15:03
• $a(m,n) = \delta(n-10) \implies A(l,k) = \delta(l) e^{-2i \pi \, 10 k /K}$ – reuns Oct 27 '16 at 15:07
• @rahul : $$F(u,v)=\sum_{x=0}^{M-1} f(x)e^{-j2\pi ux/M} \sum_{y=0}^{N-1} f(y)e^{-j2\pi vy/N}$$, then don't know the how the next because there are two other variables which are u and v in the equation. – Ardian Oct 27 '16 at 15:35
• @user1952009 : what about the amplitude. The amplitude should be the length of the vertical line. Can it be like that ? $$F(u,v) = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} \delta (y-10) e^{-j2\pi (ux/M+vy/N)} = \sum_{x=0}^{M-1} 1 e^{-j2\pi ux/M} \sum_{y=0}^{N-1} \delta(y-10)e^{-j2\pi vy/N}$$ Only when u=0, $$\sum_{x=0}^{M-1}1e^{-j2\pi ux/M}=M$$ and when u!=0, it will be 0. And for $$\sum_{y=0}^{N-1}\delta(y-10)e^{-j2\pi vy/N} = 1e^{-j2\pi 10u/M}$$ And finally $$F(u,v) = M 1e^{-j2\pi 10u/M}$$ Right? – Ardian Oct 27 '16 at 15:59
• A vertical line is of the form $f(x,y) = g(x)h(y)$ where $g(x)=\delta(x-x_0)$ and $h(y)=1$. Do you know the Fourier transforms of a delta function and a constant function? – Rahul Oct 27 '16 at 16:27