- #1

s3a

- 818

- 8

- Homework Statement
- "Compute the Fourier transform G_B(u,v) of the signal g_B(x,y) = 0.5 circ(x/B,y/B) - 0.5 rect(2x/B,2y/B) + 0.8 circ(4x/B,4y/B) - 0.8 circ(8x/B,8y/B), where B is a constant, using these tables ( https://docdro.id/dPaHriL ) (which is a two-page document)."

- Relevant Equations
- (i), (iv) and possibly (v) or (vi) from these tables ( https://docdro.id/dPaHriL (which is a two-page document) ).

Hello, everyone. :)

I'm trying to do a certain problem regarding Fourier transforms (but one that's supposedly easy, because of just using tables, rather than fully computing stuff), and I know how to do it, but I don't know why it works. Here's the problem statement.:

"Compute the Fourier transform G_B(u,v) of the signal g_B(x,y) = 0.5 circ(x/B,y/B) - 0.5 rect(2x/B,2y/B) + 0.8 circ(4x/B,4y/B) - 0.8 circ(8x/B,8y/B), where B is a constant, using these tables ( https://docdro.id/dPaHriL ) (which is a two-page document)."

Here ( https://docdro.id/oWMtf7i ) is the answer.

For both the rect and circ (which should probably be ellipse, right?) functions, I think I'm supposed to use (i) and (iv) (from the second page of the tables pdf document), and I get the solution by replacing all occurrences of u and v in each of the four subparts of g_B(x,y), 0.5circ(x/B,y/B), -0.5rect(2x/B,2y/B), 0.8circ(4x/B,4y/B) and -0.8circ(8x/B,8y/B), by u multiplied by [1/(B/denominator)]^(-T) (-T means inverse transpose) = B/denominator and v multiplied by [1/(B/denominator)]^(-T) = B/denominator, respectively, but by multiplying the whole frequency-domain function by 1/det [1/(B/denominator)] = B/denominator twice (as in squared, not times two), instead of once. To try to justify this, I'm thinking one has to perhaps think of the frequency-domain functions as a product of two functions and use (v) or (vi), but if I'm even on the right track, the details of that are not clicking in my mind.

If someone could help me fully understand what's going on, I would GREATLY appreciate it!

I'm trying to do a certain problem regarding Fourier transforms (but one that's supposedly easy, because of just using tables, rather than fully computing stuff), and I know how to do it, but I don't know why it works. Here's the problem statement.:

"Compute the Fourier transform G_B(u,v) of the signal g_B(x,y) = 0.5 circ(x/B,y/B) - 0.5 rect(2x/B,2y/B) + 0.8 circ(4x/B,4y/B) - 0.8 circ(8x/B,8y/B), where B is a constant, using these tables ( https://docdro.id/dPaHriL ) (which is a two-page document)."

Here ( https://docdro.id/oWMtf7i ) is the answer.

For both the rect and circ (which should probably be ellipse, right?) functions, I think I'm supposed to use (i) and (iv) (from the second page of the tables pdf document), and I get the solution by replacing all occurrences of u and v in each of the four subparts of g_B(x,y), 0.5circ(x/B,y/B), -0.5rect(2x/B,2y/B), 0.8circ(4x/B,4y/B) and -0.8circ(8x/B,8y/B), by u multiplied by [1/(B/denominator)]^(-T) (-T means inverse transpose) = B/denominator and v multiplied by [1/(B/denominator)]^(-T) = B/denominator, respectively, but by multiplying the whole frequency-domain function by 1/det [1/(B/denominator)] = B/denominator twice (as in squared, not times two), instead of once. To try to justify this, I'm thinking one has to perhaps think of the frequency-domain functions as a product of two functions and use (v) or (vi), but if I'm even on the right track, the details of that are not clicking in my mind.

If someone could help me fully understand what's going on, I would GREATLY appreciate it!