First, download the paper An Excel Spreadsheet for a One-Dimensional Fourier Map in X-ray Crystallography, by William Cleg. There is a spreadsheet under the supplement section you may want to download so you can play along (see picture below).
I had to adjust the security level in Excel in order for the macros to run (Tools -> Marcos -> Security).
I also had to restart Excel in order for the change to take effect.
1) Scroll right and you will see the labels for the first 13 rows (under W).
The Bragg reflections are listed in the first row
The phases are listed in row 3 (highlighted).
The first exercise uses a reduction in calculating the 3D fourier transformation to 1D. So, instead of ‘hkl’ you will just see ‘l’ written (see equations at the left of page 909).
Note: l=0, which is the total number of electrons in the unit cell and, therefore, the sum of all other reflections
As you toggle on and off ‘Use Correct Phases’ and ‘Random Phases’, you will see row 3 adjusting (highlighted) as well as your graph updating
Notice how the fourier synthesis is affected by these adjustments.
Now, suppose you don’t know the phases (like in the random case). How would you get to the ‘correct phases’?
We cannot simply guess the signs since there are 2^21 combinations (2 since, in this case, the phase can only be -1 or +1; and 21 because that is the number of Bragg reflections measured).
This presents a problem. These signs are not measured directly and there are, literally, millions of possibilities to generate the correct fourier synthesis. You can imagine how the problem becomes more complicated when you are dealing with 3 dimensions and thousands of atoms.