I think I see an error in his work. He says:
$ \,\! X_s(f) = FsRep_{Fs}[X(f)] $
And directly following that is the claim:
$ \,\! X_s(f) = FsX(f)*\sum_{-\infty}^{\infty}\delta(f-F_sk) $
I am pretty sure this is incorrect. From the notes and/or the posted equation sheet, the Rep function is not a summation of deltas. Rather, it is a summation of the X function, which produces copies, or repititions, hence the name. He has actually implemented a Comb function, which ultimately leads to his doom, as his answer is wrong. The equation should read:
$ \,\! X_s(f) = Fs*\sum_{-\infty}^{\infty}X(f-F_sk) $
The following equation that he put is correct:
$ \,\! X(w) = X_s((\frac{w}{2\pi})F_s) $
Thus, we can substitute, and we get:
$ \,\! X_s(f) = Fs*\sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s-F_sk) $
Combining the summation and the shift into one Rep produces the following:
$ \,\! X_s(f) = FsRep_{Fs}[X(((\frac{w}{2\pi})F_s)] $
Which is what is in the notes of Professor A. Alternatively, we can just substitute into the original equation with the Rep functions, without converting to a summation and then converting back. This would be a one-step process and would produce the same result, which leaves me a bit curious. Is it really this simple? One step?
