(New page: I think I see an error in his work. He says: <math>\,\! X_s(f) = FsRep_{Fs}[X(f)]</math> And directly following that is the claim: <math>\,\! X_s(f) = FsX(f)*\sum_{-\infty}^{\infty}\del...) |
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Combining the summation and the shift into one Rep produces the following: | Combining the summation and the shift into one Rep produces the following: | ||
| − | <math>\,\! X_s(f) = FsRep_{Fs}[X | + | <math>\,\! X_s(f) = FsRep_{Fs}[X((\frac{w}{2\pi})F_s)]</math> |
| − | 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 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: |
| + | |||
| + | <math>\,\! X_s(f) = FsRep_{Fs}[X((\frac{w}{2\pi})F_s)]</math> | ||
| + | |||
| + | Which leaves me a bit curious. Is it really this simple? One step? Anyone have any ideas? | ||
Revision as of 19:15, 17 February 2009
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:
$ \,\! X_s(f) = FsRep_{Fs}[X((\frac{w}{2\pi})F_s)] $
Which leaves me a bit curious. Is it really this simple? One step? Anyone have any ideas?
