Starting with some $ \,\! X(f) $, we want to derive a mathematical expression for $ \,\! X(w) $
Though we already know that it's just some shift/scale version with period 2*pi, here is the math behind it.
We know $ \,\! X_s(f) = FsRep_{Fs}[X(f)] $ from the discussion of $ \,\!x_s(t) = comb_t(x(t)) $
From the notes, we also know the relationship between $ \,\! X(w) $ and $ \,\! X_s(f) $
- $ \,\! X(w) = X_s((\frac{w}{2\pi})F_s) $
Rewriting $ \,\! X_s(f) $
- $ \,\! X_s(f) = X(f)*\sum_{-\infty}^{\infty}\delta(f-F_sk) $
Substituting known relation
- $ \,\! X(w) = X((\frac{w}{2\pi})F_s)*\sum_{-\infty}^{\infty}\delta((\frac{w}{2\pi})F_s-F_sk) $
Using delta properties
