View source for Getting X(w) from X(f) - Rhea

[[Category:ECE438Spring2009mboutin]] 

Starting with some <math>\,\! X(f)</math>, we want to derive a mathematical expression for <math>\,\! X(w)</math>

Though we already know that it's just some shift/scale version with period 2*pi, here is the math behind it.


We know <math>\,\! X_s(f) = FsRep_{Fs}[X(f)]</math> from the discussion of <math> \,\!x_s(t) = comb_t(x(t))</math>


From the notes, we also know the relationship between <math>\,\! X(w)</math> and <math>\,\! X_s(f)</math>

*<math>\,\! X(w) = X_s((\frac{w}{2\pi})F_s)</math>

Rewriting <math>\,\! X_s(f)</math>

*<math>\,\! X_s(f) = X(f)*\sum_{-\infty}^{\infty}\delta(f-F_sk)</math>

Substituting known relation 
*<math>\,\! X(w) = X((\frac{w}{2\pi})F_s)*\sum_{-\infty}^{\infty}\delta((\frac{w}{2\pi})F_s-F_sk)</math>

Using delta properties
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