| Line 12: | Line 12: | ||
From the notes, we also know the relationship between <math>\,\! X(w)</math> and <math>\,\! X_s(f)</math> | From the notes, we also know the relationship between <math>\,\! X(w)</math> and <math>\,\! X_s(f)</math> | ||
| − | + | <p><math>\,\! X(w) = X_s((\frac{w}{2\pi})F_s)</math></p> | |
Rewriting <math>\,\! X_s(f)</math> | Rewriting <math>\,\! X_s(f)</math> | ||
| − | + | <math>\,\! X_s(f) = FsX(f)*\sum_{-\infty}^{\infty}\delta(f-F_sk)</math> | |
| − | Substituting known relation | + | Substituting known relation |
| − | + | <p><math>\,\! X(w) = FsX((\frac{w}{2\pi})F_s)*\sum_{-\infty}^{\infty}\delta((\frac{w}{2\pi})F_s-F_sk)</math></p> | |
Using LTI, rearrange the equation | Using LTI, rearrange the equation | ||
| − | + | <p><math>\,\! X(w) = Fs\sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((\frac{w}{2\pi})F_s-F_sk)</math></p> | |
Re-arrange the delta function | Re-arrange the delta function | ||
| − | + | <p><math>\,\! X(w) = Fs\sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((\frac{F_s}{2\pi})(w-k2\pi))</math></p> | |
Using delta properties, you can take out the <math>(\frac{F_s}{2\pi})</math> | Using delta properties, you can take out the <math>(\frac{F_s}{2\pi})</math> | ||
| − | + | <p><math>\,\! X(w) =Fs(\frac{2\pi}{F_s}) \sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((w-k2\pi))</math></p> | |
| − | + | ||
The <math>F_s</math> will cancel and employ sifting to get | The <math>F_s</math> will cancel and employ sifting to get | ||
| − | + | <p><math>\,\! X(w) =2\pi \sum_{-\infty}^{\infty}X((\frac{w-k2\pi}{2\pi})F_s)</math></p> | |
Now you can see how your X(f) is being scaled and shifted | Now you can see how your X(f) is being scaled and shifted | ||
Revision as of 11:54, 16 February 2009
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) = FsX(f)*\sum_{-\infty}^{\infty}\delta(f-F_sk) $
Substituting known relation
$ \,\! X(w) = FsX((\frac{w}{2\pi})F_s)*\sum_{-\infty}^{\infty}\delta((\frac{w}{2\pi})F_s-F_sk) $
Using LTI, rearrange the equation
$ \,\! X(w) = Fs\sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((\frac{w}{2\pi})F_s-F_sk) $
Re-arrange the delta function
$ \,\! X(w) = Fs\sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((\frac{F_s}{2\pi})(w-k2\pi)) $
Using delta properties, you can take out the $ (\frac{F_s}{2\pi}) $
$ \,\! X(w) =Fs(\frac{2\pi}{F_s}) \sum_{-\infty}^{\infty}X((\frac{w}{2\pi})F_s)*\delta((w-k2\pi)) $
The $ F_s $ will cancel and employ sifting to get
$ \,\! X(w) =2\pi \sum_{-\infty}^{\infty}X((\frac{w-k2\pi}{2\pi})F_s) $
Now you can see how your X(f) is being scaled and shifted
