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<math>\displaystyle\delta(\omega)=\delta(\frac{f}{2\pi})=2\pi\delta(f)</math> | <math>\displaystyle\delta(\omega)=\delta(\frac{f}{2\pi})=2\pi\delta(f)</math> | ||
| − | + | This may seem strange at first. I had the urge to simply replace \omega with 2\pi f as well. But that wouldn't be telling the same story. If you have an impulse located at 1 hz with some arbitrary magnitude, then the signal in radians would naturally be the same impulse located at 2\pi (because w = 2\pi f). We'll ignore the magnitude for now. Essentially all that is done going from <math>X(f)->X(w)</math> is a frequency scale where every frequency is multiplied by 2\pi to obtain the spectrum in radians. | |
==Which also means that..== | ==Which also means that..== | ||
Revision as of 16:11, 22 September 2009
Contents
Scaling of the Dirac Delta (Impulse Function)
$ \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0 $
Mini Proof
$ \int_{-\infty}^{\infty}\delta(x)dx = 1 $
$ \displaystyle Let\;\;\;y=\alpha x\;\;\;\;\;\;\;\;\;\;\;\;\;dx=\frac{dy}{\alpha} $
$ \displaystyle\int_{-\infty}^{\infty}\delta(\alpha x)dx=\int_{-\infty}^{\infty}\delta(y)\frac{dy}{\alpha}=\frac{1}{\alpha} $
Hence,
$ \displaystyle\delta(\omega)=\delta(\frac{f}{2\pi})=2\pi\delta(f) $
This may seem strange at first. I had the urge to simply replace \omega with 2\pi f as well. But that wouldn't be telling the same story. If you have an impulse located at 1 hz with some arbitrary magnitude, then the signal in radians would naturally be the same impulse located at 2\pi (because w = 2\pi f). We'll ignore the magnitude for now. Essentially all that is done going from $ X(f)->X(w) $ is a frequency scale where every frequency is multiplied by 2\pi to obtain the spectrum in radians.
Which also means that..
$ P_T(f)=\frac{1}{T_s}\sum_{n=-\infty}^{\infty}\delta(f-\frac{n}{T_s})\;\;\;\;\;\;\;\;\;\;\;f_s=\frac{1}{T_s} $
$ P_T(\omega)=\frac{2\pi}{T_s}\sum_{n=-\infty}^{\infty}\delta(w-n\frac{2\pi}{T_s})\;\;\;\;\;\;\;w_s=\frac{2\pi}{T_s} $
