Difference between revisions of "Homework 3 ECE438F09" - Rhea

Latest revision as of 07:37, 25 August 2010

$\displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0$

$\int_{-\infty}^{\infty}\delta(x)dx = 1$

$\displaystyle Let\;\;\;y=\alpha x\;\;\;\;\;\;\;\;\;\;\;\;\;dx=\frac{dy}{\alpha}$

$\displaystyle\int_{-\infty}^{\infty}\delta(\alpha x)dx=\int_{-\infty}^{\infty}\delta(y)\frac{dy}{\alpha}=\frac{1}{\alpha}$

$\displaystyle \delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)$

$\displaystyle 2\pi\delta(\omega)=\delta(f)$

To convert $\delta(f)$ to radians, simply replace $\delta(f)$ with $2\pi\delta(\omega)$

$P_T(f)=\frac{1}{T_s}\sum_{n=-\infty}^{\infty}\delta(f-\frac{n}{T_s})\;\;\;\;\;\;\;\;\;\;\;f_s=\frac{1}{T_s}$

$P_T(\omega)=\frac{2\pi}{T_s}\sum_{n=-\infty}^{\infty}\delta(w-n\frac{2\pi}{T_s})\;\;\;\;\;\;\;w_s=\frac{2\pi}{T_s}$

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-[[ECE438_(BoutinFall2009)|Back to ECE438 course page]]
+[[Hw3ECE438F09boutin|Return to previous page]]
 ==Scaling of the Dirac Delta (Impulse Function)==
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 x)dx=\int_{-\infty}^{\infty}\delta(y)\frac{dy}{\alpha}=\frac{1}{\alpha}</math>
-==Hence,==
+==Therefore...==
-<math>\displaystyle\delta(\omega)=\delta(\frac{f}{2\pi})=2\pi\delta(f)</math>
+<math>\displaystyle \delta(\omega)=\delta(2\pi f)=\frac{1}{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.
+<math>\displaystyle 2\pi\delta(\omega)=\delta(f)</math>
+To convert <math>\delta(f)</math> to radians, simply replace <math>\delta(f)</math> with <math>2\pi\delta(\omega)</math>
 ==Which also means that..==
@@ Line 27: / Line 28: @@
 <math>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}</math>
+[[Hw3ECE438F09boutin|Return to previous page]]
+[[ECE438|Return to ECE438]]