Latest revision as of 11:08, 15 September 2010

How to obtain the CTFT of a shifted unit impulse in terms of f in hertz (from the formula in terms of $\omega$ )

$x(t)=\delta (t-t_0)$

$\mathcal{X}(\omega )=e^{-iwt_0}$

To obtain X(f), use the substitution

$\omega= 2 \pi f$ .

More specifically

$\begin{align} X(f)&=\mathcal{X}(2\pi f)=e^{-i2\pi ft_0} \end{align}$

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-{|
+=How to obtain the CTFT of a shifted unit impulse in terms of f in hertz (from the formula in terms of <math>\omega</math>) =
-|-
-| align="left" style="padding-left: 0em;" | CTFT of a shifted unit impulse
+<math> x(t)=\delta (t-t_0) </math>
-|-
-| <math>X(f)=\mathcal{X}(2\pi f)=e^{-i2\pi ft_0} </math>
+<math> \mathcal{X}(\omega )=e^{-iwt_0}</math>
-|-
-|}
+To obtain X(f), use the substitution
+<math>\omega= 2 \pi f </math>.
+More specifically
+<math>
+\begin{align}
+X(f)&=\mathcal{X}(2\pi f)=e^{-i2\pi ft_0}
+\end{align}
+</math>
+----
+[[ECE438_HW1_Solution|Back to Table]]