Latest revision as of 09:57, 15 September 2010

How to obtain the CT Fourier transform formula in terms of f in hertz (from the formula in terms of $\omega$ )

Recall:

$\mathcal{X}(\omega )=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt$

To obtain X(f), use the substitution

$\omega= 2 \pi f$ .

More specifically

$\begin{align} X(f) &=\mathcal{X}(2\pi f)\\ &=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt \end{align}$

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-{|
+=How to obtain the CT Fourier transform formula in terms of f in hertz (from the formula in terms of <math>\omega</math>) =
-|-
-| align="right" style="padding-right: 1em;" | CT Fourier Transform
+Recall:
-|-
-| <math> X(f)=\mathcal{X}(2\pi f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
+<math> \mathcal{X}(\omega )=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</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)\\
+ &=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt
+\end{align}
+</math>
+----
+[[ECE438_HW1_Solution|Back to Table]]