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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> = | + | =How to obtain the CT Fourier transform formula in terms of f in hertz (from the formula in terms of <math>\omega</math>) = |
Recall: | Recall: | ||
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<math> \mathcal{X}(\omega )=\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> | ||
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To obtain X(f), use the substitution | To obtain X(f), use the substitution | ||
| − | <math>\ | + | <math>\omega= 2 \pi f </math>. |
| + | |||
| + | More specifically | ||
<math> | <math> | ||
Revision as of 09:01, 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} $
