(New page: '''Differentiation''' Define a function x(t) with its Fourier transform being X(jw) Then by definition x(t) = <math>\int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt</math> The der...) |
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With the key point being made that differentiation and integration are interchangable operations. The proof of this is not | With the key point being made that differentiation and integration are interchangable operations. The proof of this is not | ||
| − | difficult however it is time consuming. The book assumes that summations and integrations can be | + | difficult however it is time consuming. The book assumes that summations and integrations can be interchanged also, |
therefore I will go without proof of the interchangability of differentiation and integration. | therefore I will go without proof of the interchangability of differentiation and integration. | ||
| − | so ''' | + | so '''Fourier transform'''(x(t)') = jwX(jw) |
Latest revision as of 01:07, 9 July 2009
Differentiation
Define a function x(t) with its Fourier transform being X(jw)
Then by definition x(t) = $ \int\limits_{-\infty}^{\infty}X(jw)e^{(-\jmath wt)}dt $
The derivative of x(t) equals
= $ \int\limits_{-\infty}^{\infty}jwX(jw)e^{(-\jmath wt)}dt $
With the key point being made that differentiation and integration are interchangable operations. The proof of this is not
difficult however it is time consuming. The book assumes that summations and integrations can be interchanged also,
therefore I will go without proof of the interchangability of differentiation and integration.
so Fourier transform(x(t)') = jwX(jw)
