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I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following.

$w(t)$ has the Fourier Transform $W(f)= \dfrac{(j\pi f)}{(1+j2\pi f)}$

Given $w_1(t) = w(t/5)$ we need to find the spectrum of $w(t)$.

So to start, we need to know what $w(t)$ is, and this can be done by taking the I.F.T. to find out what $w$ is and then we would be able to plug in $t=t/5$ to find the F.T. of that to get the spectrum for $w_1(t)$. The trouble I am having is evaluating the integral for the I.F.T. This is what I have done so far.

$ \begin{align} \displaystyle w(t) &= \int_{-\infty}^{\infty} \! W(f)e^{j\omega t}\,\mathrm{d}f \\ &= \int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}e^{j2\pi ft}\,\mathrm{d}f, \text{ where } \omega=2\pi f \\ &= \int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}\Big[\cos(2\pi f t)+j\sin(2\pi ft) \Big]\,\mathrm{d}f \end{align} $

From here, I do not know the best way to go about integrating the complex component with the complex exponential or trig functions.

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    @Mathlover: where does the $e^t$ go, that's on the outside of the integral on the RHS of the equation?2012-03-25

0 Answers 0