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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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    You can find the answer directly by using the scaling property of the Fourier Transform, namely $W_1(f) = 5W(5f)$.2012-03-23
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    @Tpofofn: Thanks, I just realized I did not need to find the inverse to get $w_1(t)$ because it was only asking for the frequency spectrum. But, how would I find the inverse of $W(f)$ to get what $w(t)$ is. Either using the definition which I started, or transform properties. I managed to get this from using the properties but not sure if it is correct: $w(t)= 1/2e^{-t}\frac{\mathrm{d}w}{\mathrm{d}t} u(t)\,$ where $u(t)$ is the unit step function.2012-03-23
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    "..not sure if it is correct..." Hint: Try taking the _forward_ Fourier transform of what you think $w(t)$ is to see if you get $W(f)$.2012-03-23
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    $\frac{d}{dt}(w(t)e^t)=\frac{d}{dt}(\int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}e^{(j2\pi f+1)t} df)$ you can try that way to cancel $1+j2\pi f$ and to do simple form of integral?2012-03-23
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    @Mathlover: Thanks. So when doing this, I get for the right hand side before evaluating the limits is: $\frac{e^{t+2 i \pi f t} (2 \pi f t+i)}{4 \pi t^2}$. Now, plugging in limits, the integral will not converge on $[-\infty,\infty]\,$. How do we go from here to fully obtain the inverse of $W(f)$?2012-03-23
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    You should get for the right hand side : $\frac{d}{dt}(w(t)e^t)=\int_{-\infty}^{\infty} j\pi f e^{(j2\pi f+1)t} df=e^t\int_{-\infty}^{\infty} j\pi f e^{j2\pi ft} df$2012-03-23
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    @Mathlover: Exactly is what I get. I had done: $\frac{d}{dt}(w(t)e^t)=\frac{d}{dt}(\int \! \frac{j\pi f}{1+j2\pi f}e^{(j2\pi f+1)t} df)=\frac{e^{t+2 i \pi f t} (2 \pi f t+i)}{4 \pi t^2}$, because I know the integral would not converge, but when doing: $\frac{d}{dt}(w(t)e^t)=\frac{d}{dt}(\int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}e^{(j2\pi f+1)t} df)$, it equal to what you have. So, for this last integral equation, this would be what $w(t)$ is? That's pretty interesting, for it to be in terms of a definite integral.2012-03-23
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    $\frac{e^{t+2 i \pi f t} (2 \pi f t+i)}{4 \pi t^2}$ be careful that right hand cannot have any f term after integral2012-03-23
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    @Mathlover: I didn't get what you meant by that. It would be in terms of $f$ correct because its indefinite and its whats the independent variable is for that case, right? So would our $w(t) = e^t\int_{-\infty}^{\infty} j\pi f e^{j2\pi ft} df$?2012-03-23
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    Another hint: $\int_{-\infty}^{\infty} j2\pi f e^{j2\pi ft} df= \delta '(t) $2012-03-23
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    @Mathlover: Ah hah, I knew that integral looked familiar. I do not have a book handy, but is this a transform pair I believe or property? I just remember seeing it sometime ago, but haven't used it before. Which now makes more sense in terms of what functional constructs that has infinite amplitude, such as the dirac. Thank You!2012-03-23
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    @nightowl, you do not have to compute the inverse to scale time. You can change the Fourier Transform directly. Note that stretching time compresses the F.T. in frequency and visa versa. You can apply this property directly without integrating. See the following http://en.wikipedia.org/wiki/Fourier_transform#Basic_properties2012-03-24
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    @Tpofofn: Thanks, I realized that. But I need the function $w(t)$ to determine the output to another system with $w(t)$ being the input. This is why I needed to integrate or use properties to transform $W(f) \leftrightarrow w(t)$.2012-03-24
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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

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