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When a function $f(t)=exp(-|t|)$ for example undergoes Fourier Transformation, it gives $F(w)=\frac{-2}{1+w^2}$

But what happens to the result if the time scale is scaled and shifted, so that $t \rightarrow\ t^* =at+b $ ?

How will the Fourier Transformation of the function change?

Edit: Following is the approach I took but is unsure about it's correctness

$Since \ t \rightarrow\ t^* =at+b \\ f(t) \rightarrow\ f(at+b) = e^{-|at+b|}) \\ therefore \\ F(w) = \frac{e^{-iwb}}{|-a|} \ * \frac{2}{1+(\frac{w}{-a})^2} $

The part I'm most uncertain about is $e^{-|at+b|}$ where there is the absolute value of at+b. I'm only treating it as a bracket at the moment, I'm not sure if that would change anything.

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Homework? Did you try to work it out? –  leonbloy Jun 20 '12 at 1:51
    
@leonbloy Just self revision, but anyway, I've added more information to what I have tried, please take a look. Thanks –  Synia Jun 20 '12 at 2:11
2  
Why don't you write out the formula for the Fourier transform of $g(t) = f(at+b)$ as $$\int_{-\infty}^{\infty}f(at+b)\exp(-i2\pi ft)\,\mathrm dt$$ and then make a change of variable $\tau = at+b$? –  Dilip Sarwate Jun 20 '12 at 3:00
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