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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
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

1 Answer 1

There is a property of the Fourier Transform which explains this. It's the time scale property $:$ $$ \mathscr{F}\{f(at)\} = \frac{1}{|α|} F\Big(j\frac{ω}{α}\Big) $$

Where $α$ is the scaling constant. For example considering the standard function$: f(t)=e^{-kt} $ and it's fourier transform which is $: F(jω)=\frac{1}{jω + k} $ if you scale that function, for example with a scaling constant of $-1$, then it's fourier transform would become$:$ $$ \mathscr{F}\{f(-t)\} = \frac{1}{|-1|} \frac{1}{1+j\frac{ω}{-1}} = \frac{1}{1-jω} $$ In this case the absolute value in your function makes no difference.

In your case we also want a time-shift of the function. and there's another Fourier property about this. If you're time-shifting a function, then you multiply the fourier transform of that function by $e^{-jωt_o}$, such that $$ \mathscr{F}\{f(t-t_o)\} = F(jω)e^{-jωt} $$

$(ω=2πf)$ So you would have to apply both properties. Be careful though that scaling and shifting are not commutative. Shifting should happen first and then scaling. So you have to apply the time-shift property first and afterwards the scaling property to get the Fourier transform properly.

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