Fourier transform

Fourier Transform

What is a Fourier Transform? A Fourier Transform is a function that changes a signal or waveform from the time domain into the frequency domain. One simple way to look at it is this: Suppose you are at the beach, watching the waves. You could say that a wave hits the shore at specific times (0 second, 2 seconds, 4 seconds, etc.) that would be describing the waveform in the time domain. If, however, you were to say that the waves hit the beach every two seconds, that would be describing it in the frequency domain. So a Fourier transform would take the data given in the time domain and convert that into the frequency domain. The function that does this is: $X(f)=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}\,dt$ .

The reverse is also possible. You can take the information from the frequency domain, and convert it into the time domain using an Inverse Fourier Transform.

From the Fourier Transform to the Inverse Fourier Transform

Lets start with the basic Fourier Transform: $X(f)=\int _{-\infty }^{\infty }x(t)e^{-j2\pi ft}\,dt$

Suppose that we have some function, say $\beta (t)$ , that is nonperiodic and finite in duration.
This means that $\beta (t)=0$ for some $T_{\alpha }<\left|t\right|$

Now let's make a periodic function $\gamma (t)$ by repeating $\beta (t)$ with a fundamental period $T_{\zeta }$ . Note that $\lim _{T_{\zeta }\to \infty }\gamma (t)=\beta (t)$
The Fourier Series representation of $\gamma (t)$ is
$\gamma (t)=\sum _{k=-\infty }^{\infty }\alpha _{k}e^{j2\pi fkt}$ where $f={1 \over T_{\zeta }}$
and $\alpha _{k}={1 \over T_{\zeta }}\int _{-{T_{\zeta } \over 2}}^{T_{\zeta } \over 2}\gamma (t)e^{-j2\pi kt}\,dt$
$\alpha _{k}$ can now be rewritten as $\alpha _{k}={1 \over T_{\zeta }}\int _{-\infty }^{\infty }\beta (t)e^{-j2\pi kt}\,dt$
From our initial identity then, we can write $\alpha _{k}$ as $\alpha _{k}={1 \over T_{\zeta }}\mathrm {B} (kf)$
and $\gamma (t)$ becomes $\gamma (t)=\sum _{k=-\infty }^{\infty }{1 \over T_{\zeta }}\mathrm {B} (kf)e^{j2\pi fkt}$
Now remember that $\beta (t)=\lim _{T_{\zeta }\to \infty }\gamma (t)$ and ${1 \over {T_{\zeta }}}=f.$
Which means that $\beta (t)=\lim _{f\to 0}\gamma (t)=\lim _{f\to 0}\sum _{k=-\infty }^{\infty }f\mathrm {B} (kf)e^{j2\pi fkt}$
Which is just to say that $\beta (t)=\int _{-\infty }^{\infty }\mathrm {B} (f)e^{j2\pi fkt}\,df$

So we have that ${\mathcal {F}}[\beta (t)]=\mathrm {B} (f)=\int _{-\infty }^{\infty }\beta (t)e^{-j2\pi ft}\,dt$
Further ${\mathcal {F}}^{-1}[\mathrm {B} (f)]=\beta (t)=\int _{-\infty }^{\infty }\mathrm {B} (f)e^{j2\pi fkt}\,df$

Some Useful Fourier Transform Pairs

${\mathcal {F}}[\alpha (t)]={\frac {1}{\mid \alpha \mid }}f({\frac {\omega }{\alpha }})$

${\mathcal {F}}[c_{1}\alpha (t)+c_{2}\beta (t)]$	$=\int _{-\infty }^{\infty }(c_{1}\alpha (t)+c_{2}\beta (t))e^{-j2\pi ft}\,dt$
	$=\int _{-\infty }^{\infty }c_{1}\alpha (t)e^{-j2\pi ft}\,dt+\int _{-\infty }^{\infty }c_{2}\beta (t)e^{-j2\pi ft}\,dt$
	$=c_{1}\int _{-\infty }^{\infty }\alpha (t)e^{-j2\pi ft}\,dt+c_{2}\int _{-\infty }^{\infty }\beta (t)e^{-j2\pi ft}\,dt=c_{1}\mathrm {A} (f)+c_{2}\mathrm {B} (f)$

${\mathcal {F}}[\alpha (t-\gamma )]=e^{-j2\pi f\gamma }\mathrm {A} (f)$
${\mathcal {F}}[\alpha (t)*\beta (t)]=\mathrm {A} (f)\mathrm {B} (f)$
${\mathcal {F}}[\alpha (t)\beta (t)]=\mathrm {A} (f)*\mathrm {B} (f)$
Some other usefull pairs can be found here: Fourier Transforms