# Waves/Fourier Transforms

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## Fourier TransformEdit

So far, you've learned how to superimpose a finite number of sinusoidal waves. However, a wave in general can't be expressed as the sum of a finite number of sines and cosines. Fortunately, we have a theorem called Fourier's theorem which basically states that under certain technical assumptions, any function, f(x) is equal to an integral over sines and cosines. In other words,

$f(x)=\int_{-\infty}^{\infty}(c_1(k)\cos(kx)+c_2(k)\sin(kx)) dk$.

Now, if we're given the wave function when t=0, φ(x,0) and the velocity of each sine wave as a function of its wave number, v(k), then we can compute φ(x,t) for any t by taking the inverse Fourier transform of φ(x,0) conducting a phase shift, and then taking the Fourier transform.

Fortunately, the inverse Fourier transform is very similar to the Fourier transform itself.

$c_1(k)=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x)\cos (kx)\,dx \quad c_2(k)=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) \sin (kx)\,dx$

This tells us that, since waves which are very spread out, like the sine wave, have a narrow range of wave numbers, wave functions whose wave numbers are very spread out will only be significant at a narrow range of positions.

## Fourier Transform PropertiesEdit

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,$
$G(\omega)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,$
$G(f)\!\equiv$

$\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,$
1 $a\cdot g(t) + b\cdot h(t)\,$ $a\cdot G(\omega) + b\cdot H(\omega)\,$ $a\cdot G(f) + b\cdot H(f)\,$ Linearity
2 $g(t - a)\,$ $e^{- i a \omega} G(\omega)\,$ $e^{- i 2\pi a f} G(f)\,$ Shift in time domain
3 $e^{ iat} g(t)\,$ $G(\omega - a)\,$ $G \left(f - \frac{a}{2\pi}\right)\,$ Shift in frequency domain, dual of 2
4 $g(a t)\,$ $\frac{1}{|a|} G \left( \frac{\omega}{a} \right)\,$ $\frac{1}{|a|} G \left( \frac{f}{a} \right)\,$ If $|a|\,$ is large, then $g(a t)\,$ is concentrated around 0 and $\frac{1}{|a|}G \left( \frac{\omega}{a} \right)\,$ spreads out and flattens
5 $G(t)\,$ $g(-\omega)\,$ $g(-f)\,$ Duality property of the Fourier transform. Results from swapping "dummy" variables of $t \,$ and $\omega \,$.
6 $\frac{d^n g(t)}{dt^n}\,$ $(i\omega)^n G(\omega)\,$ $(i 2\pi f)^n G(f)\,$ Generalized derivative property of the Fourier transform
7 $t^n g(t)\,$ $i^n \frac{d^n G(\omega)}{d\omega^n}\,$ $\left (\frac{i}{2\pi}\right)^n \frac{d^n G(f)}{df^n}\,$ This is the dual to 6
8 $(g * h)(t)\,$ $\sqrt{2\pi} G(\omega) H(\omega)\,$ $G(f) H(f)\,$ $g * h\,$ denotes the convolution of $g\,$ and $h\,$ — this rule is the convolution theorem
9 $g(t) h(t)\,$ $(G * H)(\omega) \over \sqrt{2\pi}\,$ $(G * H)(f)\,$ This is the dual of 8
10 For a purely real even function $g(t)\,$ $G(\omega)\,$ is a purely real even function $G(f)\,$ is a purely real even function
11 For a purely real odd function $g(t)\,$ $G(\omega)\,$ is a purely imaginary odd function $G(f)\,$ is a purely imaginary odd function

## Fourier Transform PairsEdit

Time Domain Frequency Domain
$x(t) = \mathcal{F}^{-1}\left\{ X(\omega) \right\}$ $X(\omega) = \mathcal{F} \left\{ x(t) \right\}$
1 $X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$ $x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega$
2 $1 \,$ $2\pi\delta(\omega) \,$
3 $-0.5 + u(t) \,$ $\frac{1}{j \omega} \,$
4 $\delta (t) \,$ $1 \,$
5 $\delta (t-c) \,$ $e^{-j \omega c} \,$
6 $u(t) \,$ $\pi \delta(\omega)+\frac{1}{j \omega} \,$
7 $e^{-bt}u(t) \, (b > 0)$ $\frac{1}{j \omega+b} \,$
8 $\cos \omega_0 t \,$ $\pi \left[ \delta(\omega+\omega_0)+\delta(\omega-\omega_0) \right] \,$
9 $\cos (\omega_0 t + \theta) \,$ $\pi \left[ e^{-j \theta}\delta(\omega+\omega_0)+e^{j \theta}\delta(\omega-\omega_0) \right] \,$
10 $\sin \omega_0 t \,$ $j \pi \left[ \delta(\omega +\omega_0)-\delta(\omega-\omega_0) \right] \,$
11 $\sin (\omega_0 t + \theta) \,$ $j \pi \left[ e^{-j \theta}\delta(\omega +\omega_0)-e^{j \theta}\delta(\omega-\omega_0) \right] \,$
12 $\mbox{rect} \left( \frac{t}{\tau} \right) \,$ $\tau \mbox{sinc} \left( \frac{\tau \omega}{2 \pi} \right) \,$
13 $\tau \mbox{sinc} \left( \frac{\tau t}{2 \pi} \right) \,$ $2 \pi \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,$
14 $\left( 1-\frac{2 |t|}{\tau} \right) \mbox{rect} \left( \frac{ t }{ \tau } \right) \,$ $\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau \omega}{4 \pi} \right) \,$
15 $\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau t}{4 \pi} \right) \,$ $2 \pi \left( 1-\frac{2|\omega|}{\tau} \right) \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,$
16 $e^{-a|t|}, \Re\{a\}>0 \,$ $\frac{2a}{a^2 + \omega^2} \,$
Notes:
1. $\mbox{sinc}(x)=\sin(x)/x$
2. $\mbox{rect} \left( \frac{ t }{ \tau } \right)$ is the rectangular pulse function of width $\tau$
3. $u(t)$ is the Heavyside step function
4. $\delta (t)$ is the Dirac delta function