# Waves/Fourier Transforms

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## Fourier Transform

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,

${\displaystyle f(x)=\int _{-\infty }^{\infty }(c_{1}(k)\cos(kx)+c_{2}(k)\sin(kx))dk}$.

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.

${\displaystyle c_{1}(k)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }f(x)\cos(kx)\,dx\quad c_{2}(k)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }f(x)\sin(kx)\,dx}$

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 Properties

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

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

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

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
1 ${\displaystyle a\cdot g(t)+b\cdot h(t)\,}$ ${\displaystyle a\cdot G(\omega )+b\cdot H(\omega )\,}$ ${\displaystyle a\cdot G(f)+b\cdot H(f)\,}$ Linearity
2 ${\displaystyle g(t-a)\,}$ ${\displaystyle e^{-ia\omega }G(\omega )\,}$ ${\displaystyle e^{-i2\pi af}G(f)\,}$ Shift in time domain
3 ${\displaystyle e^{iat}g(t)\,}$ ${\displaystyle G(\omega -a)\,}$ ${\displaystyle G\left(f-{\frac {a}{2\pi }}\right)\,}$ Shift in frequency domain, dual of 2
4 ${\displaystyle g(at)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,}$ If ${\displaystyle |a|\,}$ is large, then ${\displaystyle g(at)\,}$ is concentrated around 0 and ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ spreads out and flattens
5 ${\displaystyle G(t)\,}$ ${\displaystyle g(-\omega )\,}$ ${\displaystyle g(-f)\,}$ Duality property of the Fourier transform. Results from swapping "dummy" variables of ${\displaystyle t\,}$ and ${\displaystyle \omega \,}$.
6 ${\displaystyle {\frac {d^{n}g(t)}{dt^{n}}}\,}$ ${\displaystyle (i\omega )^{n}G(\omega )\,}$ ${\displaystyle (i2\pi f)^{n}G(f)\,}$ Generalized derivative property of the Fourier transform
7 ${\displaystyle t^{n}g(t)\,}$ ${\displaystyle i^{n}{\frac {d^{n}G(\omega )}{d\omega ^{n}}}\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}G(f)}{df^{n}}}\,}$ This is the dual to 6
8 ${\displaystyle (g*h)(t)\,}$ ${\displaystyle {\sqrt {2\pi }}G(\omega )H(\omega )\,}$ ${\displaystyle G(f)H(f)\,}$ ${\displaystyle g*h\,}$ denotes the convolution of ${\displaystyle g\,}$ and ${\displaystyle h\,}$ — this rule is the convolution theorem
9 ${\displaystyle g(t)h(t)\,}$ ${\displaystyle (G*H)(\omega ) \over {\sqrt {2\pi }}\,}$ ${\displaystyle (G*H)(f)\,}$ This is the dual of 8
10 For a purely real even function ${\displaystyle g(t)\,}$ ${\displaystyle G(\omega )\,}$ is a purely real even function ${\displaystyle G(f)\,}$ is a purely real even function
11 For a purely real odd function ${\displaystyle g(t)\,}$ ${\displaystyle G(\omega )\,}$ is a purely imaginary odd function ${\displaystyle G(f)\,}$ is a purely imaginary odd function

## Fourier Transform Pairs

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

Waves : 1 Dimensional Waves
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