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,

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

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 Properties

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$ $G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$ $G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$ 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 $g(t-a)\,$ $e^{-ia\omega }G(\omega )\,$ $e^{-i2\pi af}G(f)\,$ 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 $g(at)\,$ ${\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,$ ${\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,$ If $|a|\,$ is large, then $g(at)\,$ 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 ${\frac {d^{n}g(t)}{dt^{n}}}\,$ $(i\omega )^{n}G(\omega )\,$ $(i2\pi f)^{n}G(f)\,$ 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 Pairs

Time Domain Frequency Domain
$x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}$ $X(\omega )={\mathcal {F}}\left\{x(t)\right\}$ 1 $X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt$ $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$ 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. ${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ 2. ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ 3. $u(t)$ is the Heaviside step function
4. $\delta (t)$ is the Dirac delta function