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:	${\mbox{sinc}}(x)=\sin(\pi x)/(\pi x)$ ${\mbox{rect}}\left({\frac {t}{\tau }}\right)$ is the rectangular pulse function of width $\tau$ $u(t)$ is the Heaviside step function $\delta (t)$ is the Dirac delta function

Further reading

• Engineering Analysis/Multi-Dimensional Fourier Series
• The Feyman lectures Chapter 50 Harmonics.

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Waves : 1 Dimensional Waves
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