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

* Engineering Analysis/Multi-Dimensional Fourier Series
* The Feyman lectures Chapter 50 Harmonics.
Waves : 1 Dimensional Waves 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 Examples - Problems - Solutions - Terminology