Last modified on 6 July 2008, at 06:30

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

further readingEdit


Waves : 1 Dimensional Waves
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Examples - Problems - Solutions - Terminology