Fourier Transform

An approach for computing what frequencies are present in a Wave, in particular the Spectrum, entirely from its dependence on time. Allows us to transform a function between time an frequency domains. Allows us to quantitivatively define the Spectrum of a Wave A Gaussian’s transform is another Gaussian.

• Frequency is conjugate to time
• Spatial frequency is conjugate to position Anytime the Fourier Transform is involved, so is the Scale Theorem, and the Uncertainty Principle

Equation

$$F(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$$

• Where $F(\omega )$ is called the Fourier Transform of $f(t)$, containing equivalent information to that in $f(t)$
• $f(t)$ lives in the time domain
• $F(\omega )$ lives in the frequency domain

With respect to space

A spatial fourier transform. Spatial frequency $k$ is the conjugate of position $x$.

$$F(k)\equiv {\int }_{-\infty }^{\infty }f(x)\exp (-ikx)dx$$

Derivation from Fourier Series

Let $F(m)\equiv {\mathbb{E}}_m-i{\mathbb{O}}_m$ where these values are defined in Even Function and Odd Function and also Fourier Series

$$\begin{array}{r}\\ F(m)\equiv {\mathbb{E}}_m-i{\mathbb{O}}_m=\\ \int f(t)\cos (mt)dt-i\int f(t)\sin (mt)dt=\\ \int f(t)e^{-imt}dt\end{array}$$

Allow t to range from $-\infty$ to $\infty$, $m$ to be continuous Frequency $\omega$, i.e. for a continuous range of frequencies

$$F(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$$

Inverse Equation

$$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{-i\omega t}d\omega$$

Derivation from Fourier Series

Note that the $i$ gets factored in for consistency with the above derivation.

$$\begin{array}{r}f(t)=\\ \mathbb{E}(t)+\mathbb{O}(t)=\\ \frac{1}{\pi }\sum _{m=0}^{\infty }{\mathbb{E}}_m\cos (mt)+\frac{1}{\pi }\sum _{m=0}^{\infty }{\mathbb{O}}_m\sin (mt)=\\ \frac{1}{2}\left(\frac{1}{\pi }{\int }_{-\infty }^{\infty }{\mathbb{E}}_m\cos (mt)dt+\frac{1}{\pi }{\int }_{-\infty }^{\infty }{\mathbb{O}}_m\sin (mt)dt\right)=\\ \frac{1}{2}\left(\frac{1}{\pi }{\int }_{-\infty }^{\infty }\left({\mathbb{E}}_m\cos (mt)+i{\mathbb{O}}_m\sin (mt)\right)dt\right)=\\ \frac{1}{2}\left(\frac{1}{\pi }{\int }_{-\infty }^{\infty }\left(F(\omega )\cos (mt)+iF(\omega )\sin (mt)\right)dt\right)=\\ \frac{1}{2}\left(\frac{1}{\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{-i\omega t}dt\right)=\\ \frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{-i\omega t}d\omega \end{array}$$

With respect to space

$$f(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(k)\exp (ikx)dk$$

Notation

If original function lowercase, use above notation

• $f(t)$ for the function
• $F(\omega )$ for the transform If the original function is uppercase:
• $E(t)$ for the function, arbitrarily
• For the transform
  • $\mathcal{F}\left(E(t)\right)$
  • $\tilde{E}(\omega )$

Examples

Find $\mathcal{F}(\text{rect}(t))$

$$\begin{array}{r}\\ F(\omega )={\int }_{-0.5}^{0.5}\exp (-i\omega t)dt=\\ \frac{1}{-i\omega }(\exp (-i\omega t){)}_{-0.5}^{0.5}=\\ \frac{1}{-i\omega }\left(\exp \left(-\frac{i\omega }{2}\right)-\exp \left(\frac{i\omega }{2}\right)\right)=\\ \frac{1}{\frac{\omega }{2}}\frac{\exp \left(\frac{i\omega }{2}\right)-\exp \left(-\frac{i\omega }{2}\right)}{2i}=\\ \frac{\sin \left(\frac{w}{2}\right)}{\frac{\omega }{2}}\equiv \\ \text{sinc}\left(\frac{\omega }{2}\right)\end{array}$$

See sinc for more info

Fourier Transform of the Complex Conjugate of a Function

$$\mathcal{F}(f^{\ast }(t))=F^{\ast }(\omega )=F^{\ast }(-\omega )$$

Scale Theorem

Modulation Theorem

Fourier Transform of the Sum of Two Functions

$$\mathcal{F}(a\cdot f(t)+b\cdot g(t))=a\mathcal{F}(f(t))+b\mathcal{F}(g(t))$$

Shift Theorem

2D Fourier Transform