Fourier transform

• categories: Functional analysis, Definition, Signal processing

Definition

The Fourier Transform is a mathematical operation that transforms a function of time (or space) into its frequency-domain representation. For a function $f(t)$, its Fourier Transform $\mathcal{F}(f)(\omega )$ is defined as:

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

where:

• $t$ is the time (or spatial) variable,
• $\omega$ is the angular frequency,
• $e^{-i\omega t}$ represents the complex sinusoidal basis functions.

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Inverse Fourier Transform

The original function $f(t)$ can be recovered from its Fourier transform $F(\omega )$ using the inverse Fourier transform:

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

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Key Properties

1. Linearity:
   The Fourier transform is linear:

   $$\mathcal{F}(af(t)+bg(t))=a\mathcal{F}(f)(\omega )+b\mathcal{F}(g)(\omega ).$$

2. Time Shifting:
   Shifting $f(t)$ in time results in a phase shift in the frequency domain:

   $$\mathcal{F}(f(t-t_0))=e^{-i\omega t_0}F(\omega ).$$

3. Frequency Shifting:
   Multiplying $f(t)$ by a complex exponential shifts the frequency:

   $$\mathcal{F}(f(t)e^{i{\omega }_{0}t})=F(\omega -{\omega }_0).$$

4. Scaling:
   Scaling $f(t)$ compresses or stretches the frequency domain:

   $$\mathcal{F}(f(at))=\frac{1}{|a|}F\left(\frac{\omega }{a}\right).$$

5. Parseval’s Theorem:
   The total energy in the time domain equals the total energy in the frequency domain:

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

6. Duality:
   Interchanging the roles of $f(t)$ and $F(\omega )$ gives:

   $$\mathcal{F}(F(t))=2\pi f(-\omega ).$$

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Fourier Transform of Common Functions

1. Delta Function: $$\mathcal{F}(\delta (t))=1.$$
2. Constant Function: $$\mathcal{F}(1)=2\pi \delta (\omega ).$$
3. Gaussian Function:
   If $f(t)=e^{-t^2/2}$, then: $$\mathcal{F}(f(t))=\sqrt{2\pi }{e}^{-{\omega }^2/2}.$$
4. Sine and Cosine:
   • $f(t)=\cos ({\omega }_{0}t)$: $$\mathcal{F}(f(t))=\pi [\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0)].$$
   • $f(t)=\sin ({\omega }_{0}t)$: $$\mathcal{F}(f(t))=\frac{\pi }{i}[\delta (\omega -{\omega }_0)-\delta (\omega +{\omega }_0)].$$

Example

Function:

Let $f(t)=e^{-t^2}$ (a Gaussian).

Fourier Transform:

$$\mathcal{F}(f(t))={\int }_{-\infty }^{\infty }{e}^{-t^2}{e}^{-i\omega t}dt=\sqrt{\pi }{e}^{-{\omega }^2/4}.$$

This result demonstrates that the Fourier transform of a Gaussian is also a Gaussian.