1. The Fourier Transform
The Fourier transform has some nice properties. Assume and are integrable functions:
- Linearity: For , if , then .
- Translation: For , if , then .
- Modulation: For , if , then .
- Scaling: For , if , then .
Here we use the notation is used .
An important feature of the Fourier Transform is convolution. Suppose two given functions and and let the convolution be defined as , then and .
2. Discrete Fourier Transform