# 10/10,13,16,17 - Fourier Transform Properties

From Class Wiki

## Properties of the Fourier Transform

### Linearity

### Time Invariance (Delay)

Let and | ||

Why isn't this |

### Frequency Shifting

### Double Sideband Modulation

### Differentiation in Time

Thus is a linear filter with transfer function |

### The Game (frequency domain)

- You can play the game in the frequency or time domain, but it's not advisable to play it in both at same time

Input | LTI System | Output | Reason |

Given | |||

Proportionality | |||

Superposition | |||

Time Invariance | |||

Proportionality | |||

Superposition |

- Having trouble seeing

### The Game (Time Domain??)

Input | LTI System | Output | Reason |

Proportionality | |||

from 10/3,6 - The Game | |||

Proportionality | |||

Superposition |

### Relation to the Fourier Series

Let and reverse the order of summation | ||

Note that is the complex conjugate of | ||

- How can we assume that the answer exists in the real domain?

### Aside: Polar coordinates

Remember from 10/02 - Fourier Series that

- Rectangular coordinates:
- Polar coordinates:

### Building up to

Euler's Identity | ||

Real odd function of t | ||

& | ||

= Real odd. Integrates out over symmetric limits. | ||

Imaginary Odd function of | ||

Real even function of t | ||

& | ||

= Real odd. Integrates out over symmetric limits. | ||

Real Even function of |

### Definitions

Can't x(t) have parts that aren't even or odd? You can break any function down into a Taylor series. There are even and odd powers in the series. | ||