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

From Class Wiki
Jump to navigation Jump to search

Properties of the Fourier Transform

Linearity

Time Invariance (Delay)

Let and

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 not both at the same time
    • Then how can you use the Fourier Transform, but can't build up to it?
Input LTI System Output Reason
Given
Proportionality
Superposition
Time Invariance
Proportionality
Superposition
  • Having trouble seeing
  • Since we were dealing in the frequency domain, is that the reason why multiplying one side did not result in a convolution on the other?


The Game (Time Domain??)

Input LTI System Output Reason
Proportionality
Why d lambda instead of dt?
Proportionality, Why isn't this a convolution?
Superposition, Not X(f_0)H(f_0)?


Relation to the Fourier Series

Let and reverse the order of summation
Assume that
  • Does the server reset every hour?
  • How can we assume that the answer exists in the real domain?

Remember from 10/02 - Fourier Series that

  • ?
  • Rest of page

Building up to

Real odd function of t
Euler's Identity: Why is this not negative in the notes?
Integrating cosine over symmetric limits is 0. Reasoning??
Imaginary Odd function of
Real even function of t
Euler's Identity: Why is this not negative in the notes?
Why??
Real Even function of