|
|
Line 187: |
Line 187: |
|
*<math>\alpha_n=\left| \alpha_n \right|\,e^{j\theta}</math>? |
|
*<math>\alpha_n=\left| \alpha_n \right|\,e^{j\theta}</math>? |
|
*Rest of page |
|
*Rest of page |
|
|
|
|
|
===Building up to <math>F\left[u(t)\right]</math>=== |
|
|
{| border="0" cellpadding="0" cellspacing="0" |
|
|
|- |
|
|
|<math>r_0(t)\,\!</math> |
|
|
|<math>=-r_0(-t)\,\!</math> |
|
|
|Real odd function of t |
|
|
|- |
|
|
| |
|
|
|<math>=n^2 + 2n + 1</math> |
|
|
|} |
Revision as of 07:12, 24 November 2008
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
|
|
|