10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
Jump to navigation
Jump to search
Line 102: | Line 102: | ||
|Superposition |
|Superposition |
||
|- |
|- |
||
|<math> |
|<math> \int_{-\infty}^{\infty}\delta (t-\lambda)\,e^{-j\,2\,\pi f\,t}\,dt = F[\delta(t-\lambda)]=1\cdot e^{-j\,2\,\pi f\,\lambda}</math> |
||
|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
||
|<math> H(f)\cdot e^{j\,2\,\pi f\,\lambda}</math> |
|<math> H(f)\cdot e^{-j\,2\,\pi f\,\lambda}</math> |
||
|Time Invariance |
|Time Invariance |
||
|- |
|- |
||
|<math> x(\lambda)\cdot 1\cdot e^{j\,2\,\pi f\,\lambda}</math> |
|<math> x(\lambda)\cdot 1\cdot e^{-j\,2\,\pi f\,\lambda}</math> |
||
|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
||
|<math> x(\lambda)\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda}</math> |
|<math> x(\lambda)\cdot H(f)\cdot e^{-j\,2\,\pi f\,\lambda}</math> |
||
|Proportionality |
|Proportionality |
||
|- |
|- |
||
|<math> \int_{-\infty}^{\infty}x(\lambda)\cdot 1\cdot e^{j\,2\,\pi f\,\lambda}\,d\lambda=X |
|<math> \int_{-\infty}^{\infty}x(\lambda)\cdot 1\cdot e^{j\,2\,\pi f\,\lambda}\,d\lambda=X(F)</math> |
||
|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
||
|<math> \int_{-\infty}^{\infty} |
|<math> \int_{-\infty}^{\infty}x(\lambda)\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda}\,d\lambda=X(F)\,H(f)</math> |
||
|Superposition |
|Superposition |
||
|} |
|} |
Revision as of 21:56, 23 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 |