10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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|<math>\int_{-\infty}^{\infty} h(\lambda)\cdot e^{j\,2\,\pi f_0\,(t-\lambda)}\,d\lambda</math> |
|<math>\int_{-\infty}^{\infty} h(\lambda)\cdot e^{j\,2\,\pi f_0\,(t-\lambda)}\,d\lambda</math> |
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|<math>d\lambda\,\!</math> from [[10/3,6 - The Game]] |
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|Why d lambda instead of dt? |
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|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
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|<math>X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,H(f_0)</math> |
|<math>X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,H(f_0)</math> |
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|Proportionality |
|Proportionality |
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|<math> \int_{-\infty}^{\infty}X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=x(t)</math> |
|<math> \int_{-\infty}^{\infty}X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=x(t)</math> |
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|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
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|<math>\int_{-\infty}^{\infty}X(f_0)H(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=F^{-1}\left[X(f)H(f)\right]</math> |
|<math>\int_{-\infty}^{\infty}X(f_0)H(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=F^{-1}\left[X(f)H(f)\right]</math> |
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|Superposition |
|Superposition |
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===Relation to the Fourier Series=== |
===Relation to the Fourier Series=== |
Revision as of 03:24, 25 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 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? You can break any function down into a Taylor series. There are even and odd powers in the series.
Remember from 10/02 - Fourier Series that
- ?
- Rest of page
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? | ||