10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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|Proportionality |
|Proportionality |
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|<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> |
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|<math> \Longrightarrow </math> |
|<math> \Longrightarrow </math> |
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|<math> \int_{-\infty}^{\infty}x(\lambda)\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda}\,d\lambda=X( |
|<math> \int_{-\infty}^{\infty}x(\lambda)\cdot H(f)\cdot e^{j\,2\,\pi f\,\lambda}\,d\lambda=X(f)\,H(f)</math> |
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|Superposition |
|Superposition |
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*Having trouble seeing <math>F\left[x(t)*h(t)\right]=X(f)\cdot H(f)</math> |
*Having trouble seeing <math>F\left[x(t)*h(t)\right]=X(f)\cdot H(f)</math> |
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*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? |
*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? |
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===The Game (Time Domain??)=== |
===The Game (Time Domain??)=== |
Revision as of 22:09, 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 | ||
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? | ||