10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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===Building up to <math>F\left[u(t)\right]</math>=== |
===Building up to <math>F\left[u(t)\right]</math>=== |
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|<math>e^{i \pi}\,\!</math> |
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|<math>= \cos \pi + i \sin \pi.\,\!</math> |
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|Euler's Identity |
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|<math>r_0(t)\,\!</math> |
|<math>r_0(t)\,\!</math> |
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|<math>=\int_{-\infty}^{\infty}r_0(t)\,\left[ \cos (-2\pi\,f\,t) + j \sin (-2\pi\,f\,t) \right ]\,dt</math> |
|<math>=\int_{-\infty}^{\infty}r_0(t)\,\left[ \cos (-2\pi\,f\,t) + j \sin (-2\pi\,f\,t) \right ]\,dt</math> |
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|Euler's Identity: <math>e^{i \pi} = \cos \pi + i \sin \pi.\,\!</math> Why is this not negative in the notes? |
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|<math>=\int_{-\infty}^{\infty}r_0(t)\,\left[ \cos (2\pi\,f\,t) - j \sin (2\pi\,f\,t) \right ]\,dt</math> |
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|<math>\cos(-x) = \cos(x)\,\!</math> & <math>\sin(-x) = -\sin(x)\,\!</math> |
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|<math>=j\int_{-\infty}^{\infty}r_0(t)\, \sin (-2\pi\,f\,t)\,dt</math> |
|<math>=j\int_{-\infty}^{\infty}r_0(t)\, \sin (-2\pi\,f\,t)\,dt</math> |
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|<math>r_0(t)\cdot j \sin (-2\pi\,f\,t)</math> = Real odd. Integrates out over symmetric limits. |
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|Integrating cosine over symmetric limits is 0. Reasoning?? |
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|<math>=\int_{-\infty}^{\infty}r_e(t)\,\left[ \cos (-2\pi\,f\,t) + j \sin (-2\pi\,f\,t) \right ]\,dt</math> |
|<math>=\int_{-\infty}^{\infty}r_e(t)\,\left[ \cos (-2\pi\,f\,t) + j \sin (-2\pi\,f\,t) \right ]\,dt</math> |
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|Euler's Identity: <math>e^{i \pi} = \cos \pi + i \sin \pi.\,\!</math> Why is this not negative in the notes? |
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|<math>=\int_{-\infty}^{\infty}r_e(t)\,\left[ \cos (2\pi\,f\,t) - j \sin (2\pi\,f\,t) \right ]\,dt</math> |
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|<math>\cos(-x) = \cos(x)\,\!</math> & <math>\sin(-x) = -\sin(x)\,\!</math> |
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|<math>=\int_{-\infty}^{\infty}r_e(t)\, \cos(-2\pi\,f\,t)\,dt</math> |
|<math>=\int_{-\infty}^{\infty}r_e(t)\, \cos(-2\pi\,f\,t)\,dt</math> |
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|<math>r_e(t)\cdot \cos (-2\pi\,f\,t)</math> = Real odd. Integrates out over symmetric limits. |
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|Why?? |
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|<math>=\,\!</math>Real Even function of <math>f\,\!</math> |
|<math>=\,\!</math>Real Even function of <math>f\,\!</math> |
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===Definitions=== |
===Definitions=== |
Revision as of 21:40, 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
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? | ||