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|<math>=\,\!</math>Real Even function of <math>f\,\!</math> |
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|<math>=\,\!</math>Real Even function of <math>f\,\!</math> |
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===Definitions=== |
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{| border="0" cellpadding="0" cellspacing="0" |
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|<math>x(t)\,\!</math> |
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|<math>=x_e(t)+x_(o)t\,\!</math> |
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|Can't x(t) have parts that aren't even or odd? |
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|<math>x_e(t)\,\!</math> |
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|<math>=\frac{x(t)+x(-t)}{2}</math> |
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|<math>x_o(t)\,\!</math> |
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|<math>=\frac{x(t)-x(-t)}{2}</math> |
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|<math>u(t)\,\!</math> |
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|<math>=\frac{1+\sgn (t)}{2}</math> |
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|<math>\sgn (t) = \begin{cases} 1, & t>0 \\ 0, & t=0 \\ -1, & t<0\end{cases}</math> |
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|<math>u_e(t)\,\!</math> |
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|<math>=\frac{1}{2}</math> |
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|<math>u_o(t)\,\!</math> |
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|<math>=\frac{\sgn (t)}{2}</math> |
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Revision as of 18:56, 24 November 2008
Properties of the Fourier Transform
Linearity
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Time Invariance (Delay)
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Let and
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Frequency Shifting
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Double Sideband Modulation
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Differentiation in Time
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Thus is a linear filter with transfer function
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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
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LTI System
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Output
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Reason
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Given
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Proportionality
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Superposition
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Time Invariance
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Proportionality
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Superposition
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- 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
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LTI System
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Output
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Reason
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Proportionality
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Why d lambda instead of dt?
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Proportionality, Why isn't this a convolution?
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Superposition, Not X(f_0)H(f_0)?
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Relation to the Fourier Series
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Let and reverse the order of summation
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Assume that
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- 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
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- Rest of page
Building up to
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Real odd function of t
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Euler's Identity: Why is this not negative in the notes?
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Integrating cosine over symmetric limits is 0. Reasoning??
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Imaginary Odd function of
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Real even function of t
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Euler's Identity: Why is this not negative in the notes?
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Why??
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Real Even function of
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Definitions
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Can't x(t) have parts that aren't even or odd?
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