Class lecture notes October 5 - HW3: Difference between revisions
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Using <math>\frac{n}{T}=f \!x</math> and the information above, we can rewrite the equation.<br> |
Using <math>\frac{n}{T}=f \!x</math> and the information above, we can rewrite the equation.<br> |
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<math>x(t)=\int_{-\infty}^{\infty}(\int_{-\infty}^{\infty}x(t^')e^{-j2\pi ft^'}dt^')e^{j2\pi ft}df\!</math>, where <math>\int_{-\infty}^{\infty}x(t^')e^{-j2\pi ft^'}dt^'=X(f)\!</math><br> |
<math>x(t)=\int_{-\infty}^{\infty}(\int_{-\infty}^{\infty}x(t^')e^{-j2\pi ft^'}dt^')e^{j2\pi ft}df\!</math>, where <math>\int_{-\infty}^{\infty}x(t^')e^{-j2\pi ft^'}dt^'=X(f)\!</math><br> |
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So <math>x(t)=\int_{-\infty}^{\infty}X(f)e^{+j2\pi ft}df = <X(f)|e^{j2\pi ft}>\!</math> This is the inverse Fourier transform.<br> |
So <math>x(t)=\int_{-\infty}^{\infty}X(f)e^{+j2\pi ft}df =\ <X(f)|e^{j2\pi ft}>\!</math> This is the inverse Fourier transform.<br> |
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Also, <math>X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt = <x(t)|e^{j2\pi ft}>\!</math> This is the Fourier transform.<br> |
Also, <math>X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt =\ <x(t)|e^{j2\pi ft}>\!</math> This is the Fourier transform.<br> |
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So <math>x(t)=\mathcal{F}^{-1}[X(f)]\!</math> and <math>X(f)=\mathcal{F}[x(t)]\!</math><br> |
So <math>x(t)=\mathcal{F}^{-1}[X(f)]\!</math> and <math>X(f)=\mathcal{F}[x(t)]\!</math><br> |
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From above, <math>x(t)=\int_{-\ |
From above, <math>x(t)=\int_{-\infty_f}^{\infty}(\int_{-\infty_{t^'}}^{\infty}x(t^')e^{-j2\pi ft^'}dt^')e^{j2\pi ft}df\!</math>, so <br> |
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<math>x(t)=\int_{-\ |
<math>x(t)=\int_{-\infty_{t^'}}^{\infty}x(t^')(\int_{-\infty_{\delta(t-t^')}}^{\infty}e^{j2\pi f(t-t^')}df)dt^'\!</math>, where <math>\int_{-\infty}^{\infty}e^{j2\pi f(t-t^')}df =\ <e^{j2\pi ft}|e^{j2\pi ft^'}>\!</math> |
Revision as of 23:00, 7 October 2009
Max Woesner
Homework #3 - Class lecture notes October 5
The following notes are my interpretation of the material covered in class on October 5, 2009
Nonperiodic Signals
In real life, most systems have a finite time period and can be fairly easily evaluated as periodic. However, we do want to be able to evaluate nonperiodic signals for cases when this is not possible.
A nonperiodic signal can be thought of as periodic signal with an infinite period. To deal with such signals we can take the limit as the period goes to infinity, or
, where is the term.
We want to remove the restriction , which we can do as follows.
So
Using and the information above, we can rewrite the equation.
, where
So This is the inverse Fourier transform.
Also, This is the Fourier transform.
So and
From above, , so
, where