Class lecture notes October 5 - HW3: Difference between revisions
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:<math>\alpha_n\longrightarrow X(f) \!</math><br> |
:<math>\alpha_n\longrightarrow X(f) \!</math><br> |
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So <math>x(t)=\lim_{T \to \infty}\sum_{n=-\infty}^\infty \frac{1}{T}(\int_{-\frac{T}{2}}^{\frac{T}{2}}x(t^')e^{\frac{-j2\pi nt^'}{T}}dt^')e^{\frac{j2\pi nt}{T}}\!</math><br> |
So <math>x(t)=\lim_{T \to \infty}\sum_{n=-\infty}^\infty \frac{1}{T}(\int_{-\frac{T}{2}}^{\frac{T}{2}}x(t^')e^{\frac{-j2\pi nt^'}{T}}dt^')e^{\frac{j2\pi nt}{T}}\!</math><br> |
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Using <math>\frac{n}{T}=f \! |
Using <math>\frac{n}{T}=f \!</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> |
Revision as of 23:01, 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