Max Woesner
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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 still want to be able to work with nonperiodic signals.
A nonperiodic signal can be thought of as periodic signal with an infinite period. To deal with such signals we can take the limit of the Fourier series 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, or
Also, This is the Fourier transform, or
So and
From above, , so
, where , which is the projection with respect to .
This identity for is the defining property of the impulse function.
Similarly, , where ,so
, where , which is the projection with respect to .
The Linear Time Invariant System Game
Recall the Linear Time Invariant System Game that can be used to help us understand the impulse response of a linear time invariant system.
Input |
Linear Time Invariant System |
Output |
Reason |
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Given |
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Time Invariance |
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Proportionality |
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Superposition |
where for any and is the convolution integral.
We can expand the game further.
Input |
Linear Time Invariant System |
Output |
Reason |
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Given |
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Time Invariance |
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Proportionality |
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Superposition |
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Superposition |
Let , so and
Therefore
This tells us that is the eigenfunction and is the eigenvalue of all linear time invariant systems.
Also, the eigenvalue is , which equals the Fourier transform of , or
We can expand the game even further.
Input |
Linear Time Invariant System |
Output |
Reason |
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Given |
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Time Invariance |
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Proportionality |
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Superposition |
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Superposition |
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Proportionality |
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Superposition |
where and
This is helpful because in frequency space, when we go through a linear time invariant system, it multiplies by the transfer function, compared to time space, which convolves the impulse response, and we would all prefer to do multiplication rather than convolution.