Class lecture notes October 5 - HW3

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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
Given
Time Invariance
Proportionality
Superposition


where for any and is the convolution integral.

We can expand the game further.

Input Linear Time Invariant System Output Reason
Given
Time Invariance
Proportionality
Superposition
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
Given
Time Invariance
Proportionality
Superposition
Superposition
Proportionality
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.