|
|
| Line 84: |
Line 84: |
|
|
|
|
|
Let <math>\lambda\ = t-t_0</math>, so <math>t_0 = t-\lambda\ </math> and <math>dt_0 = -d\lambda\ </math><br> |
|
Let <math>\lambda\ = t-t_0</math>, so <math>t_0 = t-\lambda\ </math> and <math>dt_0 = -d\lambda\ </math><br> |
|
Therefore <math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0 = \int_{-\infty}^{\infty} h(\lambda)e^{j2\pi f(t-\lambda)}(-d\lambda) = e^{j2\pi ft}\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math><br> |
|
Therefore <math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0 = \int_{+\infty}^{-\infty} h(\lambda)e^{j2\pi f(t-\lambda)}(-d\lambda) = e^{j2\pi ft}\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math><br> |
|
This tells us that <math>e^{j2\pi ft}\!</math> is the eigenfunction and <math>\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math> is the eigenvalue of <b>all linear time invariant systems.</b><br> |
|
This tells us that <math>e^{j2\pi ft}\!</math> is the eigenfunction and <math>\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math> is the eigenvalue of <b>all linear time invariant systems.</b><br> |
|
This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.<br> |
|
This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.<br> |
Revision as of 19:36, 6 October 2009
Max Woesner
Homework #2 - Something interesting from class
The Linear Time Invariant System Game 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.
This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.