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| + | ==[[Lecture21ECE301S11|Lecture 21]]== | ||
| + | === Multiplication Property === | ||
| + | <math> \mathcal{F}(x_1(t) x_2(t)) = \frac{1}{2\pi}\mathcal{F}(x_1(t))*\mathcal{F}(x_2(t))</math> | ||
| + | |||
| + | === Causal LTI system defined by cst coeff diff equations === | ||
| + | <math> \sum_{k=0}^{N}a_k \frac{d^k}{dt^k}y(t) = \sum_{k=0}^{M}b_k \frac{d^k}{dt^k}x(t)</math> | ||
| + | |||
| + | What is the frequency response of this system? Recall: | ||
| + | |||
| + | <math> | ||
| + | \begin{align} | ||
| + | \mathcal{H}(\omega) &= \mathcal{F}(h(t)) \\ | ||
| + | \mathcal{Y}(\omega) &= \mathcal{H}(\omega)\mathcal{X}(\omega) | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | Steps to solve: | ||
| + | # Take the F.T. of both sides. | ||
| + | #: <math> | ||
| + | \begin{align} | ||
| + | \mathcal{F}\left(\sum_{k=0}^{N}a_k \frac{d^k}{dt^k}y(t)\right) &= \mathcal{F}\left( \sum_{k=0}^{M}b_k \frac{d^k}{dt^k}x(t) \right) \\ | ||
| + | \sum_{k=0}^{N}a_k \mathcal{F}\left(\frac{d^k}{dt^k}y(t)\right) &= \sum_{k=0}^{M}b_k \mathcal{F}\left(\frac{d^k}{dt^k}x(t)\right) \\ | ||
| + | & \text{recall: }\mathcal{F}\left(\frac{d^n}{dt^n}y(t)\right) = (j\omega)^n Y(\omega) \\ | ||
| + | \sum_{k=0}^{N}a_k \left( j \omega \right)^k \mathcal{Y}(\omega) &= \sum_{k=0}^{M}b_k \left( j \omega \right)^k \mathcal{X}(\omega) \\ | ||
| + | \mathcal{Y}(\omega) &= \frac{\sum_{k=0}^{M}b_k(j\omega)^{k}}{\sum_{k^\prime=0}^{N}a_k(j\omega)^{k^\prime}} \mathcal{X}(\omega) \\ | ||
| + | \mathcal{H}(\omega) &= \frac{\sum_{k=0}^{M}b_k(j\omega)^{k}}{\sum_{k^\prime=0}^{N}a_k(j\omega)^{k^\prime}} \\ | ||
| + | h(t) &= \mathcal{F}^{-1}\left(\mathcal{H}(\omega)\right) | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
Here are my lecture notes from ECE301 you can download both files from my dropbox account by Prof. Boutin | Here are my lecture notes from ECE301 you can download both files from my dropbox account by Prof. Boutin | ||
Revision as of 07:32, 7 March 2011
Lecture 21
Multiplication Property
$ \mathcal{F}(x_1(t) x_2(t)) = \frac{1}{2\pi}\mathcal{F}(x_1(t))*\mathcal{F}(x_2(t)) $
Causal LTI system defined by cst coeff diff equations
$ \sum_{k=0}^{N}a_k \frac{d^k}{dt^k}y(t) = \sum_{k=0}^{M}b_k \frac{d^k}{dt^k}x(t) $
What is the frequency response of this system? Recall:
$ \begin{align} \mathcal{H}(\omega) &= \mathcal{F}(h(t)) \\ \mathcal{Y}(\omega) &= \mathcal{H}(\omega)\mathcal{X}(\omega) \end{align} $
Steps to solve:
- Take the F.T. of both sides.
- $ \begin{align} \mathcal{F}\left(\sum_{k=0}^{N}a_k \frac{d^k}{dt^k}y(t)\right) &= \mathcal{F}\left( \sum_{k=0}^{M}b_k \frac{d^k}{dt^k}x(t) \right) \\ \sum_{k=0}^{N}a_k \mathcal{F}\left(\frac{d^k}{dt^k}y(t)\right) &= \sum_{k=0}^{M}b_k \mathcal{F}\left(\frac{d^k}{dt^k}x(t)\right) \\ & \text{recall: }\mathcal{F}\left(\frac{d^n}{dt^n}y(t)\right) = (j\omega)^n Y(\omega) \\ \sum_{k=0}^{N}a_k \left( j \omega \right)^k \mathcal{Y}(\omega) &= \sum_{k=0}^{M}b_k \left( j \omega \right)^k \mathcal{X}(\omega) \\ \mathcal{Y}(\omega) &= \frac{\sum_{k=0}^{M}b_k(j\omega)^{k}}{\sum_{k^\prime=0}^{N}a_k(j\omega)^{k^\prime}} \mathcal{X}(\omega) \\ \mathcal{H}(\omega) &= \frac{\sum_{k=0}^{M}b_k(j\omega)^{k}}{\sum_{k^\prime=0}^{N}a_k(j\omega)^{k^\prime}} \\ h(t) &= \mathcal{F}^{-1}\left(\mathcal{H}(\omega)\right) \end{align} $
Here are my lecture notes from ECE301 you can download both files from my dropbox account by Prof. Boutin
There are bound to be a few errors in the document, if you find them please let me know and I'll fix them ASAP.
Lecture.pdf contains all lectures after lecture 5.
