(→a) |
(→b) |
||
| Line 41: | Line 41: | ||
By the properties of integrating an absolute value and the linearity of the Fourier transform. | By the properties of integrating an absolute value and the linearity of the Fourier transform. | ||
| − | <math> F(j\times \omega) = H(j\times \omega) + K(j\times \omega)</math> | + | :<math> F(j\times \omega) = H(j\times \omega) + K(j\times \omega)</math> |
| − | <math>H(j\times \omega) = \frac{e^{-j \omega}}{(2 + j \omega)}</math> from part a. | + | :<math>H(j\times \omega) = \frac{e^{-j \omega}}{(2 + j \omega)}</math> from part a. |
| − | <math> | + | :<math> |
k(t) = e^{-2 (1-t)}\times u(1-t) | k(t) = e^{-2 (1-t)}\times u(1-t) | ||
</math> | </math> | ||
| Line 51: | Line 51: | ||
remove the time shift and time reversal | remove the time shift and time reversal | ||
| − | <math> m(t) = e^{-2(t)}\times u(t)</math> | + | :<math> m(t) = e^{-2(t)}\times u(t)</math> |
from the table 4.2: | from the table 4.2: | ||
| − | <math> M(j \omega) = \frac{1}{2 + j \omega} </math> | + | :<math> M(j \omega) = \frac{1}{2 + j \omega} </math> |
apply the time shift property from table 4.1: | apply the time shift property from table 4.1: | ||
| − | <math> M(j \omega) = \frac{e^{-j\omega}}{2 + j \omega} </math> | + | :<math> M(j \omega) = \frac{e^{-j\omega}}{2 + j \omega} </math> |
apply the time reversal property from table 4.1 '''making sure to only apply it to the FT of the base function and not to the portion added by the time shift''': | apply the time reversal property from table 4.1 '''making sure to only apply it to the FT of the base function and not to the portion added by the time shift''': | ||
| − | <math> K(j \omega) = \frac{e^{-j\omega}}{2 - j \omega} </math> | + | :<math> K(j \omega) = \frac{e^{-j\omega}}{2 - j \omega} </math> |
| − | <math> H(j \omega) + K(j \omega) = \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j\omega}}{2 - j \omega} </math> | + | :<math> H(j \omega) + K(j \omega) = \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j\omega}}{2 - j \omega} </math> |
finding common denominators: | finding common denominators: | ||
| − | <math> \frac{(2-j\omega)e^{-j \omega}}{2^2 + \omega^2} + \frac{(2+j\omega)e^{-j\omega}}{2^2 + \omega^2}</math> | + | :<math> \frac{(2-j\omega)e^{-j \omega}}{2^2 + \omega^2} + \frac{(2+j\omega)e^{-j\omega}}{2^2 + \omega^2}</math> |
in the numerator the <math> j\omega </math> terms will cancel when added yielding the final answer: | in the numerator the <math> j\omega </math> terms will cancel when added yielding the final answer: | ||
| − | <math> F(j\omega) = \frac{4e^{-j \omega}}{4 + \omega^2}</math> | + | :<math> F(j\omega) = \frac{4e^{-j \omega}}{4 + \omega^2}</math> |
==Problem 4.2== | ==Problem 4.2== | ||
Revision as of 06:45, 8 October 2008
Allen Humphreys_ECE301Fall2008mboutin
Homework 5_ECE301Fall2008mboutin
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Contents
Problem 4.1
a
$ f(t) = e^{-2(t-1)} \times u(t-1) $
remove the time shift where $ t_o = 1 $
$ g(t) = e^{-2t} \times u(t) $
the time shift property in table 4.1 says:
$ F(j\omega) = e^{-j\omega t_o} G(j\omega) $
from table 4.2 the FT of $ g(t) $ can be found
$ G(j\omega) = \frac{1}{2 + j\omega} $
then the final answer can be found substituting 1 for $ t_o $
$ F(j\omega) = e^{-j\omega} G(j\omega) = \frac{e^{-j\omega}}{2 + j\omega} $
b
- $ f(t) = e^{-2 |(t-1)|} $
- $ f(t) = \begin{cases} e^{-2 (t-1)}, t>1\\ e^{-2 (1-t)}, t<1 \end{cases} = \begin{cases} e^{-2 (t-1)}\times u(t-1) = h(t)\\ e^{-2 (1-t)}\times u(1-t) = k(t) \end{cases} $
By the properties of integrating an absolute value and the linearity of the Fourier transform.
- $ F(j\times \omega) = H(j\times \omega) + K(j\times \omega) $
- $ H(j\times \omega) = \frac{e^{-j \omega}}{(2 + j \omega)} $ from part a.
- $ k(t) = e^{-2 (1-t)}\times u(1-t) $
remove the time shift and time reversal
- $ m(t) = e^{-2(t)}\times u(t) $
from the table 4.2:
- $ M(j \omega) = \frac{1}{2 + j \omega} $
apply the time shift property from table 4.1:
- $ M(j \omega) = \frac{e^{-j\omega}}{2 + j \omega} $
apply the time reversal property from table 4.1 making sure to only apply it to the FT of the base function and not to the portion added by the time shift:
- $ K(j \omega) = \frac{e^{-j\omega}}{2 - j \omega} $
- $ H(j \omega) + K(j \omega) = \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j\omega}}{2 - j \omega} $
finding common denominators:
- $ \frac{(2-j\omega)e^{-j \omega}}{2^2 + \omega^2} + \frac{(2+j\omega)e^{-j\omega}}{2^2 + \omega^2} $
in the numerator the $ j\omega $ terms will cancel when added yielding the final answer:
- $ F(j\omega) = \frac{4e^{-j \omega}}{4 + \omega^2} $
