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| + | == Part A == | ||
| + | |||
CT LTI system: | CT LTI system: | ||
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h(t) = 10delta(t-1) | h(t) = 10delta(t-1) | ||
| − | <math>H(s) = \int_{-\infty}^{\infty} h(t)e^{-st}</math> | + | <math>s = j\omega</math> |
| + | |||
| + | <math>H(s) = \int_{-\infty}^{\infty}h(t)e^{-st}</math> | ||
| + | |||
| + | <math>H(s) = \int_{-\infty}^{\infty}10delta(t-1)e^{-st}</math> | ||
| + | |||
| + | <math>H(s) = 10\times\int_{-\infty}^{\infty}delta(t-1)e^{-st}</math> | ||
| + | |||
| + | <math>H(s) = 10e^{-s}\,</math> | ||
| + | |||
| + | |||
| + | == Part B == | ||
| + | |||
| + | <math>x(t) = 4\sin(5 \pi t) - (2 + j)\cos(3 \pi t)\,</math> | ||
| + | |||
| + | <math>y(t) = H(jw)x(t)\,</math> | ||
| + | |||
| + | <math>y(t) = 10e^{-jw}\times[4\sin(5 \pi t) - (2 + j)\cos(3 \pi t)]\,</math> | ||
| + | |||
| + | <math>y(t) = 10e^{-jw}\times [\frac{2}{j}e^{5*j\pi t} - \frac{2}{j}e^{-5*j\pi t} - \frac{2+j}{2}e^{3*j\pi t} - \frac{2+j}{2}e^{-3*j\pi t}]\,</math> | ||
| − | <math> | + | <math>y(t) = \frac{20}{j}e^{5*j\pi t}e^{-jw} - \frac{20}{j}e^{-5*j\pi t}e^{-jw} - (10+5j)e^{3*j\pi t}e^{-jw} - (10+5j)e^{-3*j\pi t}e^{-jw}\,</math> |
| − | <math> | + | <math>y(t) = \frac{20}{j}e^{5*j\pi (t-1)} - \frac{20}{j}e^{-5*j\pi (t-1)} - (10+5j)e^{3*j\pi (t-1)} - (10+5j)e^{-3*j\pi (t-1)}\,</math> |
Latest revision as of 12:27, 25 September 2008
Part A
CT LTI system:
y(t) = 10x(t-1)
plugging in delta(t) into the system we get:
h(t) = 10delta(t-1)
$ s = j\omega $
$ H(s) = \int_{-\infty}^{\infty}h(t)e^{-st} $
$ H(s) = \int_{-\infty}^{\infty}10delta(t-1)e^{-st} $
$ H(s) = 10\times\int_{-\infty}^{\infty}delta(t-1)e^{-st} $
$ H(s) = 10e^{-s}\, $
Part B
$ x(t) = 4\sin(5 \pi t) - (2 + j)\cos(3 \pi t)\, $
$ y(t) = H(jw)x(t)\, $
$ y(t) = 10e^{-jw}\times[4\sin(5 \pi t) - (2 + j)\cos(3 \pi t)]\, $
$ y(t) = 10e^{-jw}\times [\frac{2}{j}e^{5*j\pi t} - \frac{2}{j}e^{-5*j\pi t} - \frac{2+j}{2}e^{3*j\pi t} - \frac{2+j}{2}e^{-3*j\pi t}]\, $
$ y(t) = \frac{20}{j}e^{5*j\pi t}e^{-jw} - \frac{20}{j}e^{-5*j\pi t}e^{-jw} - (10+5j)e^{3*j\pi t}e^{-jw} - (10+5j)e^{-3*j\pi t}e^{-jw}\, $
$ y(t) = \frac{20}{j}e^{5*j\pi (t-1)} - \frac{20}{j}e^{-5*j\pi (t-1)} - (10+5j)e^{3*j\pi (t-1)} - (10+5j)e^{-3*j\pi (t-1)}\, $
