Question
Obtain the Z transform of x(t)= e^(-at) cos(wt), DO NOT USE LAPLACE TRANSFORM 20.09.2024 11:09
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Answer to a math question Obtain the Z transform of x(t)= e^(-at) cos(wt), DO NOT USE LAPLACE TRANSFORM 20.09.2024 11:09
Madelyn
4.788 Answers
1. **Express \(x(t)\)
x(t) = e^{-at} \cos(\omega t)
2. **Rewrite using Euler's formula**:
\cos(\omega t) = \frac{e^{j\omega t} + e^{-j\omega t}}{2}
x(t) = \frac{1}{2} \left( e^{(-a + j\omega)t} + e^{(-a - j\omega)t} \right)
3. **Find the Z-transform of each term**:
For\( e^{(-a + j\omega)t} \)
\mathcal{Z}\{e^{(-a + j\omega)t}\} = \frac{1}{1 - e^{-a + j\omega} z^{-1}}
For\( e^{(-a - j\omega)t} \)
\mathcal{Z}\{e^{(-a - j\omega)t}\} = \frac{1}{1 - e^{-a - j\omega} z^{-1}}
4. **Combine the results**:
X(z) = \frac{1}{2} \left( \frac{1}{1 - e^{-a + j\omega} z^{-1}} + \frac{1}{1 - e^{-a - j\omega} z^{-1}} \right)
The final answer is:
Z\{x(t)\} = \frac{1}{2} \left( \frac{1}{1 - e^{-(a+iw)} z^{-1}} + \frac{1}{1 - e^{-(a-iw)} z^{-1}} \right)
2. **Rewrite using Euler's formula**:
3. **Find the Z-transform of each term**:
For
For
4. **Combine the results**:
The final answer is:
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