Obtain the Z transform of x(t)= e^(-at) cos(wt), DO NOT USE LAPLACE TRANSFORM 20.09.2024 11:09

1. **Express \(x(t)\) in discrete form**:

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)