Fourier transform of the above signal x(t) = e^-a|t| is:

Explanation:

The Fourier transform of the signal \(x(t) = e^{-a|t|}\) can be calculated as follows:

Step 1: Definition of Fourier Transform

The Fourier transform of a function \(x(t)\) is defined as:

\[ X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \]

For the given signal \(x(t) = e^{-a|t|}\), we need to consider the absolute value function. Therefore, the signal can be written as:

\[ x(t) = \begin{cases} e^{at} & \text{for } t < 0 \\ e^{-at} & \text{for } t \ge 0 \end{cases} \]

Step 2: Split the Integral

We can split the integral into two parts: one for \(t < 0\) and one for \(t \ge 0\):

\[ X(j\omega) = \int_{-\infty}^{0} e^{at} e^{-j\omega t} dt + \int_{0}^{\infty} e^{-at} e^{-j\omega t} dt \]

Simplifying the exponents inside the integrals, we get:

\[ X(j\omega) = \int_{-\infty}^{0} e^{(a - j\omega)t} dt + \int_{0}^{\infty} e^{-(a + j\omega)t} dt \]

Step 3: Evaluate the Integrals

Let's evaluate each integral separately.

For the first integral \( \int_{-\infty}^{0} e^{(a - j\omega)t} dt \):

\[ \int_{-\infty}^{0} e^{(a - j\omega)t} dt = \left[ \frac{e^{(a - j\omega)t}}{a - j\omega} \right]_{-\infty}^{0} \]

Evaluating the limits:

\[ \left[ \frac{e^{(a - j\omega)t}}{a - j\omega} \right]_{-\infty}^{0} = \frac{1}{a - j\omega} - \lim_{t \to -\infty} \frac{e^{(a - j\omega)t}}{a - j\omega} \]

Since \(a > 0\), \(e^{(a - j\omega)t}\) approaches 0 as \(t\) approaches \(-\infty\):

\[ \frac{1}{a - j\omega} - 0 = \frac{1}{a - j\omega} \]

For the second integral \( \int_{0}^{\infty} e^{-(a + j\omega)t} dt \):

\[ \int_{0}^{\infty} e^{-(a + j\omega)t} dt = \left[ \frac{-e^{-(a + j\omega)t}}{a + j\omega} \right]_{0}^{\infty} \]

Evaluating the limits:

\[ \left[ \frac{-e^{-(a + j\omega)t}}{a + j\omega} \right]_{0}^{\infty} = 0 - \left( -\frac{1}{a + j\omega} \right) = \frac{1}{a + j\omega} \]

Step 4: Add the Results

Now, combining both integrals, we get:

\[ X(j\omega) = \frac{1}{a - j\omega} + \frac{1}{a + j\omega} \]

To simplify, find a common denominator:

\[ X(j\omega) = \frac{a + j\omega + a - j\omega}{(a - j\omega)(a + j\omega)} = \frac{2a}{a^2 + \omega^2} \]

Therefore, the Fourier transform of \(x(t) = e^{-a|t|}\) is:

\[ X(j\omega) = \frac{2a}{a^2 + \omega^2} \]

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: \(X(j\omega) = \frac{2a}{a+\omega}\)

This option is incorrect because the denominator should be \(a^2 + \omega^2\), not \(a + \omega\).

Option 3: \(X(j\omega) = \frac{a}{a - j\omega}\)

This option is incorrect because it does not account for the full expression obtained from the Fourier transform, and it lacks the correct form of the denominator.

Option 4: \(X(j\omega) = \frac{2a}{a + j\omega}\)

This option is incorrect because it only partially matches one of the terms derived from the Fourier transform calculation and does not include the complete expression.

Conclusion:

Understanding the Fourier transform process and carefully evaluating each integral's limits and results are essential for correctly identifying the transform of a given signal. In this case, the correct Fourier transform of \(x(t) = e^{-a|t|}\) is \(X(j\omega) = \frac{2a}{a^2 + \omega^2}\), making Option 2 the correct choice.