To find the Fourier transform of the given waveform, we observe that the signal is a combination of two exponential functions: \(A e^{at}\) and \(A e^{-at}\).

The waveform depicted in the image can be interpreted as the sum of two exponentially decaying functions:

• \(A e^{-at}\) for \(t \geq 0\)
• \(A e^{at}\) for \(t \lt 0\)

We need to calculate the Fourier transform of both components and then sum them up.

Fourier Transform of \(A e^{-at} \text{ for } t \geq 0\):

Using the standard formula, the Fourier transform is given by:

\(\int_{0}^{\infty} A e^{-at} e^{-j \omega t} dt = \frac{A}{a + j \omega}\)

Fourier Transform of \(A e^{at} \text{ for } t \lt 0\):

Similarly, using the standard formula, the Fourier transform is given by:

\(\int_{-\infty}^{0} A e^{at} e^{-j \omega t} dt = \frac{A}{a - j \omega}\)

Combining both results, the total Fourier Transform is:

\(\frac{A}{a - j \omega} + \frac{A}{a + j \omega}\)

Therefore, the correct answer is:

Correct Answer: \(\frac{A}{a-j\omega} + \frac{A}{a+j\omega}\)

Explanation of Incorrect Options:

• \(\frac{A}{a+j\omega}\) and \(\frac{A}{a-j\omega}\) are parts of the transform but not the complete answer.
• \(\frac{A}{a+j\omega} - \frac{A}{a-j\omega}\) represents the difference of the transforms, which is not applicable for this waveform.