Key Fourier Transform Property

The integral $\frac{1}{2 \pi} \int_{- \infty}^{ \infty} X(j \omega) d \omega$ is a fundamental property of the Fourier Transform. It directly corresponds to the value of the time-domain signal $x(t)$ at time $t=0$.

$ x(0) = \frac{1}{2 \pi} \int_{- \infty}^{ \infty} X(j \omega) d \omega $

Thus, the problem requires finding $x(0)$. Based on the provided options, we assume $n=1$ for the calculation.

Deriving x(t) from X(jω)

We are given the Fourier Transform in terms of frequency $f$ and angular frequency $\omega$ ($ \omega = 2 \pi f $):

$ X(j \omega) = 10 e^{-j \pi f} \left( \frac{ \sin(\pi f)}{n f} \right) $

Substitute $ f = \frac{\omega}{2 \pi} $ into the expression:

$ X(j \omega) = 10 e^{-j \pi (\omega / 2 \pi)} \left( \frac{ \sin(\pi (\omega / 2 \pi))}{n (\omega / 2 \pi)} \right) $

Simplify the expression:

$ X(j \omega) = 10 e^{-j \omega / 2} \left( \frac{ \sin(\omega / 2)}{n \omega / 2} \right) = \frac{20}{n} e^{-j \omega / 2} \left( \frac{ \sin(\omega / 2)}{\omega} \right) $

Inverse Fourier Transform Calculation

Recall the Fourier Transform pair involving the sinc function's component:

$ \frac{\sin(a\omega)}{\omega} \leftrightarrow \frac{1}{2} \text{rect}\left(\frac{t}{2a}\right) $

For $ a = 1/2 $, this becomes:

$ \frac{\sin(\omega / 2)}{\omega} \leftrightarrow \frac{1}{2} \text{rect}(t) $

The exponential term $ e^{-j \omega / 2} $ indicates a time shift $ t_0 = 1/2 $. Applying this property:

$ \mathcal{F}^{-1}\left[ e^{-j \omega / 2} \left( \frac{ \sin(\omega / 2)}{\omega} \right) \right] = \frac{1}{2} \text{rect}\left(t - \frac{1}{2}\right) $

Now, incorporating the scaling factor $ \frac{20}{n} $ gives the time-domain signal $ x(t) $:

$ x(t) = \frac{20}{n} \cdot \frac{1}{2} \text{rect}\left(t - \frac{1}{2}\right) = \frac{10}{n} \text{rect}\left(t - \frac{1}{2}\right) $

Evaluating x(0)

The rectangular function $ \text{rect}(u) $ equals 1 for $ -1/2 \le u \le 1/2 $. We need $ x(0) $:

$ x(0) = \frac{10}{n} \text{rect}\left(0 - \frac{1}{2}\right) = \frac{10}{n} \text{rect}\left(-\frac{1}{2}\right) $

Since $ \text{rect}(-1/2) = 1 $,

$ x(0) = \frac{10}{n} $

Assuming $ n=1 $ to match the correct answer option,

$ x(0) = 10 $

Therefore, the value of the integral is 10.