GATE EC 2016 Set 2 — Question 20

The energy E of a continuous-time signal $x(t)$ is given by the integral of its squared magnitude over all time:$$E = \int_{-\infty}^{\infty} |x(t)|^2 dt$$For the given signal $x(t) = \frac{\sin(4\pi t)}{4\pi t}$, the energy is:$$E = \int_{-\infty}^{\infty} \left| \frac{\sin(4\pi t)}{4\pi t} \right|^2 dt$$We can solve this using Parseval's theorem, which states that the energy of a signal is equal to the energy of its Fourier transform:$$E = \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$$where $X(f)$ is the Fourier Transform of $x(t)$.The signal $x(t)$ is a form of the sinc function. The Fourier transform of a rectangular pulse is a sinc function, and by duality, the Fourier transform of a sinc function is a rectangular pulse. The relevant Fourier transform pair is:$$A \cdot \text{sinc}(2Wt) \iff \frac{A}{2W} \text{rect}\left(\frac{f}{2W}\right)$$where the normalized sinc function is $\text{sinc}(u) = \frac{\sin(\pi u)}{\pi u}$ and the rectangular function is $\text{rect}(v) = 1$ for $|v| < 1/2$ and 0 otherwise.Let's express the given signal $x(t)$ in terms of the normalized sinc function:$$x(t) = \frac{\sin(4\pi t)}{4\pi t} = \frac{\sin(\pi(4t))}{\pi(4t)} = \text{sinc}(4t)$$Comparing this with $A \cdot \text{sinc}(2Wt)$, we have $A=1$ and $2W=4$, which means $W=2$.Now, we can find the Fourier transform $X(f)$:$$X(f) = \frac{A}{2W} \text{rect}\left(\frac{f}{2W}\right) = \frac{1}{4} \text{rect}\left(\frac{f}{4}\right)$$The function $\text{rect}(f/4)$ is 1 for $|f/4| < 1/2$, i.e., $|f| < 2$, and 0 otherwise. So, the Fourier transform is:$$X(f) = \begin{cases} 1/4 & \text{for } -2 < f < 2 \\ 0 & \text{otherwise} \end{cases}$$Now, we calculate the energy using the integral in the frequency domain:$$E = \int_{-\infty}^{\infty} |X(f)|^2 df = \int_{-2}^{2} \left(\frac{1}{4}\right)^2 df$$$$E = \int_{-2}^{2} \frac{1}{16} df = \frac{1}{16} [f]_{-2}^{2}$$$$E = \frac{1}{16} (2 - (-2)) = \frac{1}{16} (4) = \frac{4}{16} = \frac{1}{4} = 0.25$$Alternatively, we can use the standard integral formula:$$\int_{-\infty}^{\infty} \left(\frac{\sin(at)}{t}\right)^2 dt = a\pi$$Our energy integral is:$$E = \int_{-\infty}^{\infty} \left(\frac{\sin(4\pi t)}{4\pi t}\right)^2 dt = \frac{1}{(4\pi)^2} \int_{-\infty}^{\infty} \left(\frac{\sin(4\pi t)}{t}\right)^2 dt$$Here, $a = 4\pi$. Applying the formula:$$E = \frac{1}{16\pi^2} \cdot (4\pi \cdot \pi) = \frac{4\pi^2}{16\pi^2} = \frac{1}{4} = 0.25$$Both methods give the energy as 0.25.