ESD and total energy of sinc pulse

Energy spectral density

The energy spectral density (ESD) $\Psi_g(f)$ of an energy signal shows how its total energy is distributed over frequency. It is the squared magnitude of the Fourier transform:

$$\Psi_g(f)=\lvert G(f)\rvert^2\quad(\text{J/Hz}),\qquad E=\int_{-\infty}^{\infty}\Psi_g(f)\,df$$

Properties: it is real, non-negative and even (for real $g(t)$); it is the Fourier transform of the energy autocorrelation function; and through a filter $\Psi_y(f)=\lvert H(f)\rvert^2\Psi_x(f)$.

ESD of $g(t)=A\,\text{sinc}(2Wt)$

Using $\text{sinc}(x)=\frac{\sin\pi x}{\pi x}$ and the known pair $\text{rect}(t/T)\leftrightarrow T\,\text{sinc}(fT)$, duality gives

$$T\,\text{sinc}(tT)\leftrightarrow\text{rect}(f/T)$$

With $T=2W$: $2W\,\text{sinc}(2Wt)\leftrightarrow\text{rect}\left(\frac{f}{2W}\right)$. Therefore

$$G(f)=\frac{A}{2W}\,\text{rect}\left(\frac{f}{2W}\right)=\begin{cases}\dfrac{A}{2W}, & \lvert f\rvert<W\\ 0, & \lvert f\rvert>W\end{cases}$$

So the ESD is

$$\Psi_g(f)=\lvert G(f)\rvert^2=\frac{A^2}{4W^2}\,\text{rect}\left(\frac{f}{2W}\right)$$

i.e. a flat spectrum of height $A^2/4W^2$ between $-W$ and $+W$ and zero outside.

        Psi(f)
          |
 A^2/4W^2 +--------+--------+
          |        |        |
 ---------+--------+--------+---------> f
         -W        0        W

Total energy

By Rayleigh's theorem:

$$\begin{aligned} E&=\int_{-W}^{W}\frac{A^2}{4W^2}\,df=\frac{A^2}{4W^2}\times 2W\\ &=\frac{A^2}{2W} \end{aligned}$$

Check in time domain: $\int_{-\infty}^{\infty}A^2\text{sinc}^2(2Wt)dt=A^2\cdot\frac{1}{2W}$, the same result.

Answer: $\Psi_g(f)=\frac{A^2}{4W^2}\text{rect}\left(\frac{f}{2W}\right)$ and $E=\frac{A^2}{2W}$ joules.