---
title: "ESD and total energy of sinc pulse"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:esd-sinc-pulse"
source: https://www.bctengineer.com/q/semester_6/communication_systems/esd-sinc-pulse
site: BCT Engineering
---

# ESD and total energy of sinc pulse

Communication Systems · Chapter 1: Introduction

- Asked in: 2071 Bhadra (CS I), 2+3+2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/esd-sinc-pulse
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

What do you understand by Energy Spectral Density (ESD)? Find ESD and total energy for sinc pulse defined by g(t) = A sinc(2Wt).

## Short answer

ESD is |G(f)|², showing energy distribution over frequency. For g(t)=A sinc(2Wt), G(f)=(A/2W) rect(f/2W), so ESD = (A²/4W²) rect(f/2W). Total energy E = ∫ ESD df = A²/(2W) J.

## Answer

#### 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.

```text
        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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The ESD of an energy signal is:
   A. |G(f)|^2 in J/Hz
   B. |g(t)|^2 in W
   C. G(f)/T
   D. The limit of |G_T|^2/T
   Answer: A. Psi = |G|^2; the limit form is the PSD. (Key point: ESD definition)

2. The Fourier transform of A sinc(2Wt) is:
   A. A rect(f/2W)
   B. (A/2W) rect(f/2W)
   C. 2WA rect(f/2W)
   D. A sinc(f/2W)
   Answer: B. Duality gives G(f)=(A/2W) rect(f/2W), flat between -W and W. (Key point: Spectrum of sinc)

3. The ESD of A sinc(2Wt) is:
   A. A^2/(2W) over |f|<W
   B. A/(2W) over |f|<W
   C. A^2/(4W^2) over |f|<W, zero outside
   D. A^2 sinc^2(f)
   Answer: C. Squaring G(f) gives A^2/(4W^2). (Key point: ESD value)

4. The total energy is:
   A. A^2/(4W^2)
   B. A^2/(2W)
   C. A^2 W
   D. 2W A^2
   Answer: B. E = (A^2/4W^2)(2W)=A^2/(2W). (Key point: Total energy)

## Related in Communication Systems

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- [ESD of an energy signal (ideal BPF example)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-energy-signal-ideal-bpf) — Communication Systems, chapter 1
- [Sub-sampling; sampling sinc(5πt) at 10 Hz](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asubsampling-sinc5pit-10hz) — Communication Systems, chapter 4
- [ESD vs PSD; PSD of an arbitrary signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-vs-psd-arbitrary-signal) — Communication Systems, chapter 1, asked 2×
- [ESD, PSD and autocorrelation of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-psd-white-noise-autocorrelation) — Communication Systems, chapter 9, asked 2×
- [Autocorrelation and ESD as Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-esd-fourier-pair) — Communication Systems, chapter 1

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