---
title: "ESD vs PSD; PSD of an arbitrary signal"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:esd-vs-psd-arbitrary-signal"
source: https://www.bctengineer.com/q/semester_6/communication_systems/esd-vs-psd-arbitrary-signal
site: BCT Engineering
---

# ESD vs PSD; PSD of an arbitrary signal

Communication Systems · Chapter 1: Introduction

- Asked in: 2070 Magh (CS I), 2+6 marks
- Video: [What is Power Spectral Density (PSD)?](https://www.youtube.com/watch?v=DoSLMEEo1Y0) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/esd-vs-psd-arbitrary-signal
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Differentiate between energy spectral density function and power spectral density function. Derive the expression of power spectral density function of a arbitrary signal z(t).

## Short answer

ESD applies to energy signals as |G(f)|² (J/Hz). PSD applies to power signals and is S_z(f)=lim_{T→∞} |Z_T(f)|²/T (W/Hz). Derivation: truncate z(t) to z_T(t), compute average power via Rayleigh’s theorem, and take the limit.

## Answer

#### ESD vs PSD

| Energy spectral density (ESD) | Power spectral density (PSD) |
|---|---|
| Defined for energy signals ($0<E<\infty$, $P=0$) | Defined for power signals ($0<P<\infty$, $E=\infty$) |
| $\Psi_g(f)=\lvert G(f)\rvert^2$ | $S_g(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}$ |
| Unit: J/Hz | Unit: W/Hz |
| Area gives total energy $E$ | Area gives average power $P$ |
| FT of energy autocorrelation | FT of time-averaged autocorrelation |
| Example: single pulse | Example: sinusoid, random noise |

#### PSD of an arbitrary signal z(t)
A power signal $z(t)$ (periodic or random) has infinite energy, so its Fourier transform may not exist. We therefore use a **truncated** version:

$$
z_T(t)=\begin{cases} z(t), & -T/2\le t\le T/2\\ 0, & \text{otherwise}\end{cases}
$$

$z_T(t)$ has finite energy, so it has a Fourier transform $Z_T(f)$.

**Step 1 – average power.** The average power of $z(t)$ is

$$
P=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\lvert z(t)\rvert^2dt=\lim_{T\to\infty}\frac{1}{T}\int_{-\infty}^{\infty}\lvert z_T(t)\rvert^2dt
$$

**Step 2 – apply Rayleigh's energy theorem** to the energy signal $z_T(t)$:

$$
\int_{-\infty}^{\infty}\lvert z_T(t)\rvert^2dt=\int_{-\infty}^{\infty}\lvert Z_T(f)\rvert^2df
$$

**Step 3 – substitute and interchange limit and integral:**

$$
\begin{aligned}
P&=\lim_{T\to\infty}\frac{1}{T}\int_{-\infty}^{\infty}\lvert Z_T(f)\rvert^2df\\
&=\int_{-\infty}^{\infty}\left[\lim_{T\to\infty}\frac{\lvert Z_T(f)\rvert^2}{T}\right]df
\end{aligned}
$$

**Step 4 – identify PSD.** Since $P=\int S_z(f)df$, the quantity in brackets is the power per unit bandwidth:

$$
S_z(f)=\lim_{T\to\infty}\frac{\lvert Z_T(f)\rvert^2}{T}\quad\text{W/Hz}
$$

For a random signal the expectation is also taken: $S_z(f)=\lim_{T\to\infty}\frac{E[\lvert Z_T(f)\rvert^2]}{T}$.

**Wiener–Khinchin relation:** the same PSD is also the Fourier transform of the time-averaged autocorrelation function:

$$
S_z(f)=\int_{-\infty}^{\infty}R_z(\tau)e^{-j2\pi f\tau}d\tau,\qquad R_z(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}z(t)z^*(t-\tau)dt
$$

**Properties:** $S_z(f)\ge0$, it is even for real $z(t)$, the total area under $S_z(f)$ equals $R_z(0)=P$, and through an LTI filter $S_y(f)=\lvert H(f)\rvert^2S_z(f)$.

## Check yourself

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

1. PSD is defined for:
   A. Energy signals with P=0
   B. Power signals with 0<P<infinity
   C. Only periodic signals
   D. Signals with zero power
   Answer: B. Power signals have infinite energy; PSD applies to them. (Key point: Signal class)

2. Why is z(t) truncated to z_T(t)?
   A. To make it periodic
   B. To remove noise
   C. To make it band-limited
   D. It has infinite energy so its Fourier transform may not exist; z_T has finite energy
   Answer: D. The truncated version has a transform Z_T(f). (Key point: Truncation)

3. The PSD is:
   A. lim |Z_T(f)|^2 / T
   B. lim |Z_T(f)|^2 T
   C. lim |Z_T(f)| / T
   D. |Z(f)|^2
   Answer: A. Applying Rayleigh to z_T and dividing by T gives power per Hz. (Key point: PSD expression)

4. The PSD is also:
   A. The inverse transform of R_z
   B. The Fourier transform of the time-averaged autocorrelation R_z(tau)
   C. The derivative of R_z
   D. R_z(0)
   Answer: B. S_z(f)=FT of R_z(tau); area under S equals R_z(0)=P. (Key point: Wiener-Khinchin)

## Close variants set in other exams

- [PSDF of an arbitrary signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsdf-arbitrary-signal) — Communication Systems, chapter 1, asked 3×

## Related in Communication Systems

- [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
- [Energy and power spectral density; spectrum analyzer](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-spectrum-analyzer) — Communication Systems, chapter 1, asked 2×
- [PSD of a power signal and its interpretation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsdf-power-signal-interpretation) — 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×
- [PSD expression and relation with autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsd-expression-autocorrelation) — Communication Systems, chapter 1
- [ESD and total energy of sinc pulse](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-sinc-pulse) — Communication Systems, chapter 1

## Related in other subjects

- [Energy vs power spectral density](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-spectral-density) — Signals and Systems, chapter 3
- [FT of DC; meaning of ESD and PSD](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-dc-esd-psd) — Signals and Systems, chapter 3
- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1
- [Energy density spectrum of CT signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-density-spectrum) — Signals and Systems, chapter 3, asked 3×