---
title: "Short note: White noise and its PSDF"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:short-note-white-noise-psdf"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-white-noise-psdf
site: BCT Engineering
---

# Short note: White noise and its PSDF

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2070 Chaitra (CS II), 2 marks
- Video: [What is White Gaussian Noise (WGN)?](https://www.youtube.com/watch?v=QfUQMzHfbxs) — 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/short-note-white-noise-psdf
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

Write a short note on white noise and its psdf.

## Short answer

White noise has a constant power spectral density S_n(f)=N_0/2 for all f, and its autocorrelation is(N_0/2)δ(τ). It is an idealised model; thermal noise approximates it up to ~10^12 Hz, behaving as white within practical bandwidths.

## Answer

**White noise** is an idealised random noise whose power spectral density is **constant at all frequencies**, just as white light contains all colours equally. Its two-sided PSD is

$$
S_n(f) = \frac{N_0}{2} \quad \text{for all } f
$$

Its autocorrelation is the inverse Fourier transform, an impulse:

$$
R_n(\tau) = \frac{N_0}{2}\,\delta(\tau)
$$

so any two different samples are uncorrelated (and independent if Gaussian).

```text
 Sn(f)                      Rn(tau)
   | N0/2                      ^ (N0/2) delta
 --+---------------- f   ------+------ tau
```

- Total power $\int S_n df$ is infinite, so it is only a model; real noise (thermal, shot) is flat up to very high frequencies ($\sim 10^{12}$ Hz), so it acts white in any practical bandwidth.
- For thermal noise $N_0 = kT$. Noise through a filter of bandwidth $B$ has power $N_0B$.
- Usually also assumed Gaussian: **AWGN**.

## Check yourself

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

1. The two-sided power spectral density of white noise is
   A. N_0 delta(f)
   B. N_0/2 for all f
   C. N_0/2 only up to a cut-off
   D. proportional to 1/f
   Answer: B. White noise has a constant PSD N_0/2 at all frequencies. (Key point: White noise PSD)

2. The autocorrelation of white noise is
   A. a constant N_0/2
   B. an impulse (N_0/2) delta(tau), so different samples are uncorrelated
   C. a sinc function
   D. a decaying exponential
   Answer: B. The inverse Fourier transform of a constant is an impulse, so samples at different times are uncorrelated. (Key point: Autocorrelation of white noise)

3. Why is white noise only a model?
   A. Its mean is not zero
   B. Its total power, the integral of the PSD, is infinite
   C. It is not Gaussian
   D. Its PSD varies with frequency
   Answer: B. A flat PSD over all frequencies has infinite power; real noise is flat only to very high frequencies, so it acts white in practical bandwidths. (Key point: Idealisation)

## Related in Communication Systems

- [Random process; white noise PSDF and autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-white-noise) — Communication Systems, chapter 9, asked 2×
- [White noise: relation between PSDF and AC function](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Awhite-noise-psdf-ac-relation) — Communication Systems, chapter 9
- [White noise PSDF and AC; properties of autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Awhite-noise-psdf-ac-properties) — Communication Systems, chapter 9
- [Autocorrelation properties; AC of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-properties-white-noise) — Communication Systems, chapter 9, 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×
- [Optimum detector impulse response in additive white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response) — Communication Systems, chapter 9, asked 2×