---
title: "PSD expression and relation with autocorrelation"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:psd-expression-autocorrelation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/psd-expression-autocorrelation
site: BCT Engineering
---

# PSD expression and relation with autocorrelation

Communication Systems · Chapter 1: Introduction

- Asked in: 2073 Bhadra (CS I), 2+4+4 marks
- Video: [WIENER-KHINCHIN RELATION [Relationship Between Power Spectrum and Auto Correlation Function]](https://www.youtube.com/watch?v=Yt7ZhjeNQHQ) — PRAKASAM TUTORIALS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/psd-expression-autocorrelation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define power Spectral Density (PSD). Find expression for PSD and its relationship with autocorrelation function.

## Short answer

Power spectral density S_x(f) shows power distribution over frequency (W/Hz). It is S_x(f) = lim T→∞ (|X_T(f)|^2/T). The Wiener–Khinchine theorem states S_x(f) and autocorrelation R_x(τ) are Fourier transform pairs: S_x(f) = F{R_x(τ)}. Average power P = ∫ S_x(f) df = R_x(0).

## Answer

#### Power spectral density (PSD)
The **power spectral density** $S_x(f)$ of a power signal $x(t)$ is a function that shows how the signal's average power is distributed over frequency (W/Hz). The total average power is

$$
P = \int_{-\infty}^{\infty} S_x(f)\,df
$$

#### Expression for PSD
Truncate $x(t)$ to the interval $-T/2 \le t \le T/2$ to get $x_T(t)$, which has finite energy and a Fourier transform $X_T(f)$. By Rayleigh's theorem,

$$
\int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}|X_T(f)|^2df
$$

Divide by $T$ and let $T \to \infty$:

$$
P = \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}\lim_{T\to\infty}\frac{|X_T(f)|^2}{T}\,df
$$

Comparing with $P = \int S_x(f)df$:

$$
S_x(f) = \lim_{T\to\infty}\frac{|X_T(f)|^2}{T}
$$

#### Relationship with autocorrelation (Wiener–Khinchine theorem)
The time-averaged autocorrelation of a power signal is

$$
R_x(\tau) = \lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}x(t)\,x^*(t-\tau)\,dt
$$

For the truncated signal, $\frac1T\int x_T(t)x_T^*(t-\tau)dt$ is a time-autocorrelation of an energy signal divided by $T$. For an energy signal we know (shown below) $\mathcal{F}\{\int x_T(t)x_T^*(t-\tau)dt\} = |X_T(f)|^2$:

$$
\begin{aligned}
\int\!\!\int x_T(t)x_T^*(t-\tau)e^{-j2\pi f\tau}dt\,d\tau
&= \int x_T(t)e^{-j2\pi ft}\left[\int x_T^*(\lambda)e^{j2\pi f\lambda}d\lambda\right]dt \\
&= X_T(f)\,X_T^*(f) = |X_T(f)|^2
\end{aligned}
$$

(using $\lambda = t - \tau$). Dividing by $T$ and taking the limit:

$$
\mathcal{F}\{R_x(\tau)\} = \lim_{T\to\infty}\frac{|X_T(f)|^2}{T} = S_x(f)
$$

Therefore the PSD and the autocorrelation function form a Fourier transform pair:

$$
S_x(f) = \int_{-\infty}^{\infty}R_x(\tau)e^{-j2\pi f\tau}d\tau, \qquad R_x(\tau) = \int_{-\infty}^{\infty}S_x(f)e^{j2\pi f\tau}df
$$

At $\tau = 0$: $R_x(0) = \int S_x(f)df = P$, the average power.

**Example:** $x(t) = A\cos 2\pi f_0t$ gives $R_x(\tau) = \frac{A^2}{2}\cos 2\pi f_0\tau$ and hence $S_x(f) = \frac{A^2}{4}[\delta(f-f_0) + \delta(f+f_0)]$, with $P = R_x(0) = A^2/2$.

## Check yourself

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

1. The PSD of a power signal is:
   A. lim |X_T(f)|^2 / T
   B. lim |X_T(f)| T
   C. X(f)/T
   D. |x(t)|^2
   Answer: A. Divide Rayleigh's energy of the truncated signal by T. (Key point: PSD expression)

2. The Wiener-Khinchine theorem says:
   A. PSD equals autocorrelation
   B. PSD is the derivative of R
   C. PSD and autocorrelation are a Fourier transform pair
   D. They are unrelated
   Answer: C. S_x(f) = F{R_x(tau)}. (Key point: Wiener-Khinchine)

3. R_x(0) equals:
   A. Average power P
   B. Zero
   C. Energy
   D. Peak PSD
   Answer: A. R_x(0) = integral S_x df = P. (Key point: R(0))

## Related in Communication Systems

- [Parseval's theorem; PSD–autocorrelation relation for white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-psd-autocorrelation-white-noise) — Communication Systems, chapter 9
- [PSD–autocorrelation relation; stereo FM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsd-autocorrelation-stereo-fm) — Communication Systems, chapter 3
- [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×
- [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×
- [Short note: Autocorrelation function and properties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-autocorrelation-properties) — Communication Systems, chapter 9, asked 2×

## Related in other subjects

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- [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