--- title: "ESD, PSD and autocorrelation of white noise" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:esd-psd-white-noise-autocorrelation" source: https://www.bctengineer.com/q/semester_6/communication_systems/esd-psd-white-noise-autocorrelation site: BCT Engineering --- # ESD, PSD and autocorrelation of white noise Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2075 Bhadra (CS I), 4+4 marks; 2074 Bhadra (CS I), 4+4 marks (asked 2 times) - Video: [Autocorrelation and Power Spectral Density (PSD) Examples in Digital Communications](https://www.youtube.com/watch?v=XWytSLZZP1A) — 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-psd-white-noise-autocorrelation - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Define energy spectral density and power spectral density function of a signal and hence derive the auto correlation function of white noise utilizing power spectral density along with necessary diagrams. ## Short answer Energy spectral density is |G(f)|² for energy signals; power spectral density for power signals is S(f)=lim_{T→∞}|G_T(f)|²/T. White noise has a flat PSD S_W(f)=N_0/2. Its autocorrelation, by inverse Fourier transform, is R(τ)=(N_0/2)δ(τ), showing any two distinct samples are uncorrelated. ## Answer #### Energy and power spectral density **Energy spectral density (ESD):** for an energy signal $g(t)\leftrightarrow G(f)$, $\Psi_g(f)=\lvert G(f)\rvert^2$ (J/Hz). It shows how the energy is spread over frequency: $$ E=\int_{-\infty}^{\infty}\lvert g(t)\rvert^2dt=\int_{-\infty}^{\infty}\Psi_g(f)\,df $$ and $\Psi_g(f)$ is the Fourier transform of the energy autocorrelation $R_g(\tau)$. **Power spectral density (PSD):** for a power signal, take a truncated version $g_T(t)$ (length $T$) with transform $G_T(f)$: $$ S_g(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}\ \ \text{(W/Hz)},\qquad P=\int_{-\infty}^{\infty}S_g(f)\,df $$ For a WSS random process, $S_X(f)=\lim_{T\to\infty}E[\lvert X_T(f)\rvert^2]/T$. Both ESD and PSD are real, even and non-negative. **Wiener–Khinchin relation (PSD $\leftrightarrow$ AC):** for a power signal, $R_g(\tau)=\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}g_T(t)g_T(t+\tau)dt$. The time-domain integral is $g_T(\tau)*g_T(-\tau)$ scaled by $1/T$, and $g_T(-t)\leftrightarrow G_T^*(f)$, so $$ \begin{aligned} \mathcal{F}\{R_g(\tau)\} &= \lim_{T\to\infty}\frac{1}{T}G_T(f)G_T^*(f)=\lim_{T\to\infty}\frac{\lvert G_T(f)\rvert^2}{T}=S_g(f) \end{aligned} $$ Hence $$ S_g(f)=\int_{-\infty}^{\infty}R_g(\tau)e^{-j2\pi f\tau}d\tau,\qquad R_g(\tau)=\int_{-\infty}^{\infty}S_g(f)e^{j2\pi f\tau}df $$ and at $\tau=0$: $R_g(0)=\int S_g(f)df=P$. #### Autocorrelation of white noise from its PSD **White noise** is an idealised random noise whose power spectral density is **flat (constant) over all frequencies**, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed **Gaussian, zero-mean** (AWGN). **PSD (two-sided):** $$ S_W(f)=\frac{N_0}{2}\quad\text{for all } f $$ where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature. **Autocorrelation** (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$): $$ R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau) $$ ```text S_W(f) R_W(tau) | ^ (N0/2) delta N0/2+----------------- | | | ----+-----------------> f ------+------> tau 0 0 ``` Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so **any two samples of white noise, however close, are uncorrelated** (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite. **Band-limited white noise** (more practical): if $S(f)=\frac{N_0}{2}$ for $\lvert f\rvert\le B$ and zero elsewhere, $$ R(\tau)=\int_{-B}^{B}\frac{N_0}{2}e^{j2\pi f\tau}df=N_0B\,\frac{\sin2\pi B\tau}{2\pi B\tau} $$ Its power is finite, $R(0)=N_0B$, and samples spaced $1/2B$ apart are uncorrelated. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the energy spectral density of g(t) equal? A. |G(f)|^2, in J/Hz B. |g(t)|^2 C. G(f) D. lim |G_T(f)|^2 / T Answer: A. ESD is |G(f)|^2 and its integral over f gives energy E. (Key point: ESD) 2. How is PSD defined for a power signal? A. |G(f)|^2 without limit B. Integral of g^2 C. G_T(f) T D. lim |G_T(f)|^2 / T as T goes to infinity, in W/Hz Answer: D. PSD divides the truncated energy spectrum by T. (Key point: PSD definition) 3. What does Wiener-Khinchin state? A. PSD is the inverse of R B. R is the square of PSD C. PSD is the Fourier transform of the autocorrelation D. They are independent Answer: C. F{R(tau)} = S(f), and R(0) = power. (Key point: Wiener-Khinchin) 4. Taking the inverse transform of S = N0/2, what is R_W(tau)? A. N0/2 B. (N0/2) delta(tau) C. N0 B sinc D. 0 for all tau Answer: B. The inverse transform of a constant is an impulse. (Key point: White noise result) 5. For S(f) = N0/2 over |f| <= B, what is R(0) and where are samples uncorrelated? A. R(0) = N0 B; samples spaced 1/2B apart B. R(0) = infinite; any spacing C. R(0) = N0/2; spaced 1/B D. R(0) = 0; spaced 2B Answer: A. R(tau) = N0 B sinc(2 B tau) has zeros at multiples of 1/2B. (Key point: Band-limited noise) ## 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 - [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 - [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× - [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× - [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× - [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 ## 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