--- title: "Autocorrelation and ESD as Fourier transform pair" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:autocorrelation-esd-fourier-pair" source: https://www.bctengineer.com/q/semester_6/communication_systems/autocorrelation-esd-fourier-pair site: BCT Engineering --- # Autocorrelation and ESD as Fourier transform pair Communication Systems · Chapter 1: Introduction - Asked in: 2076 Baisakh (CS I), 1+3+4 marks - Video: [Relation between ESD and auto correlation || Signals and Systems || important question](https://www.youtube.com/watch?v=CHT5YQJBybo) — Friend's Explanation - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/autocorrelation-esd-fourier-pair - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Define auto-correlation function. Write its properties for energy signal. Prove that for an energy signal x(t) the auto-correlation function and energy spectral density form Fourier transform pair. ## Short answer The autocorrelation of an energy signal x(t) is R_x(τ)=∫ x(t)x*(t-τ) dt. Properties: R_x(0) equals energy, it is even (real signals), maximum at origin, and tends to zero as |τ|→∞. Taking Fourier transform of R_x(τ) yields Ψ_x(f)=|X(f)|^2, proving they are a Fourier pair. ## Answer #### Auto-correlation function The **auto-correlation function** measures the similarity between a signal and a time-shifted copy of itself. For an energy signal $x(t)$: $$ R_x(\tau) = \int_{-\infty}^{\infty} x(t)\,x^*(t-\tau)\,dt $$ where $\tau$ is the time shift (lag). #### Properties (energy signal) 1. **Value at origin equals energy:** $R_x(0) = \int |x(t)|^2dt = E$. 2. **Conjugate symmetry:** $R_x(\tau) = R_x^*(-\tau)$; for a real signal it is an **even** function, $R_x(\tau) = R_x(-\tau)$. 3. **Maximum at origin:** $|R_x(\tau)| \le R_x(0)$ for all $\tau$. 4. **Fourier pair with ESD:** $R_x(\tau) \leftrightarrow \Psi_x(f) = |X(f)|^2$. 5. $R_x(\tau) \to 0$ as $|\tau| \to \infty$ for energy signals. #### Proof: $R_x(\tau)$ and ESD form a Fourier transform pair Take the Fourier transform of $R_x(\tau)$: $$ \begin{aligned} \mathcal{F}\{R_x(\tau)\} &= \int_{-\infty}^{\infty} R_x(\tau)e^{-j2\pi f\tau}\,d\tau \\ &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} x(t)\,x^*(t-\tau)\,e^{-j2\pi f\tau}\,dt\,d\tau \end{aligned} $$ Change the variable in the inner integral: let $\lambda = t - \tau$, so $\tau = t - \lambda$ and $d\tau = -d\lambda$ (limits swap, cancelling the sign): $$ \begin{aligned} \mathcal{F}\{R_x(\tau)\} &= \int_{-\infty}^{\infty} x(t)\left[\int_{-\infty}^{\infty} x^*(\lambda)e^{-j2\pi f(t-\lambda)}\,d\lambda\right]dt \\ &= \int_{-\infty}^{\infty} x(t)e^{-j2\pi ft}\,dt \cdot \left[\int_{-\infty}^{\infty} x(\lambda)e^{-j2\pi f\lambda}\,d\lambda\right]^* \\ &= X(f)\,X^*(f) = |X(f)|^2 = \Psi_x(f) \end{aligned} $$ Hence $$ R_x(\tau) \;\xleftrightarrow{\ \mathcal{F}\ }\; \Psi_x(f) = |X(f)|^2 $$ i.e. the auto-correlation function and the energy spectral density are a Fourier transform pair. **Check:** putting $\tau = 0$ in the inverse transform, $R_x(0) = \int \Psi_x(f)\,df = \int |X(f)|^2df = E$, which is Rayleigh's energy theorem. **Example:** for $x(t) = e^{-at}u(t)$, $R_x(\tau) = \frac{e^{-a|\tau|}}{2a}$ and $\Psi_x(f) = \frac{1}{a^2 + 4\pi^2f^2}$, which are indeed a Fourier pair. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The autocorrelation function of an energy signal measures: A. The total bandwidth of the signal B. The similarity between the signal and a time-shifted copy of itself C. The energy in frequency band D. The phase of the signal Answer: B. R_x(tau) = integral x(t)x*(t-tau)dt compares x with its delayed version. (Key point: Autocorrelation meaning) 2. R_x(0) equals: A. Zero B. The average power C. The total energy E of the signal D. The peak amplitude Answer: C. R_x(0) = integral |x|^2 dt = E. (Key point: Value at origin) 3. In the proof that R_x and ESD form a Fourier pair, which substitution is used? A. Differentiating R_x with respect to tau B. lambda = t - tau, which separates the double integral into X(f) and X*(f) C. t = 2 tau D. lambda = f tau Answer: B. Setting lambda = t - tau splits the integral into X(f) times the conjugate X*(f). (Key point: Key step in proof) 4. The Fourier transform of R_x(tau) is |X(f)|^2. What follows at tau = 0? A. R_x(0) = maximum of |X(f)| B. R_x(0) = integral of |X(f)|^2 df = E, i.e. Rayleigh's energy theorem C. R_x(0) = X(0) D. R_x(0) = 0 Answer: B. Inverse transform at tau = 0 gives energy from the ESD area. (Key point: Result and consequence) ## Related in Communication Systems - [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× - [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 - [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1 - [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× - [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× - [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× ## 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 - [Derive the CT Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-derivation) — Signals and Systems, chapter 3, asked 3× - [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3 - [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3