--- title: "Impulse train sampling, aliasing, reconstruction" subject: "Signals and Systems" chapter: 4 key: "semester_4/signals_and_systems:impulse-train-aliasing-reconstruction" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/impulse-train-aliasing-reconstruction site: BCT Engineering --- # Impulse train sampling, aliasing, reconstruction Signals and Systems · Chapter 4: Sampling - Asked in: 2072 Asoj, 1+2+3 marks - Video: [Sampling , Aliasing and Reconstruction | Solved Problem | Nyquist Criteria | Impulse Train](https://www.youtube.com/watch?v=KYZLjXIE6JY) — Fakhar STEM Sphere - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/impulse-train-aliasing-reconstruction - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4 ## Question What is impulse train sampling? Explain the aliasing effect that may occur in impulse train sampling. How do you reconstruct original signal from its sample? ## Short answer Impulse-train sampling multiplies x(t) by an impulse train Σδ(t−nT) to produce weighted impulses x_p(t)=Σx(nT)δ(t−nT). Aliasing occurs when ω_s < 2ω_M, causing spectral copies to overlap and high frequencies to fold back. Reconstruction is achieved by passing x_p(t) through an ideal low-pass filter with gain T and cutoff ω_M < ω_c < ω_s−ω_M, yielding sinc interpolation. ## Answer #### Impulse-train sampling Impulse-train sampling is the mathematical model of ideal sampling: $x(t)$ is multiplied by a periodic train of unit impulses of period $T$: $$ x_p(t) = x(t)\sum_{n=-\infty}^{\infty}\delta(t-nT) = \sum_{n=-\infty}^{\infty} x(nT)\,\delta(t-nT) $$ Each impulse carries the sample value $x(nT)$ as its area. #### Aliasing in impulse-train sampling The spectrum of the sampled signal is $$ X_p(j\omega) = \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big), \qquad \omega_s = \frac{2\pi}{T} $$ i.e. copies of $X(j\omega)$ at every multiple of $\omega_s$. - If $\omega_s > 2\omega_M$, the copies are separate. - If $\omega_s < 2\omega_M$, the copy at $\omega_s$ begins at $\omega_s - \omega_M$, which is below $\omega_M$, so the copies **overlap**. Frequencies above $\omega_s/2$ fold back into the band and add to the true components. This is **aliasing**; the original spectrum can no longer be separated. ```text ws < 2wM: ____ ____ ____ / \ / \ / \ ---/------\-/------\-/------\---> w -ws \/ 0 \/ ws overlap = aliasing ``` #### Reconstruction from samples If $\omega_s > 2\omega_M$, pass $x_p(t)$ through an **ideal low-pass filter** with gain $T$ and cutoff $\omega_c$, where $\omega_M < \omega_c < \omega_s - \omega_M$ (usually $\omega_c = \omega_s/2$). The filter keeps only the $k=0$ copy, so its output is $x(t)$. In time, $$ h(t) = \frac{T\sin\omega_c t}{\pi t}, \qquad x_r(t) = \sum_{n} x(nT)\,\frac{T\sin\omega_c(t-nT)}{\pi(t-nT)} $$ This is **band-limited (sinc) interpolation**. Practical systems use simpler approximations: zero-order hold (staircase) or first-order hold (straight-line interpolation) followed by smoothing. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Each impulse in x_p(t) has area A. 1 B. T C. x(t) D. The sample value x(nT) Answer: D. x_p = sum x(nT) δ(t - nT). (Key point: Impulse sampling) 2. Aliasing arises when ωs < 2ωM because A. The copy at ωs starts at ωs - ωM < ωM and overlaps B. The copies move apart C. The impulses vanish D. The gain T becomes zero Answer: A. Overlapping copies add. (Key point: Aliasing cause) 3. Reconstruction uses an ideal LPF with A. Gain 1/T and ωc > ωs B. Gain T and ωM < ωc < ωs - ωM C. Gain T and ωc < ωM D. HPF with gain T Answer: B. It keeps only the k = 0 copy. (Key point: Reconstruction filter) 4. The time-domain result is A. Zero-order hold B. Linear interpolation C. Band-limited sinc interpolation D. Quadratic fitting Answer: C. h(t) = T sin(ωc t)/(πt). (Key point: Interpolation) ## Related in Signals and Systems - [Impulse train sampling and reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-sampling-reconstruction) — Signals and Systems, chapter 4 - [Aliasing via impulse-train sampling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-impulse-train-frequency) — Signals and Systems, chapter 4 - [Sampling theorem via impulse train sampling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-impulse-train-frequency) — Signals and Systems, chapter 4 - [Prove sampling theorem; aliasing effect](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-aliasing) — Signals and Systems, chapter 4 - [Sampling theorem; aliasing cause; Nyquist](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-aliasing-cause-nyquist) — Signals and Systems, chapter 4 - [Aliasing in a band-limited signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-bandlimited-w) — Signals and Systems, chapter 4 ## Related in other subjects - [Sampling theorem; aliasing and aperture effect remedies](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-theorem-aliasing-aperture-remedy) — Communication Systems, chapter 4 - [Prove sampling theorem; aliasing and aperture effect](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-prove-aliasing-aperture) — Communication Systems, chapter 4 - [Ideal sampling and reconstruction; Nyquist rate of ½π cos(400πt)cos(1000πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aideal-sampling-reconstruction-nyquist) — Communication Systems, chapter 4 - [Reconstruction of signal from samples](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Areconstruction-from-sampled-signal) — Digital Control System, chapter 1