--- title: "Aliasing via impulse-train sampling" subject: "Signals and Systems" chapter: 4 key: "semester_4/signals_and_systems:aliasing-impulse-train-frequency" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/aliasing-impulse-train-frequency site: BCT Engineering --- # Aliasing via impulse-train sampling Signals and Systems · Chapter 4: Sampling - Asked in: 2070 Bhadra, 1+5 marks - Video: [Sampling Signals](https://www.youtube.com/watch?v=AcuQnIXiZ2A) — 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_4/signals_and_systems/aliasing-impulse-train-frequency - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4 ## Question What do you mean by aliasing? Explain with the help of frequency domain analysis for impulse-train sampling. ## Short answer Aliasing is spectrum overlap when sampling below the Nyquist rate, causing high frequencies to appear as lower ones. Impulse-train sampling repeats the signal's spectrum every ω_s; if ω_s < 2ω_M, adjacent copies overlap, corrupting the baseband and preventing recovery. ## Answer #### Aliasing **Aliasing** is the overlapping of spectral copies that happens when a signal is sampled below its Nyquist rate ($\omega_s < 2\omega_M$); high-frequency components then appear as lower frequencies and the signal cannot be recovered. #### Frequency-domain analysis of impulse-train sampling Let $x(t)$ be band-limited to $\omega_M$. Impulse-train sampling multiplies it by a periodic impulse train of period $T$: $$ p(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT), \qquad x_p(t) = x(t)p(t) = \sum_{n=-\infty}^{\infty} x(nT)\,\delta(t-nT) $$ The Fourier transform of the impulse train is also an impulse train, with spacing $\omega_s = 2\pi/T$: $$ P(j\omega) = \frac{2\pi}{T}\sum_{k=-\infty}^{\infty}\delta(\omega - k\omega_s) $$ Multiplication in time is convolution in frequency: $$ \begin{aligned} X_p(j\omega) &= \frac{1}{2\pi}\, X(j\omega) * P(j\omega) \\ &= \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big) \end{aligned} $$ So the spectrum of the sampled signal is the original spectrum, scaled by $1/T$, repeated at every multiple of $\omega_s$. **Case 1: $\omega_s > 2\omega_M$ (no aliasing)** ```text /\ /\ /\ / \ / \ / \ --/----\----/----\----/----\---> w -ws 0 ws ``` The copies are separate. An ideal LPF (gain $T$, cutoff $\omega_M < \omega_c < \omega_s - \omega_M$) recovers $X(j\omega)$ exactly. **Case 2: $\omega_s < 2\omega_M$ (aliasing)** ```text ____ ____ ____ / \ / \ / \ ---/------\/------\/------\---> w -ws 0 ws overlapped regions add ``` The $k=1$ copy starts at $\omega_s - \omega_M < \omega_M$, so it overlaps the baseband copy. In the band $\omega_s - \omega_M < |\omega| < \omega_M$, the spectrum is a sum of two copies. A component at $\omega_0 > \omega_s/2$ shows up at $\omega_s - \omega_0$. **Example:** $x(t) = \cos(2\pi\cdot 700t)$ sampled at $f_s = 1000$ Hz. The copy at $1000 - 700 = 300$ Hz lies inside the LPF band, so the output of reconstruction is a 300 Hz tone instead of 700 Hz. Aliasing is prevented by sampling at $\omega_s > 2\omega_M$ and using an anti-aliasing filter before sampling. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is aliasing? A. Loss of amplitude due to quantization B. Addition of noise by the filter C. Overlap of spectral copies when ωs < 2ωM, making high frequencies appear as low ones D. Gaps between spectral copies due to oversampling Answer: C. Under-sampling makes the copies overlap so frequencies fold back. (Key point: Aliasing definition) 2. Impulse-train sampling in time produces what in frequency? A. A single narrower spectrum B. The original spectrum scaled by 1/T repeated at every multiple of ωs C. The original spectrum shifted once by ωs D. The spectrum multiplied by ωs Answer: B. Xp(jω) = (1/T) Σ X(j(ω - kωs)). (Key point: Sampled spectrum shape) 3. When ωs > 2ωM, how is X(jω) recovered? A. Ideal LPF with gain T and cutoff between ωM and ωs - ωM B. Ideal HPF with gain 1 C. Differentiator D. Not possible without anti-aliasing Answer: A. Copies are separate, so an LPF isolates the baseband copy. (Key point: No-aliasing recovery) 4. With aliasing, a component at ω0 > ωs/2 appears at: A. ω0 + ωs B. ω0 / 2 C. 2ω0 D. ωs - ω0 Answer: D. The k=1 copy folds the component to ωs - ω0. (Key point: Alias location) ## Related in Signals and Systems - [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 - [Impulse train sampling, aliasing, reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-aliasing-reconstruction) — Signals and Systems, chapter 4 - [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 - [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 - [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 ## Related in other subjects - [Nyquist-Kotelnikov sampling theorem; aliasing and aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-kotelnikov-aliasing-aperture) — Communication Systems, chapter 4 - [Hilbert transform in frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-frequency-domain) — Communication Systems, chapter 2 - [Aliasing and aperture effects; types of sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaliasing-aperture-sampling-types) — Communication Systems, chapter 4 - [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