--- title: "Nyquist-Kotelnikov sampling theorem; aliasing and aperture" subject: "Communication Systems" chapter: 4 key: "semester_6/communication_systems:nyquist-kotelnikov-aliasing-aperture" source: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-kotelnikov-aliasing-aperture site: BCT Engineering --- # Nyquist-Kotelnikov sampling theorem; aliasing and aperture Communication Systems · Chapter 4: Pulse Modulation - Asked in: 2071 Shrawan (CS II), 6+2 marks - Video: [SAMPLING THEOREM in digital communication - sampling rate and Nyquist rate](https://www.youtube.com/watch?v=wgS5OglyB80) — Electronics Subjectified - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-kotelnikov-aliasing-aperture - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4 ## Question State and explain Nyquist-Kotelnikov sampling theorem with time domain and frequency domain analysis. Define aliasing and aperture effect. ## Short answer A band-limited signal sampled at Nyquist rate can be exactly reconstructed using sinc interpolation. Aliasing is spectral overlap from under-sampling, prevented by anti-aliasing filters. Aperture effect is high-frequency attenuation due to finite sampling pulse width. ## Answer #### Nyquist–Kotelnikov sampling theorem A signal $g(t)$ band-limited to $W$ Hz is completely described by its instantaneous values taken at uniform intervals $T_s \le \dfrac{1}{2W}$, and can be exactly reconstructed from them. The minimum rate $f_s = 2W$ is the Nyquist rate. (Kotelnikov proved it independently in 1933; it is also called the Kotelnikov theorem or the Whittaker–Kotelnikov–Shannon (WKS) theorem.) #### Time-domain analysis Ideal sampling multiplies $g(t)$ by a periodic impulse train: $$ g_\delta(t) = g(t)\sum_n\delta(t - nT_s) = \sum_n g(nT_s)\,\delta(t - nT_s) $$ Reconstruction by an ideal LPF of bandwidth $W$ (with $f_s = 2W$ and $\text{sinc}(x) = \sin x/x$, $h(t) = \text{sinc}(2\pi Wt)$) gives $$ g(t) = \sum_{n=-\infty}^{\infty} g(nT_s)\,\text{sinc}\big(2\pi W(t - nT_s)\big) $$ At $t = kT_s$ only the $k$-th sinc is nonzero (all the others are at their zeros), so the curve passes exactly through every sample, and between samples the sincs interpolate the signal. ```text g(t) .-~~-. .' | '. _.' | | | '._ | | | | | | samples every Ts -2Ts -Ts 0 Ts 2Ts ``` #### Frequency-domain analysis The impulse train has Fourier series $\delta_{T_s}(t) = f_s\sum_n e^{j2\pi nf_st}$, so $$ G_\delta(f) = f_s\sum_{n=-\infty}^{\infty}G(f - nf_s) $$ ```text /\ /\ /\ ___/ \______/ \______/ \___ f -fs -W 0 W fs ``` - Copies of $G(f)$ sit at $0, \pm f_s, \pm 2f_s, \ldots$ - They do not overlap if $f_s - W \ge W$, i.e. $f_s \ge 2W$. - An ideal LPF $H(f) = T_s$ for $|f| < W$ then gives $G(f)$ back. - If $f_s < 2W$ the copies overlap and $G(f)$ is lost. #### Aliasing and aperture effect - **Aliasing:** overlap of spectral copies when $f_s < 2W$; high frequencies appear as false low frequencies. It is prevented by an anti-aliasing filter and $f_s > 2W$. - **Aperture effect:** in flat-top sampling with pulse width $\tau$, the spectrum is multiplied by $H(f) = \tau\,\text{sinc}(\pi f\tau)$, which attenuates high frequencies. It is corrected with an equalizer $1/H(f)$ or by keeping $\tau$ small. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The Nyquist-Kotelnikov theorem says a signal band-limited to W Hz can be exactly reconstructed if sampled with: A. Ts >= 1/2W B. Ts <= 1/2W C. Ts = 1/W only D. Ts <= 2W Answer: B. Ts <= 1/(2W), i.e. fs >= 2W. (Key point: Sampling theorem) 2. Ideal sampling in frequency domain gives: A. A single scaled copy of G(f) B. Copies of G(f) at multiples of fs scaled by fs C. G(f) shifted by W D. Impulses at W Answer: B. G_delta(f) = fs sum G(f - n fs); copies do not overlap if fs >= 2W. (Key point: Frequency-domain view) 3. In the reconstruction formula, why does the curve pass exactly through each sample? A. Each sinc is non-zero at all sample instants B. At t = kTs only the k-th sinc is non-zero; others are at zeros C. The filter gain is infinite D. Sinc functions are all equal Answer: B. Sinc pulses are zero at all other sampling instants. (Key point: Interpolation) 4. Which pairing is correct? A. Aliasing: sinc droop; aperture: overlap of copies B. Aliasing: overlap when fs < 2W; aperture: sinc attenuation due to pulse width C. Both are quantization noise D. Both are removed by increasing bits Answer: B. Aliasing is spectral overlap; the aperture effect is high-frequency attenuation from finite pulse width, corrected by 1/H(f) equalizer. 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