--- title: "Natural and flat top sampling; Nyquist rate of sin²(400πt)" subject: "Communication Systems" chapter: 4 key: "semester_6/communication_systems:natural-flat-top-sin2-400pit" source: https://www.bctengineer.com/q/semester_6/communication_systems/natural-flat-top-sin2-400pit site: BCT Engineering --- # Natural and flat top sampling; Nyquist rate of sin²(400πt) Communication Systems · Chapter 4: Pulse Modulation - Asked in: 2076 Asoj (CS II), 6+4 marks - Video: [Sampling Types in digital communication - Ideal, Natural, Flat top Sampling - Aperture effect](https://www.youtube.com/watch?v=sFiCgEVpTWs) — 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/natural-flat-top-sin2-400pit - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4 ## Question Define Sampling. Explain with proper illustration the Natural Sampling and Flat Top Sampling. Find the Nyquist rate and the interval for m(t) = sin²(400πt) ## Short answer Sampling converts continuous signal to discrete. Natural sampling tops follow signal; flat-top sampling holds value constant. m(t)=sin²(400πt) has max frequency 400 Hz, Nyquist rate 800 Hz, interval 1.25 ms. ## Answer #### Sampling **Sampling** is the process of converting a continuous-time signal into a discrete-time signal by taking its values at regular intervals $T_s$ (sampling rate $f_s = 1/T_s$). It is the first step of analog-to-digital conversion. #### Natural sampling The signal is multiplied by a train of rectangular pulses of width $\tau$; during each pulse the top **follows the signal**. $$ s(t) = g(t)c(t), \qquad S(f) = \frac{\tau}{T_s}\sum_n \text{sinc}(nf_s\tau)\,G(f - nf_s) $$ Every spectral copy has the exact shape of $G(f)$, so a LPF recovers $g(t)$ without distortion. #### Flat-top sampling The instantaneous sample value is **held constant** for width $\tau$ (sample-and-hold): $$ s(t) = \sum_n g(nT_s)h(t - nT_s), \qquad S(f) = f_s H(f)\sum_n G(f - nf_s) $$ with $H(f) = \tau\,\text{sinc}(f\tau)e^{-j\pi f\tau}$. The copies are multiplied by $H(f)$, causing **aperture effect** (high-frequency loss), corrected by an equalizer $1/H(f)$. ```text Signal .-~~~-. .' '. Natural |‾\ /‾\ /‾‾\| tops follow curve Flat-top |‾| |‾| |‾‾| tops flat (held) Ts 2Ts 3Ts ``` | Natural | Flat-top | |---|---| | Tops follow signal | Tops constant | | No aperture distortion | Aperture effect, needs equalizer | | Hard to quantize | Easy to quantize (used in PCM) | #### Nyquist rate and interval of $m(t) = \sin^2(400\pi t)$ Using $\sin^2\theta = \tfrac{1}{2}(1 - \cos 2\theta)$: $$ m(t) = \frac{1}{2} - \frac{1}{2}\cos(800\pi t) $$ It has a dc term and a single tone at $f = 800\pi/(2\pi) = 400$ Hz, so $f_{max} = 400$ Hz. $$ \begin{aligned} f_N &= 2f_{max} = 800\ \text{Hz} \\ T_N &= \frac{1}{800} = 1.25\ \text{ms} \end{aligned} $$ **Answer: Nyquist rate = 800 Hz, Nyquist interval = 1.25 ms.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Sampling is the process of: A. Rounding amplitudes to discrete levels B. Taking values of a continuous-time signal at regular intervals Ts C. Compressing the dynamic range D. Coding samples into bits Answer: B. Sampling converts continuous time to discrete time; quantization and coding come later. (Key point: Sampling definition) 2. Which pair correctly describes the two sampling types? A. Natural: flat tops; flat-top: tops follow the signal B. Natural: tops follow the signal; flat-top: tops held constant C. Both have flat tops D. Both follow the signal Answer: B. Natural sampling tops follow the curve; flat-top holds the sample value for width tau. (Key point: Natural vs flat-top) 3. Which type suffers aperture effect that needs an equalizer? A. Natural sampling B. Flat-top sampling C. Neither D. Both equally Answer: B. Flat-top spectrum is multiplied by H(f) = tau sinc(f tau), attenuating high frequencies. (Key point: Aperture effect) 4. For m(t) = sin^2(400 pi t), the Nyquist rate is: A. 400 Hz B. 800 Hz C. 1600 Hz D. 1000 Hz Answer: B. sin^2 = 1/2 - 1/2 cos(800 pi t), so the tone is 400 Hz, fmax = 400 Hz and the Nyquist rate is 800 Hz (interval 1.25 ms). (Key point: Nyquist rate of sin^2) ## Related in Communication Systems - [Flat top vs natural sampling; Nyquist rate of sin(800πt)/πt + 10cos2000t](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-vs-natural-nyquist-rate) — Communication Systems, chapter 4 - [Nyquist sampling theorem; flat top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-flat-top) — 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 - [Nyquist sampling theorem; natural sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-natural-sampling) — Communication Systems, chapter 4 - [Nyquist rate and interval for 6cos50πt + 20sin300πt − 10cos100πt](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-rate-6cos50-512) — Communication Systems, chapter 4, asked 2× - [Merits and demerits of flat-top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-sampling-merits) — Communication Systems, chapter 4 ## Related in other subjects - [Sampling theorem; Nyquist of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 5× - [Aliasing; Nyquist rate of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 4×