--- title: "Ideal sampling and reconstruction; Nyquist rate of ½π cos(400πt)cos(1000πt)" subject: "Communication Systems" chapter: 4 key: "semester_6/communication_systems:ideal-sampling-reconstruction-nyquist" source: https://www.bctengineer.com/q/semester_6/communication_systems/ideal-sampling-reconstruction-nyquist site: BCT Engineering --- # Ideal sampling and reconstruction; Nyquist rate of ½π cos(400πt)cos(1000πt) Communication Systems · Chapter 4: Pulse Modulation - Asked in: 2075 Asoj (CS II), 6+4 marks - Video: [Examples of Nyquist rate and Sampling Theory in Digital Communication by Engineering Funda](https://www.youtube.com/watch?v=6a8pmz-HPH4) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/ideal-sampling-reconstruction-nyquist - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4 ## Question Illustrate and explain the ideal sampling and reconstruction of sampled signal. Find the Nyquist rate and the interval for ½π Cos(400πt) Cos(1000πt). ## Short answer Ideal sampling multiplies signal by impulse train, replicating spectrum at multiples of fs. Reconstruction uses ideal LPF with cutoff W ≤ B ≤ fs-W to recover original if fs ≥ 2W. For 1/(2π) cos(400πt) cos(1000πt), product-to-sum gives frequencies 300 Hz and 700 Hz. Maximum frequency is 700 Hz, so Nyquist rate = 1400 Hz, Nyquist interval = 0.714 ms. ## Answer #### Ideal sampling In ideal (impulse) sampling the message $g(t)$, band-limited to $W$, is multiplied by an impulse train of period $T_s$: $$ g_\delta(t) = g(t)\sum_n \delta(t - nT_s) = \sum_n g(nT_s)\,\delta(t - nT_s) $$ The impulse train has the Fourier series $\delta_{T_s}(t) = f_s\sum_n e^{j2\pi nf_st}$, so $$ G_\delta(f) = f_s\sum_{n=-\infty}^{\infty} G(f - nf_s) $$ The spectrum is $G(f)$ repeated every $f_s$ (scaled by $f_s$). ```text G(f) /\ ____/ \____ -W W Gd(f) /\ /\ /\ ____/ \__/ \__/ \___ (fs > 2W) -fs 0 fs |<-LPF->| ``` #### Reconstruction - If $f_s \ge 2W$, the copies do not overlap. An ideal LPF of gain $T_s$ and cut-off $B$ ($W \le B \le f_s - W$) passes only the $n = 0$ copy, giving back $G(f)$. - In the time domain, the LPF impulse response is $h(t) = 2BT_s\,\text{sinc}(2\pi Bt)$, where here $\text{sinc}\,x = \sin x/x$. With $f_s = 2W$ and $B = W$, $h(t) = \text{sinc}(2\pi Wt)$, and $$ g(t) = \sum_n g(nT_s)\,\text{sinc}\big(2\pi W(t - nT_s)\big) $$ This is the **interpolation formula**: each sample is replaced by a sinc pulse, and the sum is exactly $g(t)$. - If $f_s < 2W$, copies overlap (aliasing) and $g(t)$ cannot be recovered. - In practice a guard band ($f_s > 2W$) is used since ideal LPFs do not exist. #### Nyquist rate of $\tfrac{1}{2\pi}\cos(400\pi t)\cos(1000\pi t)$ (The constant in front only scales amplitude, so it does not affect the answer.) Using $\cos A\cos B = \tfrac{1}{2}[\cos(A - B) + \cos(A + B)]$: $$ \frac{1}{2\pi}\cos(400\pi t)\cos(1000\pi t) = \frac{1}{4\pi}\big[\cos(600\pi t) + \cos(1400\pi t)\big] $$ Frequencies: $300$ Hz and $700$ Hz, so $f_{max} = 700$ Hz. $$ \begin{aligned} f_N &= 2 \times 700 = 1400\ \text{Hz} \\ T_N &= \frac{1}{1400} = 0.714\ \text{ms} \end{aligned} $$ **Answer: Nyquist rate = 1400 Hz, Nyquist interval = 0.714 ms.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Ideal sampling of g(t) gives Gd(f) equal to: A. G(f) only B. fs times the sum of G(f - n fs) C. tau sinc(f tau) G(f) D. G(f)/fs Answer: B. The impulse train repeats the spectrum at multiples of fs, scaled by fs. (Key point: Ideal sampling spectrum) 2. To recover g(t), the ideal LPF cut-off B must satisfy: A. B <= W B. W <= B <= fs - W C. B >= fs D. B = 2 fs Answer: B. B must pass the whole baseband copy (>= W) but reject the first image starting at fs - W. (Key point: Reconstruction filter) 3. The time-domain reconstruction formula replaces each sample by: A. A rectangular pulse B. A sinc pulse C. An impulse D. A triangle Answer: B. g(t) is the sum of g(nTs) times sinc terms, since the ideal LPF has a sinc impulse response. (Key point: Interpolation formula) 4. (1/2pi) cos(400 pi t) cos(1000 pi t) has Nyquist rate: A. 1000 Hz B. 1400 Hz C. 2800 Hz D. 600 Hz Answer: B. The product gives tones at 300 and 700 Hz. fmax = 700 Hz, so the rate is 1400 Hz and T = 0.714 ms. (Key point: Nyquist rate of product) ## Related in Communication Systems - [Natural and flat top sampling; Nyquist rate of sin²(400πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anatural-flat-top-sin2-400pit) — 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× - [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 - [Aperture and aliasing; sampling 10cos(20πt)cos(200πt) at 250 samples/s](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-aliasing-10cos20pit-250) — Communication Systems, chapter 4 - [Reconstruction from samples taken at Nyquist rate](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Areconstruction-nyquist-samples) — Communication Systems, chapter 4 - [TDM of 24 voice signals: PAM bandwidth and PCM sampling rate](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Atdm-24-voice-pam-pcm) — Communication Systems, chapter 5 ## 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× - [Prove sampling theorem; Nyquist of product](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-nyquist) — Signals and Systems, chapter 4, asked 2× - [Nyquist rate of (1/2π)cos4000πt·cos1000πt](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-product) — Signals and Systems, chapter 4, asked 2×