--- title: "Nyquist zero-ISI criteria; one pulse shaping method" subject: "Communication Systems" chapter: 6 key: "semester_6/communication_systems:nyquist-pulse-shaping-method" source: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-pulse-shaping-method site: BCT Engineering --- # Nyquist zero-ISI criteria; one pulse shaping method Communication Systems · Chapter 6: Baseband Digital Data Transmission - Asked in: 2070 Asar (CS II), 6 marks - Video: [DC#52 nyquist criterion for zero isi || EC Academy](https://www.youtube.com/watch?v=lsIT1Ngzrow) — EC Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-pulse-shaping-method - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6 ## Question State Nyquist pulse shaping criteria for Zero ISI. Discuss any one pulse shaping method of ISI reduction. ## Short answer Nyquist criterion: overall pulse p(nTb)=1 for n=0 and 0 otherwise, achieved if folded spectrum is constant. The raised cosine pulse satisfies this with roll-off factor α, bandwidth B_T = (Rb/2)(1+α), and provides fast tail decay, reducing ISI even with timing jitter. ## Answer #### Nyquist criterion for zero ISI For an overall pulse $p(t)$ sampled every $T_b$ seconds, there is no ISI if $$ p(nT_b) = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases} \quad\Longleftrightarrow\quad \sum_{n=-\infty}^{\infty} P\left(f - \frac{n}{T_b}\right) = T_b $$ The ideal solution is the rectangular spectrum of bandwidth $B_0 = R_b/2$, $p(t) = \text{sinc}(t/T_b)$, but it is not realizable and its slowly decaying tails make it very sensitive to timing errors. #### Raised cosine pulse shaping The raised cosine spectrum keeps the zero-ISI property but rolls off gradually: $$ P(f) = \begin{cases} \dfrac{1}{2B_0}, & 0 \le |f| < f_1 \\[2mm] \dfrac{1}{4B_0}\left[1 + \cos\dfrac{\pi(|f| - f_1)}{2B_0 - 2f_1}\right], & f_1 \le |f| < 2B_0 - f_1 \\[2mm] 0, & |f| \ge 2B_0 - f_1 \end{cases} $$ with $B_0 = \frac{R_b}{2}$, **roll-off factor** $\alpha = 1 - \frac{f_1}{B_0}$ ($0 \le \alpha \le 1$). **Transmission bandwidth:** $$ B_T = B_0(1 + \alpha) = \frac{R_b}{2}(1+\alpha) $$ **Pulse:** $$ p(t) = \text{sinc}(2B_0t)\,\frac{\cos(2\pi\alpha B_0 t)}{1 - 16\alpha^2B_0^2t^2} $$ The sinc factor keeps zeros at $t = nT_b$ (zero ISI); the second factor makes the tails decay as $1/|t|^3$. ```text P(f) |---------. alpha = 0 |----. '. . alpha = 0.5 | '. '. '. alpha = 1 | '. '. '. +----------+-----+----+--> f 0 Rb/2 3Rb/4 Rb ``` | $\alpha$ | Bandwidth | Remarks | |---|---|---| | 0 | $R_b/2$ | Ideal Nyquist, unrealizable | | 0.5 | $0.75R_b$ | Common practical choice | | 1 | $R_b$ | Full cosine roll-off, easy filters, tails decay fastest | **Example:** $R_b = 10$ kbps, $\alpha = 0.5$ gives $B_T = 5000 \times 1.5 = 7.5$ kHz. Advantages: realizable filters, small ISI under timing jitter, wide eye opening. Cost: extra bandwidth $\alpha B_0$. The filter is often split equally as root raised cosine filters at transmitter and receiver. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the ideal rectangular spectrum (sinc pulse) not used in practice? A. It needs too much bandwidth B. It is not realizable and its slowly decaying tails make it sensitive to timing errors C. It causes ISI D. It has a DC component Answer: B. The brick-wall filter cannot be built and sinc tails decay slowly as 1/t. (Key point: Ideal Nyquist limitation) 2. What is the transmission bandwidth of a raised cosine pulse with roll-off alpha? A. Rb/2 B. (Rb/2)(1 + alpha) C. Rb(1 + alpha) D. 2 Rb alpha Answer: B. B_T = B0(1 + alpha) with B0 = Rb/2. (Key point: Raised cosine bandwidth) 3. For Rb = 10 kbps and alpha = 0.5, what is B_T? A. 5 kHz B. 7.5 kHz C. 10 kHz D. 15 kHz Answer: B. 5000 x 1.5 = 7.5 kHz. (Key point: Example calculation) 4. What does raised cosine shaping give in return for extra bandwidth? A. Tails decaying faster (1/|t|^3), realizable filters B. A narrower bandwidth than Rb/2 C. More ISI D. No zero crossings Answer: A. Zero ISI is kept (sinc zeros), tails decay faster, but bandwidth increases by alpha B0. (Key point: Tail decay and cost) ## Related in Communication Systems - [ISI and Nyquist pulse shaping criterion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-nyquist-pulse-shaping) — Communication Systems, chapter 6 - [Nyquist criteria for zero ISI; duobinary difficulties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-zero-isi-duobinary-difficulties) — Communication Systems, chapter 6 - [ISI; Nyquist criteria; duobinary with example](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-nyquist-duobinary-example) — Communication Systems, chapter 6, asked 2× - [ISI, Nyquist zero-ISI conditions and practical constraints](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-nyquist-zero-isi-constraints) — Communication Systems, chapter 6 - [ISI; solutions for zero ISI; duobinary with precoder](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-zero-isi-duobinary-precoder) — Communication Systems, chapter 6 ## Related in other subjects - [Pulse shaping circuit and output waveform](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Apulse-shaping-circuit) — Basic Electronics Engineering, chapter 2 - [Nyquist criterion from given pole-zero and Nyquist plot](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-criteria-from-plot) — Control Systems, chapter 6 - [Nyquist criteria, aliasing; 30sin100πt + …](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-criteria-aliasing-example) — Signals and Systems, chapter 4, asked 3×