--- title: "ISI and Nyquist pulse shaping criterion" subject: "Communication Systems" chapter: 6 key: "semester_6/communication_systems:isi-nyquist-pulse-shaping" source: https://www.bctengineer.com/q/semester_6/communication_systems/isi-nyquist-pulse-shaping site: BCT Engineering --- # ISI and Nyquist pulse shaping criterion Communication Systems · Chapter 6: Baseband Digital Data Transmission - Asked in: 2081 Bhadra (CS II), 2+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/isi-nyquist-pulse-shaping - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6 ## Question Define Inter Symbol Interference. Elaborate Nyquist pulse shaping criterion for mitigation of ISI. ## Short answer Inter Symbol Interference (ISI) is distortion where a symbol's energy spreads into adjacent time slots. Nyquist's zero-ISI criterion requires the overall pulse p(t) to satisfy p(nTb)=1 for n=0 and 0 otherwise; equivalently, folded spectrum sum equals Tb. Practical pulse shaping uses raised cosine with roll-off α, bandwidth B = (Rb/2)(1+α), split as root-raised-cosine. ## Answer #### Inter symbol interference **Inter symbol interference (ISI)** is distortion in which the energy of one symbol spreads into the time slots of nearby symbols. It happens because the channel and filters have limited bandwidth, so pulses get wider in time. At a sampling instant, the received sample then holds tails of other pulses as well as the wanted symbol: $$ y(iT_b) = \mu a_i + \underbrace{\mu \sum_{k \ne i} a_k\, p\big((i-k)T_b\big)}_{\text{ISI}} + n(iT_b) $$ #### Nyquist criterion for zero ISI If $p(t)$ is the overall pulse (transmit filter, channel and receive filter), ISI is zero when $$ p(nT_b) = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases} $$ In the frequency domain this is $$ \sum_{k=-\infty}^{\infty} P\!\left(f - \frac{k}{T_b}\right) = T_b \quad \text{(constant)} $$ That is, copies of the pulse spectrum shifted by multiples of $R_b = 1/T_b$ must add up to a flat spectrum. #### Nyquist pulse shaping for mitigation of ISI **1. Ideal Nyquist channel (minimum bandwidth).** Choose a rectangular spectrum of width $B_0 = R_b/2$: $$ P(f) = \frac{1}{R_b}\,\text{rect}\!\left(\frac{f}{R_b}\right) \;\Rightarrow\; p(t) = \frac{\sin \pi R_b t}{\pi R_b t} $$ The sinc pulse is 1 at $t = 0$ and 0 at every other $nT_b$, so ISI is zero. $B_0 = R_b/2$ is the **Nyquist bandwidth**, and $R_b = 2B_0$ is the **Nyquist rate**. In practice it fails because the brick-wall filter cannot be built and the tails decay only as $1/|t|$, so any timing jitter adds up to large ISI. **2. Raised-cosine pulse shaping (practical).** The spectrum rolls off smoothly with a cosine shape, with odd symmetry about $B_0$, so the folded spectrum is still flat: $$ p(t) = \text{sinc}(2B_0 t)\,\frac{\cos(2\pi\alpha B_0 t)}{1 - 16\alpha^2B_0^2t^2} $$ $$ B_T = B_0(1 + \alpha) = \frac{R_b}{2}(1 + \alpha) $$ ```text P(f) |______ | \ alpha = 0 : brick wall, B = Rb/2 | \ alpha = 0.5 : B = 0.75 Rb | \__ alpha = 1 : B = Rb +--------+--+---> f B0 B0(1+a) ``` | Roll-off $\alpha$ | Bandwidth $B_T$ | Tail decay | Practicality | |---|---|---|---| | 0 | $R_b/2$ | $1/t$ | Not realisable | | 0.5 | $0.75 R_b$ | $1/t^3$ | Commonly used | | 1 | $R_b$ | $1/t^3$, fastest | Easy; least timing-sensitive | The pulse shaping is usually split equally between transmitter and receiver as **root-raised-cosine** filters, which also gives matched filtering. Any ISI that remains because the channel is not ideal is removed with **equalisers**. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What causes ISI? A. Limited bandwidth making pulses spread into neighbouring slots B. Quantisation C. Too little signal power D. Excess bandwidth Answer: A. Channel and filters of limited bandwidth widen pulses so tails fall in other symbols' sampling instants. (Key point: ISI cause) 2. What does the Nyquist zero-ISI criterion require of p(t)? A. p(t) is rectangular B. p(0)=1 and p(nTb)=0 for n not 0 C. p(t) is positive everywhere D. p(nTb)=1 for all n Answer: B. The pulse must be 1 at its own sampling instant and zero at all other sampling instants. (Key point: Time-domain criterion) 3. What bandwidth does the ideal sinc (rectangular spectrum) pulse need? A. Rb B. Rb/2 C. 2Rb D. Rb/4 Answer: B. B0 = Rb/2 is the Nyquist bandwidth; it is not practical because of the brick-wall filter and 1/t tails. (Key point: Ideal Nyquist channel) 4. For raised cosine with roll-off alpha, the bandwidth is: A. Rb/2 (1 + alpha) B. Rb/2 alpha C. 2Rb alpha D. Rb (1 - alpha) Answer: A. B_T = B0(1 + alpha); alpha = 0.5 gives 0.75 Rb and alpha = 1 gives Rb. (Key point: Raised cosine) ## Related in Communication Systems - [Nyquist zero-ISI criteria; one pulse shaping method](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-pulse-shaping-method) — Communication Systems, chapter 6 - [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; 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× - [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; modified duobinary for 10110011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-modified-duobinary-10110011) — 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 - [Short note: Nyquist stability criterion](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-nyquist-criterion) — Control Systems, chapter 6 - [Sampling theorem, Nyquist criterion and aliasing](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Asampling-nyquist-aliasing) — Instrumentation I, chapter 6 - [ISI mitigation and M-branch RAKE receiver](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aisi-rake-m-branch) — Wireless Communication, chapter 5