---
title: "ISI, Nyquist zero-ISI conditions and practical constraints"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:isi-nyquist-zero-isi-constraints"
source: https://www.bctengineer.com/q/semester_6/communication_systems/isi-nyquist-zero-isi-constraints
site: BCT Engineering
---

# ISI, Nyquist zero-ISI conditions and practical constraints

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2080 Bhadra (CS II), 2+2+4 marks
- Video: [How to Avoid ISI in Digital Communications: Nyquist Zero ISI Theorem](https://www.youtube.com/watch?v=sgyTlI9BsKc) — Iain Explains Signals, Systems, and Digital Comms
- 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-zero-isi-constraints
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

Define Inter Symbol Interference (ISI). State Nyquist conditions for zero ISI. What are the physical constraints in implementing Nyquist pulse shaping criteria?

## Short answer

ISI occurs when pulse tails invade other symbol intervals. Nyquist zero-ISI: p(nTb)=1 for n=0 else 0; ∑ P(f - n/Tb) = Tb. Physical constraints: ideal brick-wall filter unrealizable, sinc pulse decays only as 1/|t| making timing jitter severe, exact channel unknown, requiring excess bandwidth with raised cosine.

## Answer

#### Inter Symbol Interference (ISI)

**ISI** is the distortion in which the pulse of one symbol spreads into the time slots of neighbouring symbols, so that the sample taken at the decision instant contains contributions from earlier and later symbols. It is caused by the limited bandwidth and dispersion of the channel and filters. The received sample at $t = kT_b$ is

$$
y(kT_b) = \mu a_k + \underbrace{\mu\sum_{n\ne k} a_n\, p\big((k-n)T_b\big)}_{\text{ISI}} + n(kT_b)
$$

ISI raises the bit error rate even when there is no noise, and it closes the eye in the eye pattern.

#### Nyquist conditions for zero ISI

Let $p(t)$ be the overall pulse (transmit filter, channel and receive filter) with $p(0)=1$.

- **Time domain:** $p(nT_b) = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}$, i.e. the pulse must pass through zero at all other sampling instants.
- **Frequency domain:** $\displaystyle\sum_{n=-\infty}^{\infty} P\left(f - \frac{n}{T_b}\right) = T_b$ (constant), i.e. the shifted copies of the spectrum must add to a flat value.

The simplest solution is the ideal Nyquist channel $P(f) = T_b\,\text{rect}(fT_b)$, bandwidth $B_0 = \frac{R_b}{2}$, giving $p(t) = \text{sinc}(t/T_b)$.

#### Physical constraints in implementing the Nyquist criterion

1. **Unrealizable filter:** the ideal response needs a flat spectrum up to $R_b/2$ and an abrupt fall to zero (brick-wall). Such a filter is non-causal and cannot be built.
2. **Infinite, slowly decaying pulse:** the sinc pulse lasts from $-\infty$ to $\infty$ and its tails decay only as $1/|t|$. Any truncation brings back ISI.
3. **Timing sensitivity:** because the tails are large, a small timing error (jitter) in the sampling instant gives large ISI; the sum of ISI terms can even diverge.
4. **Channel uncertainty:** the criterion applies to the overall response, but the real channel response is not known exactly and may change with time, so fixed filters cannot meet it exactly.
5. **Bandwidth penalty in practice:** practical solutions such as the raised cosine pulse need extra bandwidth, $B = \frac{R_b}{2}(1+\alpha)$ with roll-off $0 \le \alpha \le 1$, and equalizers or partial response (duobinary) coding are needed to handle the remaining problems.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The Nyquist frequency-domain condition requires:
   A. Shifted spectra copies sum to a constant Tb
   B. P(f) is zero
   C. P(f) is rectangular only
   D. Spectrum is narrower than Rb/2
   Answer: A. Sum over n of P(f - n/Tb) = Tb. (Key point: Frequency criterion)

2. Why is the ideal Nyquist channel not realizable?
   A. It is nonlinear
   B. It needs a brick-wall, non-causal filter
   C. It needs too little bandwidth
   D. It has no tails
   Answer: B. A flat response with abrupt cutoff and an infinite sinc pulse cannot be built. (Key point: Unrealizable)

3. Why is timing jitter harmful with the sinc pulse?
   A. It has no zeros
   B. It raises power
   C. Tails decay only as 1/|t|, so ISI terms add to large values
   D. Pulses are too short
   Answer: C. Slowly decaying tails make sampling-time errors cause large ISI. (Key point: Timing sensitivity)

4. What do practical solutions such as raised cosine require?
   A. Infinite tails
   B. Extra bandwidth Rb/2 (1+alpha)
   C. Less bandwidth
   D. No filters
   Answer: B. Raised cosine with roll-off alpha uses B = Rb/2(1+alpha), plus equalizers for remaining ISI. (Key point: Practical solution)

## 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 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 in baseband systems; ideal and practical solutions](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-ideal-practical-solutions) — Communication Systems, chapter 6, asked 2×