--- title: "ISI in baseband systems; ideal and practical solutions" subject: "Communication Systems" chapter: 6 key: "semester_6/communication_systems:isi-ideal-practical-solutions" source: https://www.bctengineer.com/q/semester_6/communication_systems/isi-ideal-practical-solutions site: BCT Engineering --- # ISI in baseband systems; ideal and practical solutions Communication Systems · Chapter 6: Baseband Digital Data Transmission - Asked in: 2074 Asoj (CS II), 4+3+3 marks; 2072 Chaitra (CS II), 2+6 marks (asked 2 times) - Video: [LECT-41: Nyquist solution & Raised Cosine roll off filter for Zero ISI.](https://www.youtube.com/watch?v=gqFs_GbXYFM) — EPOV CHANNEL - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/isi-ideal-practical-solutions - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6 ## Question Explain intersymbol interference (ISI) in baseband digital communication system with derivations. Also explain the ideal and practical solutions of ISI. ## Short answer ISI is the overlap of pulse tails causing decision errors; zero ISI demands p(nTb)=1 for n=0 and 0 otherwise. The ideal sinc pulse meets this at minimum bandwidth but is impractical. The raised-cosine filter provides zero ISI with a practical roll-off, trading extra bandwidth for timing robustness. ## Answer #### Intersymbol interference (ISI) **ISI** is the spreading of each transmitted pulse beyond its own symbol interval because the channel's bandwidth is limited. Tails of earlier and later pulses then add to the sample of the current symbol, which can cause decision errors. **Derivation.** The baseband PAM signal at the receiver filter output is $$ y(t) = \mu \sum_{k} a_k \, p(t - kT_b) + n(t) $$ where $a_k$ are the symbols, $p(t)$ is the overall pulse shape (normalised so $p(0) = 1$) and $n(t)$ is noise. Sampling at $t_i = iT_b$: $$ y(t_i) = \underbrace{\mu a_i}_{\text{wanted}} + \underbrace{\mu \sum_{k \ne i} a_k \, p\big((i-k)T_b\big)}_{\text{ISI}} + n(t_i) $$ The middle term is the ISI. It is zero only when $p(nT_b) = 0$ for all $n \ne 0$. ```text pulse k-1 pulse k pulse k+1 /\ /\ /\ ___/ \___...___/ \___...___/ \___ tails overlap at sampling instants -> ISI ``` #### 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. #### Ideal solution: Nyquist (sinc) pulse The ideal solution uses an ideal rectangular spectrum of bandwidth $B_0 = R_b/2$: $$ P(f) = \frac{1}{2B_0}\,\text{rect}\!\left(\frac{f}{2B_0}\right) \;\Rightarrow\; p(t) = \text{sinc}(2B_0 t) = \frac{\sin 2\pi B_0 t}{2\pi B_0 t} $$ This pulse is zero at every $t = nT_b$ ($n \ne 0$), so ISI is zero. It needs the minimum bandwidth $R_b/2$ (the Nyquist bandwidth). It is not practical because: - the brick-wall filter cannot be built, and - the sinc tails decay only as $1/t$, so a small timing error causes large ISI. #### Practical solution: raised-cosine spectrum The spectrum is given a gradual cosine roll-off from $f_1$ to $2B_0 - f_1$, with roll-off factor $\alpha = 1 - f_1/B_0$ ($0 \le \alpha \le 1$): $$ P(f) = \begin{cases} \dfrac{1}{2B_0}, & |f| < f_1 \\[4pt] \dfrac{1}{4B_0}\left[1 + \cos\dfrac{\pi(|f| - f_1)}{2B_0 - 2f_1}\right], & f_1 \le |f| < 2B_0 - f_1 \\[4pt] 0, & |f| \ge 2B_0 - f_1 \end{cases} $$ $$ p(t) = \text{sinc}(2B_0 t)\,\frac{\cos 2\pi\alpha B_0 t}{1 - 16\alpha^2 B_0^2 t^2}, \qquad B_T = B_0(1 + \alpha) = \frac{R_b}{2}(1+\alpha) $$ - It still has zeros at all $nT_b$, so ISI is zero. - The tails decay as $1/t^3$, so it tolerates timing error well. - The filter can be built. The cost is extra bandwidth: $\alpha = 0.5$ needs $0.75R_b$, and $\alpha = 1$ needs $R_b$. Other practical remedies are **equalisers** (e.g. a tapped-delay-line zero-forcing equaliser) to correct channel distortion, and **correlative (duobinary) coding**, which allows controlled ISI at the Nyquist bandwidth. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What causes ISI in a baseband system? A. Limited channel bandwidth spreads pulses so tails overlap neighbouring symbols B. Too high a signal power C. Using binary instead of M-ary signalling D. Absence of noise Answer: A. Pulse tails from earlier and later symbols add to the sample of the current symbol. (Key point: Cause of ISI) 2. What is the Nyquist condition for zero ISI in the time domain? A. p(nTb) = 1 for all n B. p(nTb) = 1 for n = 0 and 0 for n not equal to 0 C. p(t) = 0 for all t D. p(nTb) = 0 for n = 0 only Answer: B. The overall pulse must be zero at all other sampling instants so the ISI summation vanishes. (Key point: Zero-ISI condition) 3. Why is the ideal sinc pulse with bandwidth Rb/2 not practical? A. It has ISI B. It needs more than Rb bandwidth C. It needs a brick-wall filter and its 1/t tails make timing errors cause large ISI D. It is not zero at nTb Answer: C. The sinc is zero at nTb with minimum bandwidth, but the filter is unrealisable and slow-decaying tails are timing sensitive. (Key point: Ideal solution drawbacks) 4. For a raised-cosine pulse with roll-off factor alpha, what is the bandwidth? A. Rb(1 + alpha) B. (Rb/2)(1 + alpha) C. Rb/2 for any alpha D. 2 Rb alpha Answer: B. BT = B0(1 + alpha) = (Rb/2)(1 + alpha); tails decay as 1/t^3 and it is realisable. Alpha = 1 gives Rb. (Key point: Raised cosine) ## Related in Communication Systems - [Short note: M-ary baseband communication system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-mary-baseband) — Communication Systems, chapter 6, asked 3× - [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 - [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 and two practical methods to minimize it](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-practical-methods) — Communication Systems, chapter 6 - [Duplex digital communication system vs analog](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adigital-comm-duplex-vs-analog) — Communication Systems, chapter 1 - [Function of basic components of a digital communication system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adcs-basic-components) — Communication Systems, chapter 1 ## Related in other subjects - [Filter importance in communication; ideal vs practical](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Afilter-communication-ideal-practical) — Filter Design, chapter 1, asked 2× - [Analog and digital communication systems](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Aanalog-digital-comm-systems) — Instrumentation II, chapter 5 - [Block diagram of digital communication system](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ablock-diagram-digital-comm) — Data Communication, chapter 1, asked 8× - [Ideal vs practical filters and scaling](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aideal-practical-filters-scaling) — Filter Design, chapter 1, asked 2×