--- title: "ISI; Nyquist criteria; duobinary with example" subject: "Communication Systems" chapter: 6 key: "semester_6/communication_systems:isi-nyquist-duobinary-example" source: https://www.bctengineer.com/q/semester_6/communication_systems/isi-nyquist-duobinary-example site: BCT Engineering --- # ISI; Nyquist criteria; duobinary with example Communication Systems · Chapter 6: Baseband Digital Data Transmission - Asked in: 2072 Kartik (CS II), 2+2+4 marks; 2071 Shrawan (CS II), 2+2+6 marks (asked 2 times) - 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-duobinary-example - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6 ## Question What is ISI? State Nyquist pulse shaping criteria for zero ISI. Explain duo-binary encoding with example. ## Short answer Intersymbol interference (ISI) is pulse overlap distorting samples. Nyquist criterion for zero ISI: p(nTb)=1 if n=0 else 0; spectrum copies sum to constant. Duobinary adds controlled ISI via c_k = a_k + a_{k-1}. Precoding d_k = b_k XOR d_{k-1} prevents error propagation; decode |c_k|=0→1, else→0. Example: input 1011001 yields output 1011001. ## Answer #### ISI **Intersymbol interference (ISI)** is the overlap of a pulse with the neighbouring symbol intervals. A band-limited channel (or filter) spreads each pulse in time, so the tails of nearby pulses add to the sample of the current symbol: $$ y(iT_b) = \mu a_i + \mu \sum_{k \ne i} a_k\, p\big((i-k)T_b\big) + n(iT_b) $$ The summation term is the ISI. It closes the eye diagram and raises the bit error rate. #### 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 Nyquist pulse:** $p(t) = \text{sinc}(R_b t)$ with bandwidth $B_0 = R_b/2$ (the minimum possible). It cannot be built, and its slow tails are sensitive to timing error. - **Raised-cosine pulse:** bandwidth $B_T = \frac{R_b}{2}(1+\alpha)$, $0 \le \alpha \le 1$. It still gives zero ISI, its tails decay fast, and it can be built. #### Duobinary encoding **Duobinary signalling** (a correlative-level coding scheme) adds each binary symbol to the one before it. This brings in a **controlled amount of ISI** that the receiver knows and can remove, and in return the data can be sent at $R_b$ b/s in the minimum Nyquist bandwidth $R_b/2$ with realisable filters. **Encoder:** with $a_k = \pm 1$, $$ c_k = a_k + a_{k-1}, \qquad c_k \in \{-2, 0, +2\} $$ This is equal to passing $a_k$ through $H(f) = 1 + e^{-j2\pi f T_b}$. With the ideal Nyquist filter, the overall response becomes $$ H_I(f) = 2\cos(\pi f T_b)\, e^{-j\pi f T_b}, \quad |f| \le \frac{1}{2T_b} $$ This is a smooth half-cosine, which is easy to approximate with a real filter. ```text b_k->[Precoder]->[2d-1]-a_k-+---->(+)-c_k->[LPF]-> out d_k=b_k XOR d_k-1 | ^ +-[Tb]-+ ``` **Decoder (without precoding):** $\hat a_k = c_k - \hat a_{k-1}$. One wrong decision then spreads to the bits that follow (error propagation). **Precoding** removes this. Use $d_k = b_k \oplus d_{k-1}$ and send $a_k = 2d_k - 1$. Then the receiver decides each bit on its own: $|c_k| = 0 \Rightarrow b_k = 1$, and $|c_k| = 2 \Rightarrow b_k = 0$. **Example:** input $b_k$ = 1011001, with initial reference $d_{-1} = 0$ (so $a_{-1} = -1$). | $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---|---| | $b_k$ | 1 | 0 | 1 | 1 | 0 | 0 | 1 | | $d_k = b_k \oplus d_{k-1}$ | 1 | 1 | 0 | 1 | 1 | 1 | 0 | | $a_k$ | +1 | +1 | −1 | +1 | +1 | +1 | −1 | | $c_k = a_k + a_{k-1}$ | 0 | +2 | 0 | 0 | +2 | +2 | 0 | | Decision ($c_k = 0 \to 1$, else 0) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | The decoded sequence 1011001 equals the input. Each bit is decided on its own, so there is no error propagation. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the frequency domain, what must the sum of shifted pulse spectra satisfy for zero ISI? A. Sum of P(f - k/Tb) = Tb (constant) B. Sum of P(f - k/Tb) = 0 C. P(f) = 1 only at f = 0 D. P(f) must be infinite Answer: A. Copies of P(f) shifted by multiples of Rb must add to a flat spectrum equal to Tb. (Key point: Nyquist frequency criterion) 2. How does a duobinary encoder form its output with a_k = +/-1? A. c_k = a_k - a_(k-1) B. c_k = a_k x a_(k-1) C. c_k = a_k + a_(k-1), giving levels -2, 0, +2 D. c_k = a_k only Answer: C. Adding the current and previous symbol introduces controlled ISI with three levels. (Key point: Duobinary encoding) 3. What is the main advantage of duobinary signalling? A. It removes all ISI and needs more bandwidth B. It allows Rb to be sent in the minimum Nyquist bandwidth Rb/2 with realisable filters C. It doubles the noise margin D. It needs no filter Answer: B. The overall response is a smooth half-cosine, easy to approximate, while the controlled ISI is known and removed at the receiver. (Key point: Benefit of duobinary) 4. With precoding, how is the bit decided at the receiver? A. |c_k| = 0 means b_k = 1; |c_k| = 2 means b_k = 0 B. |c_k| = 0 means b_k = 0; |c_k| = 2 means b_k = 1 C. By subtracting the previous decision D. By comparing with the next bit Answer: A. Precoding d_k = b_k XOR d_(k-1) lets each bit be decided alone, which prevents error propagation. (Key point: Precoding decision) ## 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 - [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 and Nyquist pulse shaping criterion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-nyquist-pulse-shaping) — 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 - [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; modified duobinary for 10110011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-modified-duobinary-10110011) — Communication Systems, chapter 6