---
title: "Reducing ISI; modified duobinary with precoder"
subject: "Communication Systems"
chapter: 6
key: "semester_6/communication_systems:isi-reduce-modified-duobinary"
source: https://www.bctengineer.com/q/semester_6/communication_systems/isi-reduce-modified-duobinary
site: BCT Engineering
---

# Reducing ISI; modified duobinary with precoder

Communication Systems · Chapter 6: Baseband Digital Data Transmission

- Asked in: 2079 Bhadra (CS II), 4+6 marks
- Video: [Problem on duo binary system with and without precoder#JNTUA June/July 2019](https://www.youtube.com/watch?v=8mWuq8QiCZU) — PRAKASAM TUTORIALS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/isi-reduce-modified-duobinary
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-6

## Question

Illustrate the methods to reduce Intersymbol Interference. Explain modified Duobinary coding technique with precoder, illustrating with input sequence 01001101.

## Short answer

ISI reduction methods include Nyquist pulse shaping (raised cosine), correlative coding (duobinary, modified duobinary), equalization, eye pattern monitoring, and reducing signaling rate. Modified duobinary uses c_k = a_k - a_{k-2} with precoder d_k = b_k ⊕ d_{k-2}, yielding three-level output (-2,0,+2) and no DC component. For input 01001101, the decoded sequence matches the input exactly.

## Answer

#### Methods to reduce Intersymbol Interference

1. **Nyquist pulse shaping (raised cosine filtering):** shape the overall response so that each pulse has zero value at all other sampling instants. The raised cosine spectrum with roll-off $\alpha$ meets the Nyquist criterion, needs bandwidth $B = \frac{R_b}{2}(1+\alpha)$, and its tails decay as $1/t^3$, so timing errors cause little ISI.
2. **Correlative (partial response) coding:** add a controlled, known amount of ISI (duobinary, modified duobinary) so that the signal fits in the minimum bandwidth $R_b/2$ with realizable filters; the receiver removes the known ISI.
3. **Equalization:** a transversal (tapped delay line) filter after the channel cancels the channel's distortion. Zero-forcing equalizers force ISI to zero at the sampling points; adaptive (LMS) equalizers track changing channels.
4. **Eye pattern monitoring and correct sampling:** sample at the instant of maximum eye opening using good timing recovery.
5. **Reducing the signalling rate** or using more channel bandwidth, so pulses spread less.

#### Modified duobinary coding

Modified duobinary is a correlative coding scheme that correlates the present symbol with the symbol **two** bit intervals earlier:

$$
c_k = a_k - a_{k-2}, \qquad a_k = \pm 1
$$

Transfer function:

$$
H(f) = 1 - e^{-j4\pi fT_b} = 2j\sin(2\pi fT_b)\,e^{-j2\pi fT_b}, \quad |f| \le \frac{1}{2T_b}
$$

Impulse response: $h(t) = \text{sinc}\left(\frac{t}{T_b}\right) - \text{sinc}\left(\frac{t-2T_b}{T_b}\right)$.

$|H(f)|$ is zero at $f = 0$ and at $f = 1/(2T_b)$, so the signal has **no DC component** (useful for transformer-coupled lines and SSB) and is easy to filter. The output has three levels: $-2, 0, +2$.

**Precoder:** to avoid error propagation, the data are precoded as $d_k = b_k \oplus d_{k-2}$, then $a_k = +1$ if $d_k = 1$ and $-1$ if $d_k = 0$. Decision rule: $|c_k| > 1 \Rightarrow \hat b_k = 1$; $|c_k| < 1 \Rightarrow \hat b_k = 0$.

```text
b_k -->(XOR)---+--> level map --> a_k --+------------->(+)--> c_k
         ^     | d_k                    |               ^ (-)
         |     v                        |               |
         +--[2Tb delay]                 +--[2Tb delay]--+
```

#### Illustration for input 01001101

Initial precoder bits assumed $d_{-1} = d_0 = 0$ (so $a_{-1} = a_0 = -1$).

| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| $b_k$ | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| $d_{k-2}$ | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| $d_k = b_k \oplus d_{k-2}$ | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| $a_k$ | −1 | +1 | −1 | +1 | +1 | −1 | +1 | +1 |
| $a_{k-2}$ | −1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 |
| $c_k = a_k - a_{k-2}$ | 0 | +2 | 0 | 0 | +2 | −2 | 0 | +2 |
| Decoded $\hat b_k$ | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |

The decoded sequence **01001101** equals the input. Each bit is decided from its own sample, so a single error does not propagate.

## Check yourself

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

1. How does a raised cosine pulse reduce ISI?
   A. It makes each pulse zero at the sampling instants of all other symbols
   B. It increases the signalling rate
   C. It removes the need for a channel
   D. It adds noise to hide ISI
   Answer: A. Raised cosine meets the Nyquist criterion: zero value at other sampling instants, with tails decaying as 1/t^3. (Key point: Nyquist pulse shaping)

2. In modified duobinary coding, c_k equals:
   A. a_k + a_(k-1)
   B. a_k - a_(k-1)
   C. a_k - a_(k-2)
   D. a_k + a_(k-2)
   Answer: C. It correlates the present symbol with the one two intervals earlier, c_k = a_k - a_(k-2). (Key point: Modified duobinary rule)

3. Why does modified duobinary have no DC component?
   A. Its output has only two levels
   B. H(f) is proportional to sin(2 pi f Tb), which is zero at f = 0
   C. It uses a precoder
   D. It uses RZ pulses
   Answer: B. |H(f)| = 0 at f = 0 (and at 1/(2Tb)), so there is no DC, suiting transformer-coupled lines. (Key point: No DC component)

4. With the precoder d_k = b_k XOR d_(k-2), how is bit b_k decided?
   A. b = 1 if |c_k| > 1, b = 0 if |c_k| < 1
   B. b = 1 if c_k = 0, b = 0 if |c_k| = 2
   C. b = c_k - previous b
   D. b = 1 only if c_k = +2
   Answer: A. |c_k| = 2 means b = 1 and c_k = 0 means b = 0, so each bit is decided from its own sample with no error propagation. (Key point: Precoder and decision)

## Related in Communication Systems

- [ISI; duobinary with precoder for 0010110](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aisi-duobinary-precoder-0010110) — 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
- [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 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×
- [Correlative coding; minimizing error in duobinary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acorrelative-coding-duobinary-error) — 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

## Related in other subjects

- [Line coding in PCM and the right technique](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aline-coding-pcm-solution) — Telecommunications, chapter 3