---
title: "Octal-to-binary encoder; A+B'C canonical form"
subject: "Digital Logic"
chapter: 4
key: "semester_3/digital_logic:octal-binary-encoder-canonical"
source: https://www.bctengineer.com/q/semester_3/digital_logic/octal-binary-encoder-canonical
site: BCT Engineering
---

# Octal-to-binary encoder; A+B'C canonical form

Digital Logic · Chapter 4: Data Processing Circuits

- Asked in: 2075 Asoj, 3+3 marks
- Video: [Lec-7: Canonical Sum of Product (SOP) with example](https://www.youtube.com/watch?v=DgsvuhG7ShI) — Gate Smashers
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/octal-binary-encoder-canonical
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-4

## Question

Explain the operation of octal to binary encoder with necessary diagrams. Convert A+B'C in to canonical form.

## Short answer

Octal-to-binary encoder has outputs Y2 = D4+D5+D6+D7, Y1 = D2+D3+D6+D7, Y0 = D1+D3+D5+D7, assuming only one input active. Canonical SOP of A+B'C is Σm(1,4,5,6,7); POS is ΠM(0,2,3) = (A+B+C)(A+B'+C)(A+B'+C').

## Answer

#### Octal-to-binary encoder

An **octal-to-binary (8-to-3) encoder** has eight inputs $D_0$–$D_7$, one for each octal digit, and three outputs $Y_2Y_1Y_0$ giving the 3-bit binary code of the active input. Only one input is assumed to be active at a time.

| Active input | $Y_2$ | $Y_1$ | $Y_0$ |
|---|---|---|---|
| $D_0$ | 0 | 0 | 0 |
| $D_1$ | 0 | 0 | 1 |
| $D_2$ | 0 | 1 | 0 |
| $D_3$ | 0 | 1 | 1 |
| $D_4$ | 1 | 0 | 0 |
| $D_5$ | 1 | 0 | 1 |
| $D_6$ | 1 | 1 | 0 |
| $D_7$ | 1 | 1 | 1 |

Each output is 1 for the inputs whose code has a 1 in that position:

$$
Y_2 = D_4 + D_5 + D_6 + D_7,\quad Y_1 = D_2 + D_3 + D_6 + D_7,\quad Y_0 = D_1 + D_3 + D_5 + D_7
$$

```text
 D4 D5 D6 D7 --[OR 4]--> Y2
 D2 D3 D6 D7 --[OR 4]--> Y1
 D1 D3 D5 D7 --[OR 4]--> Y0
 (D0 is not connected: it gives 000)
```

Limitations: $D_0$ active and "no input active" both give 000, and two active inputs give a wrong code (e.g. $D_3 + D_5$ → 111). A priority encoder solves these.

#### Canonical form of $A + B'C$

Expand each term with the missing variables using $X + X' = 1$:

$$
\begin{aligned} A &= A(B + B')(C + C') = ABC + ABC' + AB'C + AB'C' \\ B'C &= B'C(A + A') = AB'C + A'B'C \end{aligned}
$$

Combine and remove the repeated term $AB'C$:

$$
A + B'C = A'B'C + AB'C' + AB'C + ABC' + ABC
$$

$$
= \Sigma m(1, 4, 5, 6, 7)
$$

The canonical POS form uses the remaining combinations:

$$
A + B'C = \Pi M(0, 2, 3) = (A + B + C)(A + B' + C)(A + B' + C')
$$

## Check yourself

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

1. Which equation gives output Y0 of the octal-to-binary encoder?
   A. D4 + D5 + D6 + D7
   B. D2 + D3 + D6 + D7
   C. D1 + D3 + D5 + D7
   D. D0 + D2 + D4 + D6
   Answer: C. Y0 is the LSB, which is 1 for odd codes 1, 3, 5, 7. (Key point: Encoder equations)

2. Which is a limitation of the simple octal-to-binary encoder?
   A. It cannot encode D7
   B. Two active inputs give a wrong code and D0 equals no input (both 000)
   C. It needs a clock
   D. It outputs 4 bits
   Answer: B. It assumes one active input; D0 and no input both give 000, and simultaneous inputs mix codes (D3 + D5 gives 111). (Key point: Encoder limitation)

3. What is A + B'C in canonical sum-of-minterms form?
   A. Σm(1,4,5,6,7)
   B. Σm(0,2,3)
   C. Σm(1,4,5)
   D. Σm(4,5,6,7)
   Answer: A. A gives 4,5,6,7 and B'C gives 1 and 5; the union is 1,4,5,6,7. (Key point: Canonical SOP)

4. What is the canonical POS form of A + B'C?
   A. ΠM(1,4,5,6,7)
   B. ΠM(0,2,3)
   C. ΠM(4,5,6)
   D. ΠM(0,1,2)
   Answer: B. The maxterms are the minterm numbers missing from the SOP: 0, 2, 3. (Key point: Canonical POS)

## Related in Digital Logic

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- [Encoder and octal-to-binary (8-to-3) encoder](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aoctal-to-binary-encoder) — Digital Logic, chapter 4, asked 2×
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- [SOP/POS forms and canonical form of A+BC+ABC](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Asop-pos-canonical-form) — Digital Logic, chapter 3
- [Canonical form of A(B'+C)+BD; minterm/maxterm](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Acanonical-form-minterm-maxterm) — Digital Logic, chapter 3

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