---
title: "Hadamard matrix H8"
subject: "Wireless Communication"
chapter: 6
key: "semester_7/wireless_communication:hadamard-matrix-h8"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/hadamard-matrix-h8
site: BCT Engineering
---

# Hadamard matrix H8

Wireless Communication · Chapter 6: Speech and Channel Coding

- Asked in: 2080 Bhadra, 4 marks
- Video: [Hadamard Code ( Linear Block Code )  (हिन्दी )](https://www.youtube.com/watch?v=3JeatK8vAS4) — LEARN AND GROW
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/hadamard-matrix-h8
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-6

## Question

Construct Hadamard matrix for H8.

## Short answer

Using Sylvester construction, H8 = [H4 H4; H4 -H4] in ±1 form, with H4 = [1 1 1 1; 1 -1 1 -1; 1 1 -1 -1; 1 -1 -1 1]. Rows are mutually orthogonal, so H8 H8^T = 8I.

## Answer

A **Hadamard matrix** of order N is a square matrix whose rows are mutually orthogonal. It is constructed recursively by the Sylvester method:

H₂ₙ = [ Hₙ  Hₙ ; Hₙ  −Hₙ ] in ±1 form, or H₂ₙ = [ Hₙ  Hₙ ; Hₙ  Hₙ′ ] in 0/1 form, where Hₙ′ is the complement of Hₙ.

**Step 1:** H₁ = [+1]

**Step 2:**

```text
 H2 = | +1 +1 |
      | +1 -1 |
```

**Step 3:**

```text
 H4 = | H2  H2 | = | +1 +1 +1 +1 |
      | H2 -H2 |   | +1 -1 +1 -1 |
                   | +1 +1 -1 -1 |
                   | +1 -1 -1 +1 |
```

**Step 4:**

```text
 H8 = | H4  H4 |
      | H4 -H4 |

      | +1 +1 +1 +1 +1 +1 +1 +1 |
      | +1 -1 +1 -1 +1 -1 +1 -1 |
      | +1 +1 -1 -1 +1 +1 -1 -1 |
      | +1 -1 -1 +1 +1 -1 -1 +1 |
      | +1 +1 +1 +1 -1 -1 -1 -1 |
      | +1 -1 +1 -1 -1 +1 -1 +1 |
      | +1 +1 -1 -1 -1 -1 +1 +1 |
      | +1 -1 -1 +1 -1 +1 +1 -1 |
```

In binary form (+1 → 0, −1 → 1):

```text
         c1 c2 c3 c4 c5 c6 c7 c8
 W0:      0  0  0  0  0  0  0  0
 W1:      0  1  0  1  0  1  0  1
 W2:      0  0  1  1  0  0  1  1
 W3:      0  1  1  0  0  1  1  0
 W4:      0  0  0  0  1  1  1  1
 W5:      0  1  0  1  1  0  1  0
 W6:      0  0  1  1  1  1  0  0
 W7:      0  1  1  0  1  0  0  1
```

**Check:** the inner product of any two different rows is 0 (e.g. row 2 · row 3 = 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 = 0), and H8·H8ᵀ = 8·I. These 8 rows are the Walsh codes used for channelization in CDMA.

## Check yourself

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

1. In +1/-1 form, how is H2n built from Hn?
   A. [Hn Hn; -Hn -Hn]
   B. [Hn -Hn; Hn -Hn]
   C. [Hn Hn; Hn -Hn]
   D. [Hn 0; 0 Hn]
   Answer: C. H2n = [Hn Hn; Hn -Hn] (bottom-right block negated). (Key point: Sylvester recursion)

2. What does H8 times its transpose equal?
   A. I
   B. 8 I
   C. H8
   D. 0
   Answer: B. Rows are orthogonal and each has squared norm 8, so H H^T = 8 I. (Key point: H8 inner property)

3. In the binary form of H8, how are entries mapped?
   A. +1 to 1 and -1 to 0
   B. +1 to 0 and -1 to 1
   C. Both to 0
   D. +1 to 2
   Answer: B. +1 maps to 0 and -1 maps to 1. (Key point: Binary mapping)

4. What are the 8 rows of H8 used for in CDMA?
   A. Walsh codes for channelization
   B. Pitch codes
   C. Error-correcting parity bits
   D. Interleaver patterns
   Answer: A. Orthogonal Walsh codes separate channels in CDMA. (Key point: Use in CDMA)

## Related in Wireless Communication

- [Turbo coding and 8×8 Hadamard code](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aturbo-coding-hadamard) — Wireless Communication, chapter 6
- [Hadamard code conditions and 8×8 construction](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Ahadamard-code-conditions) — Wireless Communication, chapter 6

## Related in other subjects

- [Construct a (7,4) Hamming code](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahamming-74-construct) — Data Communication, chapter 8