--- title: "Turbo coding and 8×8 Hadamard code" subject: "Wireless Communication" chapter: 6 key: "semester_7/wireless_communication:turbo-coding-hadamard" source: https://www.bctengineer.com/q/semester_7/wireless_communication/turbo-coding-hadamard site: BCT Engineering --- # Turbo coding and 8×8 Hadamard code Wireless Communication · Chapter 6: Speech and Channel Coding - Asked in: 2082 Bhadra, 2+3 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/wireless_communication/turbo-coding-hadamard - Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-6 ## Question Define turbo coding. Construct 8 by 8 Hadamard code. ## Short answer Turbo coding uses two recursive systematic convolutional encoders in parallel with an interleaver and iterative soft-in soft-out decoding, performing within 0.5–1 dB of the Shannon limit. The 8x8 Hadamard matrix by Sylvester construction has binary rows 00000000, 01010101, 00110011, 01100110, 00001111, 01011010, 00111100, 01101001. ## Answer #### Turbo coding **Turbo coding** is a powerful forward error-correction scheme that uses **two (or more) recursive systematic convolutional (RSC) encoders in parallel**, separated by an **interleaver**, and is decoded **iteratively** by two soft-in soft-out decoders that exchange information. It performs within about 0.5–1 dB of the Shannon limit and is used in 3G (WCDMA, CDMA2000) and 4G LTE. ```text d --+---------------------> systematic bits +-->[RSC encoder 1]---> parity 1 +-->[interleaver]->[RSC encoder 2]-> parity 2 ``` #### 8 × 8 Hadamard code A Hadamard (Walsh) matrix is built recursively (Sylvester construction), where H′ is the complement of H: ```text H1 = [0] H2 = | H1 H1 | = | 0 0 | | H1 H1'| | 0 1 | H4 = | H2 H2 | = | 0 0 0 0 | | H2 H2'| | 0 1 0 1 | | 0 0 1 1 | | 0 1 1 0 | H8 = | H4 H4 | (H' = complement of H) | H4 H4'| ``` Writing out H8: ```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 ``` Each row is one 8-bit Walsh codeword. Any two rows differ in exactly 4 positions (in ±1 form, with 0 → +1 and 1 → −1, their inner product is 0), so the rows are **orthogonal**, as required for CDMA spreading codes. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the structure of a turbo encoder? A. A single non-recursive encoder B. Two parallel RSC encoders separated by an interleaver C. Two cascaded block codes with no interleaver D. A Viterbi decoder and an equalizer Answer: B. Turbo codes use parallel recursive systematic convolutional encoders with an interleaver between them. (Key point: Turbo encoder structure) 2. How is a turbo code decoded? A. With a single hard-decision pass B. By ML exhaustive search C. With a RAKE receiver D. Iteratively by two soft-in soft-out decoders exchanging information Answer: D. Two SISO decoders pass information back and forth iteratively. (Key point: Turbo decoding) 3. In the 0/1 Sylvester construction, what is the bottom-right block of H2n? A. Hn B. Complement of Hn C. Zero matrix D. Transpose of Hn Answer: B. H2n = [Hn Hn; Hn Hn'] where Hn' is the complement. (Key point: Hadamard recursion) 4. How do any two rows of H8 relate? A. They differ in exactly 4 positions B. They are identical in 4 positions only for adjacent rows C. They differ in 1 position D. They differ in 8 positions Answer: A. Any two rows differ in N/2 = 4 positions, so they are orthogonal. (Key point: Row orthogonality) ## Related in Wireless Communication - [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 - [Short note: Turbo coding](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aturbo-coding-note) — Wireless Communication, chapter 6 - [Hadamard matrix H8](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Ahadamard-matrix-h8) — 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 - [Importance of coding and ASCII code](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Acoding-importance-ascii) — Digital Logic, chapter 1 - [Variable length coding; entropy and fixed code efficiency](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Avariable-length-coding-entropy-six) — Communication Systems, chapter 6 - [Huffman code for five-symbol DMS vs fixed length](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahuffman-dms-five-symbols) — Communication Systems, chapter 6