Polar Codes over Wireless Fading Channels

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# Polar Codes over Wireless Fading Channels - PowerPoint PPT Presentation

Polar Codes over Wireless Fading Channels. Siddharth Dangi Arjun Singh. Polar codes. Introduced by Erdal Arikan Achieve the symmetric capacity of any binary-input discrete memoryless channel (B-DMC) examples of B-DMCs: BEC, BSC Complexity O(N log N) for both encoder and decoder.

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### Polar Codes over Wireless Fading Channels

Siddharth Dangi

Arjun Singh

Polar codes
• Introduced by Erdal Arikan
• Achieve the symmetric capacity of any binary-input discrete memoryless channel (B-DMC)
• examples of B-DMCs: BEC, BSC
• Complexity O(N log N) for both encoder and decoder
Channel polarization
• N independent and identical B-DMCs with symmetric capacity C
• end up with (for large N)
• NC channels with symmetric capacity ≈ 1
• N(1-C) channels with symmetric capacity ≈ 0
• Send data through channels with capacity ≈ 1
Application to wireless channels
• Idea – model the channel as BSEC (“binary symmetric erasure channel”)
• declare deep fades as erasures
• other error events cause bit flips
• N channels are:
• in time, across N symbol times
• in frequency, across N OFDM subcarriers
Notation & Terminology
• W – a B-DMC with input x and output y
• W(y|x) – transition probability
• “symmetric capacity” (rate)
• highest rate achievable using input symbols with equal frequency
• “Bhattacharyya parameter” (reliability)
Polar Encoder (general)
• block length N = 2n
• 3 stages of WN
• form s from u
• “reverse shuffle”
• 2 N/2 polar encoders
• linear operation!
Polar Encoder
• matrix representation:
• matrix for N = 4
• depending on rate R, fix some positions of u
• example: “freeze” indices 1 and 3 (R = ½)
Polar Decoder
• successive cancellation (SC) decoder

for i = 1,…,N generate decision for bit i based on: 1. received bits y1,…, yN 2. decisions for bits 1,…,i-1end

• suboptimal, but leads to efficient recursive computation for decision functions
• can still achieve symmetric capacity
Choosing Frozen Set
• choose indices for which corresponding “new” channels have either the
• highest symmetric capacities (closest to 1)
• lowest Bhattacharyya parameters (closest to 0)
• both methods achieve symmetric capacity
• second method gives explicit bound:
Calculating Bhattacharyya parameters
• nice recursive formulas if W is a BEC
• for other channels, can use approximation:
• calculate symmetric capacity C of W
• approx. W as a BEC with erasure probability 1 – C
• use BEC recursive formulas
Simulation parameters