Polar Codes over Wireless Fading Channels

Siddharth DangiArjun 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

• start with– 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

Channel polarization (ex: N = 1024)

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 (simple cases)

N = 2 N = 4

Polar Encoder (general)

• block length N = 2n

• 3 stages of WN

1. form s from u2. “reverse shuffle”3. 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) decoderfor 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:

Probability of block error

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

• Rayleigh fading channel– 2 paths– Td = 10 μs

– DS = 100 Hz

• OFDM– QPSK modulation– W = 1.25 MHz– NC = 128

Channel Simulation

Capacity vs. SNR

Implementation issues

1. Decoding in MATLAB too slow for large N2. Repeated computation in recursive formulas

for SC decoder3. Underflow in computation of likelihood ratios

for large N