Polar Codes over Wireless Fading Channels

1. Polar Codes over Wireless Fading Channels Siddharth Dangi Arjun Singh

2. 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

3. 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

4. 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

5. Channel polarization (ex: N = 1024)

6. 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)

7. Polar Encoder (simple cases) N = 2 N = 4

8. Polar Encoder (general) • block length N = 2n • 3 stages of WN • form s from u • “reverse shuffle” • 2 N/2 polar encoders • linear operation!

9. 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)

10. 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

11. 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:

12. Probability of block error

13. 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

14. Simulation parameters • Rayleigh fading channel • 2 paths • Td = 10 μs • DS = 100 Hz • OFDM • QPSK modulation • W = 1.25 MHz • NC = 128

15. Channel Simulation

16. Capacity vs. SNR

17. Implementation issues • Decoding in MATLAB too slow for large N • Repeated computation in recursive formulas for SC decoder • Underflow in computation of likelihood ratios for large N