Institute for Communications Engineering Department of Electrical and Computer Engineering Technical University of Munich
Flexible IR-HARQ Scheme for Polar-Coded Modulation Peihong Yuan, Fabian Steiner, Tobias Prinz, Georg Böcherer {peihong.yuan, fabian.steiner, tobias.prinz}@tum.de,
[email protected]
Incremental Redundancy Hybrid Automated Repeat Request (IR-HARQ) n1
n2
nt
IR-HARQ I In the t-th transmission, nt bits are transmitted.
1st Transmission
···
2nd Transmission
···
t-th Transmission
I After the t-th transmission, an (n1 + n2 + · · · + nt , k) code is decoded. • Turbo codes in LTE: I Low rate mother code (≈ 1/3) I Punctured with different patterns for several transmissions
1st decoding 2nd decoding
• LDPC codes in 5G: I Protograph-based high rate mother code I Raptor-like extension for several transmissions
t-th decoding
Preliminaries
Proposed Scheme
Polar Coding and Construction I We use the non bit-reversal representation" of 1 c = uF⊗ log2 N , where F = 1
polar # codes [1]: 0 1
I− = 1 − J I
+
=J
[J −1 (1 − I1 )]2 + [J −1 (1 − I2 )]2
q
[J −1 (I1 )]2
+
[J −1 (I2 )]2
+
U1 U2
I Density evolution with Gaussian approximation (GA) [2]: q
I Based on dynamically frozen bits [10]
F⊗ log2 N
X1 X2
I Length flexible
Y1 Y2
I
−
=I(U1 ; Y1 Y2 )
I
+
=I(U2 ; Y1 Y2 |U1 )
ntmax
Itmax .. .
I Capacity achieving under design constraint (nesting property): P (1) nt = t−1 q=1 nq ,
.. .
I2
I1 =I(X1 ; Y1 )
I Equivalent to [9] for nt is a power of two P and nt = t−1 q=1 nq , t = 2, 3, . . .
n2
(2) I + [t] ≤ I [t − 1] , t = 2, 3, . . .
n1
where I [t] denotes the design MI of the t-th transmission
I2 =I(X2 ; Y2 ) I1
I Extendable to MLPC Quasi-Uniform Puncturing (QUP) Algorithm [3] Design the t-th transmission (t 6= 1)
I (n, k, N) QUP-polar code (N is a power of two): The first N − n bits of the (k, N) regular mother code are punctured (not transmitted).
ˆi = ˆ i−1 = U i−1 for the (n1 + n2 + · · · + nt+1 , k) code via GA 1 Calculate Pr U 6 Ui |U 1 1
I For GA construction: set Ii = I(Xi ; Yi ) = 0, i = 1, . . . , N − n
2 Freeze the bits which are not in the information set of the (n1 + n2 + · · · + nt , k) code A0t
I The first N − n bits in u are frozen.
3 Put the k most reliable bits of the (n1 + n2 + · · · + nt+1 , k) code in the information set At 4 Dynamically frozen constraint is given by uA0t \At = uAt \A0t
Multilevel Polar Coding (MLPC) Comparison with “normal” QUP-Polar Codes I AWGN channel with constellation: Y = X + N, where Xi ∈ {±1, ±3, . . . , ± (2m − 1)} , Ni ∼ N (0, σ 2 )
Y → demapper for level 1 I SC decoding .. .
k1
I Set partitioning (SP) labeling [4] I
n (kj , m )
polar code for level j,
Pj
1 kj
= k, j = 1, 2, . . . , m
Y → demapper for level 2
I k = 3000, tmax = 4
10−2
I
: proposed scheme n1 = 4200, n2 = n3 = n4 = 600
I
: “normal” QUP-polar n = {4200, 4800, 5400, 6000}
10−3
I MI demapper GA construction (MI-DGA) [5] I List decoding [6] for MLPC: All survived path are passed into the demapper for next level [5].
10
−1
BLER
2m -ASK
.. .
k2
10−4
10
11
12 13 14 SNR in dB (biAWGN)
Y → demapper for level 3
I Extendable to 22m -QAM constellations
.. .
15
16
Comparison with LTE/5G solutions (tmax = 4) proposed
10−1 BLER
Related Works for Polar Coded IR-HARQ
3rd
10−2
4th
• 16 bits CRC outer code g(x ) = x 16 + x 12 + x 5 + 1
− Huge performance loss in the finite length regime
Polar Extension [9] + More polarized than [7, 8] − Length not flexible: P Nt = t−1 q=1 Nq , t = 2, 3, . . .
10−4 −6
+
−4
−2
0
2
• biAWGN
SNR in dB (biAWGN)
2nd
I k = 128 (include CRC) I n14 = (250, 250, 200, 140)
10−1 BLER
+ Length flexible
most unreliable infomation bits
Polar Codes with Incremental Freezing [7, 8] F⊗ log2 N2
F⊗ log2 N1
1st
• 8-ASK 10−2
I k = 896 (include CRC) I n14 = (1200, 600, 1200, 900)
10−3 10−4
2
4
6
8 10 12 SNR in dB (8-ASK)
14
16
Bibliography [1]
E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jul. 2009.
[2]
S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for modulation and detection,” IEEE Trans. Commun., vol. 52, no. 4, pp. 670–678, May 2004.
[3]
K. Niu, K. Chen, and J.-R. Lin, “Beyond turbo codes: Rate-compatible punctured polar codes,” IEEE Int. Conf. Commun. (ICC), pp. 3423–3427, Jun. 2013.
[4]
M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “Polar-coded modulation,” IEEE Trans. Commun., vol. 61, no. 10, pp. 4108–4119, Sep. 2013.
[5]
G. Böcherer, T. Prinz, P. Yuan, and F. Steiner, “Efficient polar code construction for higher-order modulation,” IEEE Wireless Commun. Netw. Conf. (WCNC), Mar. 2017.
[6]
I. Tal and A. Vardy, “List decoding of polar codes,” IEEE Trans. Inf. Theory, vol. 61, no. 5, pp. 2213–2226, May 2015.
[7]
B. Li, D. Tse, K. Chen, and H. Shen, “Capacity-achieving rateless polar codes,” IEEE Int. Symp. Inf. Theory (ISIT), pp. 46–50, Jul. 2016.
[8]
S.-N. Hong, D. Hui, and I. Marić, “Capacity-achieving rate-compatible polar codes,” IEEE Int. Symp. Inf. Theory (ISIT), pp. 41–45, Jul. 2016.
[9]
L. Ma, J. Xiong, Y. Wei, and M. Jiang, “An incremental redundancy HARQ scheme for polar code,” arXiv preprint arXiv:1708.09679, 2017.
[10] P. Trifonov and V. Miloslavskaya, “Polar subcodes,” IEEE J. Sel. Areas Commun., vol. 34, no. 2, pp. 254–266, Feb. 2016.
LTE-turbo
2nd
10−3
+ Capacity achieving (nesting property [7])
5G-LDPC
1st
• Avoid “heavily” punctured codes: (N + w , k, 2N) QUP, w N
