New Construction and Performance Analysis of
Polar Codes over AWGN Channels
Bashar Tahir, and Markus Rupp
Institute of Telecommunications
Technische Universit¨ at (TU) Wien
Vienna, Austria
Email: {bashar.tahir, markus.rupp}@tuwien.ac.at
Abstract—We propose a new construction algorithm for Polar
codes operating over Additive White Gaussian Noise channels
under Successive Cancellation decoding. Our approach is based
on tracking the bit error probabilities of the bit channels as
they evolve through the decoder, allowing us on the one hand to
characterize the performance of these channels, and on the other
hand, providing a solid construction algorithm. We then use our
approach to derive a modification for the density evolution (with
Gaussian approximation) based construction, providing better
accuracy and implementation.
I. I NTRODUCTION
Polar codes attracted a lot of attention since they were
introduced by Arıkan in 2008 [1]. They are the first practical
codes that are proven to achieve the channel capacity at infinite
length. The structure of polar codes is based on the channel
polarization transform, where a specific dependency between
the input bits is introduced. Such dependency can be exploited
using a Successive Cancellation (SC) based decoder, which
causes subsets of the bits to have improved reliability when the
successively fedback bits are correct. In this case, the task of
constructing polar codes involves finding the most reliable bits’
positions (bit channels) that gain the most from the successive
decoding, and use them for the transmission of information
bits. While foreknown bits (usually zeros) are transmitted over
the other unreliable channels (Frozen set), ensuring correct
feedback of the those bits since they are known by the receiver,
thus producing the coding gain.
Polar codes are non-universal, in the sense that their con-
struction is dependent on the Signal-to-Noise Ratio (SNR) of
the receiver. Such property is actually a drawback, since the
optimum transmission scheme would then require an adaptive
construction based on the SNR. Therefore, it’s no surprise
that the SNR is a design parameter when it comes to the
construction algorithms of polar codes.
The first construction algorithm was introduced by Arıkan,
where the Bhattacharyya parameters of the bit channels are
evolved through the polarization structure. A construction
based on Density Evolution with Gaussian Approximation
(DEGA) is given in [2], which is based on tracking (evolving)
the density functions of the Log-Likelihood Ratios (LLRs)
The financial support by the Austrian Federal Ministry of Science, Research
and Economy and the National Foundation for Research, Technology and
Development is gratefully acknowledged.
through the decoder. Some other constructions are given in
[3]–[6], with varying performance and complexity.
In this paper, we propose a new construction algorithm for
polar codes operating over Additive White Gaussian Noise
(AWGN) channels under SC decoding, based on tracking the
bit error probabilities of the bit channels as they evolve through
the decoder, providing also a performance characterization for
those channels. Our results show that the new algorithm deliv-
ers similar performance to that of DEGA, and because the two
algorithms are connected through the bit error probabilities, we
derive a modification for DEGA, improving its accuracy and
implementation. We finally benchmark their performance.
II. PROBABILITY ANALYSIS
In this section, we perform the probability analysis for the
successive cancellation decoder. Fig. 1 shows the SC decoder
for a polar code of length 4.
f
g
f
g
f
g
f
g
u
0
^
u
1
^
u
1
^
u
2
^
u
0
^
u
0
^
u
1
^
u
2
^
u
3
^
L
0
(0)
L
1
(0)
L
2
(0)
L
3
(0)
L
0
(2)
L
1
(2)
L
2
(2)
L
3
(2)
L
0
(1)
L
1
(1)
L
2
(1)
L
3
(1)
Fig. 1. Polar decoder of length 4.
In the LLR domain, the nodes f and g perform the following
calculations for given input LLRs L
a
and L
b
f (L
a
,L
b
) = log
e
La+L
b
+1
e
La
+ e
L
b
, (1)
g(L
a
,L
b
,u
s
)=(-1)
us
L
a
+ L
b
,
where u
s
is the Partial Sum, which is the sum of the
previously decoded bits that are fedback to the current node g.
978-1-5386-0643-8/17/$31.00 ©2017 IEEE