Successive cancellation decoding of polar codes using new hybrid processing element

International Journal of Reconfigurable and Embedded Systems (IJRES)
Vol. 11, No. 2, July 2022, pp. 157~165
ISSN: 2089-4864, DOI: 10.11591/ijres.v11.i2.pp157-165  157
Journal homepage: http://ijres.iaescore.com
Successive cancellation decoding of polar codes using new
hybrid processing element
Sujanth Roy James, Lakshminarayanan Gopalakrishnan
Department of Electronics and Communication Engineering National Institute of Technology Tiruchirappalli, Tamil Nadu, India
Article Info ABSTRACT
Article history:
Received Dec 29, 2021
Revised Feb 17, 2022
Accepted Mar 11, 2022
Polar codes are one of the best linear block codes that are capacity achieving
and incorporating along with it a simplified encoding and decoding routines.
Successive cancellation (SC) algorithm is one of the predominantly used
decoding algorithms due to its low complexity. It has broad scopes for
hardware architecture design and reformulation. For polar code, the trade-off
among the long latency and the silicon area of the SC algorithm is a
bottleneck for the design of a high throughput polar decoder. The available
prior SC polar decoder designs have higher area requirements for higher
block length. This paper introduces a unique reformulation of the processing
element (PE) block of SC decoding. The proposed reformulation leads to
two benefits: firstly, critical path and hardware complexity of the PE are
meaningfully reduced by using a unified adder block. Secondly, the silicon
area requirement and the power consumption were also reduced
considerably without any loss in performance. The proposed PE is used to
build the decoder for various block lengths. Moreover, a Gate-level analysis
of the proposed decoder has revealed that the design attains an 18% area
reduction and 38% reduction in power consumption over the conventional
one with similar performance.
Keywords:
ASIC
Channel coding
Decoder
FPGA architecture
Polar code
Successive cancellation
This is an open access article under the CC BY-SA license.
Corresponding Author:
Sujanth Roy James
Department of Electronics and Communication Engineering
National Institute of Technology Tiruchirappalli
Tanjore Main Road, NH67, near BHEL, Tiruchirappalli, Tamil Nadu 620015, India
Email: sujanthroyj@gmail.com
1. INTRODUCTION
Errors owing to interference, device reparations, and arbitrary noise corrupt the actual data stream
before they reach the receiver end. These errors need to be corrected before they are used at the receiver end.
Channel coding techniques helps to minimize the errors and rectify them appropriately to reconstruct the
original data. Channel coding primarily modifies the original data stream at the transmitter to minimize the
error occurrence and reverts them at the receiver end to reconstruct the original signal. The term encoding
denotes the operations at the transmitter and decoding indicates the processes at the receiver, respectively.
The prime focus is to develop high-performance channel codes with adequately low complexity that
diminishes the impact of errors in a communication system. This allows timely, practical implementation into
the silicon technology of the day.
Proposed by E Arikan, the polar codes stand as one of the finest capacity-achieving codes having
low encoding and decoding complexity of the order O (N log N), where N is the code length [1]. Polar codes
function on blocks of bits and are therefore classified under the block code family. Polar codes are
established on a recursive concatenation of the short core functions that convert physical channels to virtual
channels. As the virtual channel count increases, they tend to have either low reliability or high reliability i.e.,

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Int J Reconfigurable & Embedded Syst, Vol. 11, No. 2, July 2022: 157-165
158
they polarize. This feature enables the message bits to be allotted to the most reliable channel. Erdal Arikan
put forth the first version of capacity achieving error correcting codes (ECC) for binary-input discrete
memoryless channel [1]. Since then polar codes have emerged as one of the prominent model. Polar codes
are known for their exceptional encoding and decoding architecture which have low complexity and
recursive structure. They are constructed based on channel polarization, where message bits are transmitted
through the most reliable channel [1]. The widely used successive cancellation decoder (SCD) has an
attractive architecture with low hardware complexity of the order O (N log N), where N is the block length.
Several architectures have been proposed for the successive cancellation decoder (SCD). In SCD,
there are three major blocks, including the processing element (PE), partial sum generation unit (PSG) and
memory unit (MU). In this work, an area-efficient architecture for the PE and hence the decoder is proposed.
An architectural modification has been carried out on the PE, which reduces the overall area to a great extent.
SCD is the widely used algorithm that can assure exceptional error-correcting performance in polar code
decoding. The hardware architectures of the conventional SCD occupies a large silicon area. Improvements
in polar decoding techniques and systems are eagerly awaited by the 5G telecommunication industry, where
even more complex systems are being developed that require less power, area and obtain high throughput and
speed. The challenge in the 5G wireless system is to produce a suitable channel coding scheme to ensemble
increasing spectral eﬃciency. Research in SCD has been going on recently, and therefore, it is worthwhile to
come out with a new architecture for the same.
The original version of SCD was proposed in [1] and since then the derivative versions of the same
decoder has emerged with improved performance. In spite of the availability of other decoding schemes, the
SCD stands unique with its simple architecture easy construction. Yuan et al. had implemented a two bit low-
latency polar decoder by using reformulation technique [2]. Leroux et al. had used scheduling and allocation
method to reduce the hardware complexity of the decoder [3]. The logarithmic implementation [3], [4]
significantly reduced the hardware complexity. The unrolled architecture proposed by Giard et al. has
produced a high-throughput in the decoder [5]. A novel merged processing element for SCD based on one’s
complement was proposed in [6] which helped to increase the throughput. In [7], a Novel simplified merged
processing element (SMPE) architecture was proposed for SCD, and the decoding tree was constructed with
low complexity. The SCD has the greatest throughput and energy per bit [8] compared to other decoders.
Polar codes have been verified to be capacity achieving for binary-input symmetric memoryless channels [9].
The combinational logic based SCD produced a low-power performance [10]. A non-recursive method to
create a decoding schedule without affecting the performance has been portrayed in [11].
In [12], path splitting selection (PSS) strategy aided decoder was proposed to lessen the decoding
complexity with negotiable performance loss. Based on PSS, two schemes were suggested to locate flawed
information bits more precisely. Apart from the PE and PSG, the memory unit is another sub-block that can
be optimized according to the requirement [13]. The PE operates with log-likelihood ratio (LLR) instead of
likelihood ratio (LR). Hence, polar codes are preferred over the low-density parity-check codes (LDPC) on
wireless communication channels due to their better performance [14]. There are rate-less polar code
implementations [15], [16] where the number of frozen bits varies based on the information bits' length. This
feature provides the extendibility of the decoder for various code rates [17]. Polar codes are extended to non-
identically distributed channels and they were found to be capacity achieving but the latency and complexity
were sacrificed [18].
Captivatingly, the 5G standardization process of the 3rd generation partnership project (3GPP) has
preferred polar codes as the channel coding structure. Therefore, it is clear that different architectural
modifications have been proposed on the conventional SC approach to improve its performance like
increased throughput, low power, etc. In that way, this paper presents a reduced area SCD with low power
consumption. On careful examination of the 2-bit SCD [2], it was eminent that the functioning of the g node
in the PE comprises of addition and subtraction performed in parallel and one of the output is chosen based
on the previously decoded bit of the f node in the PE. It is clear that one among these is an unwanted
operation and can be suitably eliminated. The idea to eliminate the idle branch of the decoding tree prospered
to removing the subtraction operation from the PE and carrying out the same with the existing adder. The
hybrid processing element proposed here is used to build the SCD. Further, gate optimization is also carried
out in the proposed decoder and a noticeable improvement was achieved. The total area of the decoder was
reduced by 18%, and the power was reduced by 38% compared with existing recent architectures. During this
research, the functionality of the SCD was the same as the original proposed SCD in [1].
The remaining paper is arranged as follows: Section 2 highlights polar codes' encoding and
decoding process and the proposed system design consisting of the proposed hybrid processing element and
the decoder. Section 3 deliberates the implementation results, and the paper is concluded in section 4.

Int J Reconfigurable & Embedded Syst ISSN: 2089-4864 
Successive cancellation decoding of polar codes using new hybrid processing … (Sujanth Roy James)
159
2. RESEARCH METHOD
Polar codes fall under the banner of linear block codes and are one of the available forward error
correction (FEC) codes. They have low computational complexity and are capacity-achieving codes. Polar
codes with rate R=I/N have a code length of N=2^n, and I (0≤I≤N) denotes the number of information bits,
whereas N-I denotes the number of frozen bits added to the information bits. The construction of polar codes
follows a very simpler approach compared to other traditional approaches like turbo codes. Encoding occurs
in the transmitter end, where the channel would be polarized into reliable and unreliable channels. The
information bits would be transmitted on the I most reliable channels and the remaining N-I channels stand
frozen and are set to 0.
2.1. Polar code-encoding
The encoding approach of polar codes as the (1) [3].
N
n
N
n
N
N
N
N
N
B
F
u
F
B
u
G
u
x 

=
=
= 1
1
1
1 (1)
where n
N
N F
B
G 
= denotes the generator matrix, 





=
1
1
0
1
F , (.)⊗n
represents the nth Kronecker power,
N
B denotes the bit reversal vector, the input vector represented as N
N
u
u
u
u ,...
, 2
1
1 = having I information
bits and (N-I) frozen bits. Also, N
N
x
x
x
x ,...
, 2
1
1 = represents the encoded value [12]. The position of
frozen bits are identified by the method explained in [4]. The generator matrix for N=8 is given as (2) and
(3).
3
8
8

= F
B
G (2)


























=
1
1
1
1
1
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
1
0
0
0
1
0
0
0
0
1
1
1
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
8
8 B
G
(3)
Once GN is available, the input vector is encoded and prepared for transmission. For rate R = 0.5, the
number of information bits will be equal to the number of frozen bits.
2.2. Polar code-decoding
The decoding approach is carried out with the widely used Successive Cancellation decoding
algorithm [19], [20]. In the decoding scheme, the information bits N
N
u
u
u
u ,...
, 2
1
1 = are retrieved
sequentially from the received vector N
N
y
y
y
y ,...
, 2
1
1 = . The output bits at stage t can be decoded by
processing the LLR function as (4) and (5).
))
,
(
( t
i
LL
O
ui = (4)
where,







=
frozen
is
i
when
and
0
t)
LL(i,
if
0,
free
is
i
when
and
0
<
t)
LL(i,
if
1,
i
u (5)
Both encoder and decoder facilitate well-ordered processing configurations and appropriate module
sharing owing to their recursive build. Every stage is composed of the kernels f and g appropriately
scheduled to decode the data. The f and g kernels function based on (6) and (7).
|)
|
|,
min(|
)
(
)
( d
c
d
sign
c
sign
f = (6)

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d
c
g sum
u
+
−
=
ˆ
)
1
( (7)
where ûsum decides between addition and subtraction in the kernel g, c and d represent the LLR inputs. ûsum
represents the partial sum of the subset of previously decoded bits. The Conventional decoding tree is shown
in Figure 1.
Figure 1. Conventional decoding tree
In Figure 1 the highlighted path is the decoding path for bit û1. This takes three clock cycles. The
calculation of û2 occurs in the next clock cycle through the corresponding g kernel which operates with ûsum =
û2. At the end of 14 cycles u1 – u8 are decoded. The decoding schedule for a code block length of 8 is
presented in Table 1. Thus the conventional SCD takes 2N-2 clock cycles to solve a code of length N [1]. Po
et al. [21] proposed a polar code decoder with variable R and N.
Table 1. Decoding schedule of the conventional SCD for N=8
Clock cycle
Stage
Output
S1 S2 S3
1 f
2 f
3 f û1
4 g û2
5 g
6 f û3
7 g û4
8 g
9 f
10 f û5
11 g û6
12 g
13 f û7
14 g û8

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2.3. Proposed new hybrid processing element
The proposed decoder has an area optimized unique scheme of combinational logic that functions to
eliminate the unnecessary subtraction operation carried out in the PEs. The proposed PE is termed new
hybrid processing element (NHPE), which is used to design the decoder. The conventional SCD has the f
node and g node that form the decoder's processing element and are scheduled appropriately to form the
decoding tree. After the development of the merged processing element, the f and g operations are selected
using a control signal and the decoding tree is constructed accordingly. The research for better-merged
processing continued and evolved into 2b-SC decoding, 2b overlapped SC decoder, 2b precomputation
decoder and so on [2]. The architectures are optimized for area, power, speed and better architectures were
produced. This paper proposes a novel area-optimized processing element with low power consumption
compared with the existing processing elements.
On careful examination of the merged processing element in [2], it was noted that during the g node
operation, both addition and subtraction are performed in all the iterations. But for the successful functioning
of the decoder, only one operation is sufficient and it is decided by the partial sum of the previously decoded
bits as (7). The proposed NHPE uses a unique scheme of combinational logic to eliminate the subtraction
operation throughout the entire architecture along with a 2x1 multiplexer that chooses between addition and
subtraction. This modification in the NHPE is reflected in the whole decoder. In the proposed scheme, the f
node operates in the conventional way with the sign and magnitude processed separately. The proposed
modification is effected in the g node operation. As soon as the f node is executed, the partial sum is
generated which triggers the adder of the g node through the combinational block of XOR gate. When
subtraction is to be performed, the sign bit of LLR(d) is negated and addition is performed. Table 2 shows the
truth-table for the proposed idea.
Table 2. Truth table of the proposed scheme
Sign of d Value of ûsum Operation to be performed Modified sign of LLR(d)
Positive (0) 0 Addition 0
Negative (1) 0 Addition 1
Positive (0) 1 Subtraction 1
Negative (1) 1 Subtraction 0
The following example explains the operation.
If ûsum = 0, i.e. Addition to be performed.
Let's take c = +3 and d = -5.
Sign rep(c) = 0 0011, Sign rep (d) = 1 0101
New Signed bit (d) = Ex-OR of Previous sign bit (d) and ûsum = (1, 0) = 1
Now 2s_comp_rep(c) = 0 0011,
2s_comp_rep (modified d) = 1 1011,
Now c+d = 1 1110. Sign(c+d) = 1 0010 i.e -2.
If ûsum = 1, i.e. subtraction to be performed.
Let's take c = +3 and d = -5.
Sign rep(c) = 0 0011, Sign rep (d) = 1 0101
New Signed bit (d) = Ex-OR of Previous sign bit (d) and ûsum = (1, 1) = 0.
modified d = 0 0101.
Now 2s_comp_rep (c) = 0 0011,
2s_comp_rep (modified d) = 0 0101,
Now a+b = 0 1000. Sign (a+b) = 0 1000 i.e 8.
The above examples suitably explain Table 1, thus eliminating the subtraction and multiplexer
operations successfully. The NHPE is shown in Figure 2. The architecture clearly shows that the subtraction
operation is eliminated and a hybrid processing element is proposed.

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Figure 2. Proposed processing element-NHPE
2.4. Proposed SCD based on NHPE
The proposed NHPE is used to build the decoder. The proposed modifications in the NHPE when
realized in the entire decoder has a greater impact. In stage 1, all the PEs work as f node in the first cycle. In
the second cycle, all the PEs of stage 2 are f nodes, and in the third cycle, the PE in stage 3 decodes the
output message bits. This is the decoding sequence of the decoder and this process takes a total of (1.5N-2)
clock cycles. When pipelining is introduced in this process, the decoding latency can be further reduced by a
few cycles. Pipelining is introduced between the stages to speed up the entire decoding process. When
pipelining is introduced, the decoding latency can be reduced from 1.5N-2 to N-1 cycles. The f node has a
delay of Tf = 0.709ns, whereas the g node has a delay of Tg = 1.86ns. The other delays involved are Tmux =
0.136ns and Txor = 0.148ns. In stage 3, once the f node completes its operation, partial sum value, ûsum is
updated, which completes the execution of the g node. The decoding tree for a block length of 8 is shown in
Figure 3.
Figure 3. Proposed SC decoder for block length 8
The use of less complex processing element (NHPE) has resulted in reducing the overall complexity
of the decoder. The area and power have been reduced considerably in the proposed decoder. The decoding

163
schedule of the proposed decoder is shown in Table 3. NHPE(f) denotes the proposed processing element
functioning in f mode and NHPE(g) denotes the proposed processing element functioning in g mode. The
proposed scheduling enables the decoder to complete the process in 7 cycles. Thus, the proposed NHPE not
only reduces the on-chip area but also reduces the latency of the decoder.
Table 3. Decoding schedule of the proposed SCD for N=8
Clock cycle
Stage
Output
S1 S2 S3
1 NHPE(f)
2 NHPE(f)
3 NHPE(f) û1 and û2
4 NHPE(g) NHPE(g) û3 and û4
5 NHPE(g)
6 NHPE(f) û5 and û6
7 NHPE(g) û7 and û8
3. RESULTS AND DISCUSSION
The proposed SCD was modelled in Verilog HDL, implemented in Xilinx xc7vx980t FPGA and
synthesized in Cadence 45nm CMOS technology. The proposed NHPE was used to construct the decoding
tree and the decoder was tested upto a code length of 1024. A code rate of 0.5 was used in the design. The
proposed model's area and power compared to the existing models [4], [22] are given below in Table 4 and 5.
Considerable area and power reduction was achieved in the decoder architecture when the poposed NHPE
was used. The functionality of the decoder was unaltered in the proposed architecture. In Table 4 it is
emphasized that the area of the proposed model was reduced by at least 16.5% on average.
Table 4. Proposed SCD area comparison report using TSMC 45nm technology
Block length (N)
Area, µm2
Reduction (%)
Yuan and Parhi [4] Zhang and Parhi [22] Proposed
8 966.718 3621 8 966.718
64 13917.83 30141 64 13917.83
128 31980.37 60899 128 31980.37
256 72258.24 121925 256 72258.24
512 161123.60 243891 512 161123.60
1024 355478.48 444944 1024 355478.48
Table 5. Proposed SCD power comparison report using TSMC 45nm technology
Block length (N)
Power, µW
Reduction (%)
Yuan and Parhi [4] Zhang and Parhi [22] Proposed
8 82.02 217.89 49.92 39.133
64 1248.17 1725.03 763.04 38.867
128 2828.38 3567.91 1730.83 38.80
256 6620.32 7049.74 4063.27 38.62
512 14012.20 14002.70 8632.40 38.392
1024 30909.90 27220.40 19104.85 38.19
This is the improvement brought by the proposed architecture. Also, Table 5 shows a 38% power
reduction on an average which can be a considerably good improvement. Thus, the proposed processing
element, NHPE and therefore the proposed SCD have shown a reduction in area and power compared to the
available decoders. Table 6 shows the hardware performance of the proposed decoder. The performance of
the proposed model was compared to other existing architectures at the same code length and code rate. It
was evident that the performance of the proposed decoder is competent with the other existing decoders and
can provide even better implementation results. The proposed decoder produced better hardware efficiency
[23] among the other models and also good normalized throughput. The technology scaled normalized
throughput (TNST) helps to compare the throughput across various technologies and is defined by [24],














=
Technology
Target
Technology
*
Count
Gate
Throughput
TNST (8)

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This helps to scale the throughput based on the target technology. From Tables 4, 5, and 6, it is
evident that the proposed decoder stands out in significantly when compared to the state-of-the-art
architectures in terms of low area, low power and higher efficiency without altering the performance. Hence,
they can be used for area and power-efficient communication applications.
Table 6. Hardware performance of decoder with code length, N=1024
Design Yuan and Parhi [4] Leroux et al.[25] Kim et al. [6] Proposed
Message form LLR LLR LLR LLR
Code rate 0.5 0.5 0.5 0.5
Tech. (nm) 65 65 40 45
Quantization (bits) 6 5 5 5
Frequency (Mhz) 390 500 1000 400
Latency (clock cycles) 1056 2080 1023 1023
Area (µm2
) 355478.48 308693 587070 300717.77
Throughput (Mbps) 378 123 500 490
TNST 2 1.65 3.03 3.1
Efficiency (Mbps/mm2
) 1064 398 851.78 1633
4. CONCLUSION
In this paper, a low power, area-efficient architecture for SCD was proposed. a new hybrid
processing element was presented, which occupies less area and has low power consumption compared to
existing architectures. The proposed processing element was used as the basic building block to construct the
decoder. Also, the proposed architecture had better efficiency, which is also a significant feature. The
proposed architecture can be combined with look-ahead techniques to further reduce the latency and speed up
the decoder. The analysis and implementation conclude that the proposed decoder has inimitable benefits in
terms of area and power. The proposed decoder has an error-free approach and finds application in the high-
speed communication system, including 5G.
ACKNOWLEDGEMENTS
The authors thank Visvesvaraya Ph.D. Scheme (VISPHDMEITY-1713), Ministry of electronics and
information technology (MeiTY), Government of India for funding this research, and SMDP-C2SD and
DST-FIST (Grant number: DST/ETI-324/2012), Government of India for providing the lab facilities.
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BIOGRAPHIES OF AUTHORS
Sujanth Roy James is currently pursuing his doctoral research at National
Institute of technology Tiruchirappalli. His area of research includes digital architecture
design, ASIC Design and FPGA based system design. He is currently working on design of
FPGA architectures for decoders used in Polar codes. He can be contacted at email:
sujanthroyj@gmail.com.
Lakshminarayanan Gopalakrishnan is a Professor in the Dept of Electronics
and Communication Engineering at NIT-Trichy, India. His research expertise are primarily in
the areas of Wireless System Design in VLSI / Physical Layer Design, Cognitive Radio
algorithms and techniques. He is carrying out research work in VLSI Signal Processing, FPGA
based Wireless Transceivers and Reconfigurable systems He served in one of the VLSI design
companies, Sasken Communication Technologies Limited, Bangalore and was involved in the
project “FPGA implementation of USB Sub System”. He is holding one patent and filed
another one. He can be contacted at email: laksh@nitt.edu.

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